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What are the most useful books an undergraduate in math should read? I found Alock's "How to study as a math major" and "how to study for a math degree" very useful. Are there any other good readings before starting a math degree (or any other material which may come in handy, like shaum's mathematical handbook of formulas and tables?)? [EDIT: can you suggest some readings to prepare for physics courses that will be included in the degree too?] [EDIT_2: Thank you for your answers, but?] 3 Answers Shaum's is better for physics than for mathematics (for physics it is invaluable), at least that is what I found. It really depends on what your focus is, it seems like you would be best served by a good applied math book to go along with physics (I would also suggest getting really good at programming and statistics if you are doing mathematical physics). But another student would be better suited to a book more loaded towards number theory (I do not know of any at an undergrad level), or mathematical history.
Find an East Chicago MathWHence, there is a greater emphasis on multiple-step problem solving using concepts that were previously learned and concepts that will be further explored. To be brief, calculus is the mathematical study of change. Therefore it is very helpful and useful to understand its concepts.
"You got the right answer, but your method was not the one I taught you. Let me show you why my method is the one you should use, even though your method will usually get the right answer." In calculus, there's a reason you use the methods you use, but not all teachers explain that part very well, though they should (I didn't understand that until college, when we really started applying calculus to engineering concepts, and I understood why you do what you do in calc). The computer science program at that seems to be kind of lacking, but I'm looking into it for completion, and to possibly learn something new. It seems that they use Python, based on textual links. Hopefully, they'll teach me to learn and apply something new. I'll edit update. UPDATE: It teaches all of the basic concepts quite well. The program uses predefined functions to teach the most basic concepts while quietly keeping the template of the used language out of view. However, in order to practice manipulating the code it gives you, you are expected to read immense texts. For a beginner to programming, this is an excellent start, but it is best used as a tutor to review concepts as opposed to replacing the in-school teaching methods. I find Khan academy really hard to use. I want to pick up my maths skills from where I left off, a level maths (uk), but it's all split up into I belive the American school system and everything is either stupidly easy or far too advanced for me. It's not hard to learn calculus if you study a bit of physics. Kinematics at it's core is calculus, so if you learn some of the processes and equations you can come to understand it through observation rather than study That's the key in math, really. You have to be able to understand what you're doing conceptually, whether it's visualizing what a second derivative is or understanding why trig functions do what they do or whatever. You can't just memorized it as a formula to be mechanically cranked out. I don't really enjoy Khan Academy much... It's too fast and too disorganized for me (that's not to diminish the work put into that though; it's amazing). I like some of the stuff I've seen on iTunes U so maybe I'll get into that. In truth, I tested out of having to take math general education courses in college because I got credit for them in high school, but I'm jealous of my friends who are math majors. I feel inadequate :P I'm much more of a humanities person. I would try and continue anyway. I nearly failed every math class I took in highschool and decided to take AP calc anyway cause a girl I liked was in the class. Everything I had learned the last 4 years suddenly clicked when I took calculus. It was still hard, though and honestly the calculus in calculus isnt that hard. Its the algebra that will bite you in the ass. Still got a B+, the highest grade I ever got in a math class. Most people I have met found pre calc to be just as, or more difficult than calculus. That's because its a different kind of math, and also pre calc is a more "pure" math where are calculus is a math that actually explains how things work. Every time you learn something new in calculus you learn a new way the world works, its really special like that its not a "when will i ever use this" type math at all. That's why its used my engineers after all. One of the most useful things I've ever done to review precalc was google 'review precalculus for calculus' and I ended up finding a free PDF with simple, short, digestible review units. And now I'm fairly prepared for calc this coming trimester. Ugh... I aced Calc I/II, Multivarialbe, Diff Eq, Linear Algebra and Discrete Mathematics... but proofs can go fuck themselves. I transferred in, and was a drunk/lush who failed out of my original school years ago (4.0 now though). Back in my original work, I took a Probability and Statistics course for Math majors. Supposedly it was for Engr students, but apparently that cross listing was wrong. A few weeks in of nothing but proofs, someone asked how this applied to Engr and the professor realized it, but nothing could be done. The course had to be taught for math majors, and was close enough supposedly that the Engrs, who couldn't drop it at that point, would simply have to deal with it. Fucking barely got a C-, and only then because the girl I sat next to in class was a math major and we starting 'studying' together, which lead to friendly benefits, which lead to more study sessions occurring (not complaining, but we did actually study more because of it, and I barely got a C-). So now I'm at this school, and I have all gen ed's done with a lot of transfer work. I have to maintain full time though, and that means that, since I can't find 12 credits in my major for the next 3 of remaining 4 semesters, I have to take SOMETHING. I already did a Physics minor sans one course, and I transferred in enough Math that it exceeds a Math minor. Except I must take 9 Math credits at the school to get the minor. So now I'm stuck starring at a Geometric proofs course, and at least one other proofs course (and a course my degree actually requires in stat that counts), in order to get the minor. God damn proofs !!!!!!!!!!!!!!!!!!!!!!!!! I hate them so much. TL;DR - FUCK PROOFS, FUCK THE COURSE I TOOK WITH PROOFS, AND FUCK THE FACT THAT I NOW HAVE TO TAKE AT LEAST TWO MORE PROOFS COURSES. I failed algebra 2 times and passed it the third. In my school algebra 1 is a freshman course. I took it 8th 9th and 10th grade, then took "functions" which is just remedial algebra but counts for a separate credit. I took geometry alongside functions. I'm great at math, terrible at school. Twice? In my experience to actually fail a class in highschool (get an actual F), you had to either not really show up at all, or just refuse to do any of the work. Did the teacher not speak to you when it was clear you were failing and try to work with you? I was mercifully given a D- by my second calculus college professor do that I could graduate. I failed calculus the first time and was just as lost the second time. It didn't help that my second college calculus professor had an awful Indian accent and was almost unintelligible. Depending on how you want to look at it the highest I took was geometry or algebra II. Took pre-algebra 7th grade. Failed, had to repeat it 8th grade. 9th grade I took alg 1. 10th geometry. 11th grade was alg 2. Failed and failed the relevant portion of my state standardized test, so I had to take TWO math classes my senior year. Retake algebra II and some semi-remedial course for kids who failed the math portion of their PSSAs so they could still be considered up to the state mandated level of proficiency and graduate. Flash forward to being about to begin my junior year of college and I still barely comprehend fourth grade arithmetic. My grades have me shuffled around a lot. I can hardly be considered a senior at this point other than by my projected graduation date. I've taken freshman classes as a junior and vice versa. I meet some cool people in different grades that I have fun with in class. The whole highschool experience has been a big joke. While my senior friends go on to higher level math, I'm in sophomore classes having fun with these guys. But I have an expiration date and so does everybody else. Most of my friends have graduated and I haven't seen them since freshman year. My brother is off in the marines, my friends from last year are gone, and soon ill be leaving my sophomore and freshman friends too. I've taken a lot for granted. You get so caught up in the moment of having the time of your life that you forget that you and everybody else you know WILL part ways soon. You try to tell yourself you'll keep in touch but you know you won't. My debate captain wrote me a letter, telling me she knows all this, but it doesn't matter, because we'll still have known eachother. She quoted someone whose name I can't remember. A roman. I love the romand. And the greeks too. Perfect example of no matter how much you do or try in life, it all gets washed away someday. The quote read "a friendship that can end never really began". Highschool is a terrible terrible thing. They make you go somewhere you don't want to go when you're very young, so you make friends to survive. Then, just when you're getting to an age where you can really understand what friends are and how good it is yo have them, just when you're capable of really finding meaning from those around you, it all ends. You've spent 18 years falling in love with these people many whom you've known for all 18 years. Then it catches up with you. The best time of your life fades away; you grab at it to hold it all together, but piece by piece it breaks away, and you know it will never be the same. You'll fall out of love, fall out of contact, forget your first crush, distance your best friend, forget completely about everything you'd known for your entire life so you can 'grow up'. This is the place where souls go to die. Highschool was created by sadists I dunno man I really don't miss high school that much, nor the people I have fallen out of contact with. I'm still friends with a good few of my old friends and we hang out when we get the chance, but eh. Life moves on. By the time I was a senior I was no longer friends with most of the people I knew in third or fourth grade. (Of course, I also went to a high school the size of a small college, so it can be kind of hard to keep track of people if you don't have common classes, but 99% of them we had drifted apart before we even got there). College is pretty great man. What I am not looking forward to is graduating from here. Finding a 9-5. Fuck that shit. I'm enjoying my gross irresponsibility and incredibly lax schedules. This past April two of my friends and I blew off a week and a half of class to go on a road trip to Chicago, just because we could. I'm actually really good at math, getting through nearly a college minor in material (one course short) before finishing high school. Then I had personal issues, took a few years off after failing out college for being a lush, and now I'm back in school getting a 4.0 I recently took a course which required basic calculus work, and realized how much I'd forgotten through half a decade+. Everyone in the class was only done with two calculus courses, and I'd done five plus other related ones. I kept getting things confused, and had to struggle through the material until I started to really remember it all. The first few weeks though, everyone looked at me like I was a crazy person for taking the class if I didn't know these BASIC concepts. I kept wanting to throw a transcript in their face with all A's in Math classes they won't touch for two more years. In the end I got an A because I studied and the material came back to me, at least what I needed did. But the point is, I know that feeling now. I've never been not top percentile in math (one question off perfect in math when I took my SATs, and it was stupid easy so I chalk that up to test anxiety). All of a sudden though, I felt what you felt. I'd be sitting there, watching the physics lecture, and everyone was focusing on electromagnetism and relationships that they needed to learn, all the while I'm thinking to myself 'what in the fuck are all those symbols up there... I have no idea what he's writing, let alone what we're learning.' I mean I've taught math before, helped friends prepare for their GREs on subjects that are basic to me, and I was always a bit too smug about how silly easy I thought it was. My feeling now with some more humility added in (though I felt much like this before) is that most anyone can do this math. It just takes a bit longer for some. The material isn't mind breaking. The concepts are really neat if you convey them right. If you wanted to, you could know this math. Just like how when I take an English course, I hate doing the reading because I'm being forced to read this and that. Sure, I'll get an A if I spend time and apply, but it still sucks to be forced to do when it doesn't interest me. Calculus starts with the concept of the limit, and area under a curve, and if I state it THAT way people are already like "right, fuck off, don't understand those statements or what you're getting at." But if I show you what that means, instead of using words and complexities that can throw you, you'd start to get it real fast. 90% of a first course in calculus these days doesn't even touch calculus till the end. They throw a bunch of basic stuff at you so that you understand what they're getting at. Basic integration/differentiation (calculus) are easy rules. But it's hard to understand without knowing WHY they work and where they come from and what the fuck it all means. I've taught people who suck at math the basic theory of calculus, without using any math. And it's amazing to see their eyes light up. They'll never likely learn it... but the concept is really interesting just in theory without numbers. TL;DR - Anyways, I'm pretty sure if you WANTED to know it, you could get it. And there is some really cool concepts in there that change how you view things in life. You aren't, however, any less of a person for not knowing. Just because you don't know doesn't mean you can't know. It just doesn't interest you enough. People get so bitchy over this, but there's a good reason. He isn't testing you on answers, he's testing you on method. If he's trying to teach you a generally applicable method and you've skirted around it by a trick, you haven't at all proved that you've learned the concept. Generally, though, a good teacher will remember to say "solve using x method" or "you may not use y method." Yep. I was on a pretty accelerated track for math, but even so I found a lot of ways to skirt past proper methods through much of trig and precalc, and once it started to catch up to me and my shortcuts no longer worked, I had to relearn concepts and realized why they were in place to begin with. Actually, most likely it was: You got the right answer, but by using a shortcut or other non-generalizable method, rather than the method I taught you (or any other valid method), therefore the answer is invalid. In math class, answers are almost never important. What is important is that you are able to find the answer using valid mathematical reasoning (which, 99% of the time, the simplest or most easily applied method is the one that was taught in class). The point isn't to see if you can find THIS answer, but rather demonstrate that you could have found the answer to ANY similar problem. We first construct the natural numbers, starting with 0 = the empty set {} in the natural numbers, and then define successors via the successor function S that sends any a in the natural numbers to {a} U a, also a member of the natural numbers; note that S(a) is not equal to a. We define for any nonempty set A that satisfies the following: There is a successor function S' such that for any a in A, S' (a) is in A. S' is injective, that is, S' (a) = S' (b) implies a = b for all a,b in A. There is an element e in A such that for no a in A does S' (a) = e. Every element of A can be obtained by iterations of the successor function to e, that is, for any a, S'n (e) = a for some n in the natural numbers. that A is a set identical to the natural numbers. We define addition as iterations of the successor function via the following: if a and b are natural numbers, and S(c)=b, then a + b = S(a) + c, and if b = 0, then a + b = a. Well, the first part just creates this system of numbers from sets by "adding one," starting from 0, but in a way that makes each new number in essence that number (nothing is in the set represented by 0 = {}, there is only one thing in the set represented by 1 = { {} }, two things in 2 = { {{}} , {} }). The second part says that anything you call "your numbers," as long as it follows this idea of starting from zero and adding one, is identical to the (annoyingly abstract) system created previously. For instance, if you gave me a number line and said those were numbers based on how many units away from zero they are, then it's the same system as the natural numbers. We're talking about the same thing. Then you'd have to ask, what is addition, based on only these numbers and this successor function. One way is to define it recursively, transferring a one over to the left number from the right number, one at a time. Then you can finally add 1 and 4 using the clunky, bare definition of addition. Only seven of us in the class. Exam is on displacement, velocity and acceleration. We're all well along if not through Calc I & II at a minimum (I was doing Multivariable at the time). I can't remember the exact question, I just remember the way he'd set it up, it was mad easy to throw variable into the displacement equation, differentiate and give him his answer. So much so that we ALL DID IT THAT WAY. When we got the tests back, he said we all had the wrong answer, and none of us had used his method. See, he apparently didn't understand calculus enough to see how the three equations were related. His answer was based on a strange method which echo'd what we did with calculus, but without actually using it. Average velocity being equal to displacement over time etc. In any case, he not only didn't understand the calculus, but he got the question wrong. We had to figure out his method to show him how he'd screwed up his own method in order to prove our answer right, and then show him how bare calculus understanding gives the correct answer and is more general. In the end, he said we'd got it right, but that he wasn't sure our method was going to hold up for all situations, so we could use it but be warned that if we got questions wrong in the future that way, he wasn't going to give partial credit... THE FUCKING IRONY WAS PAINFUL TL;DR - High School AP Physics teacher didn't know calculus, marked us all wrong because we used calculus and he fucked up his own answer with a ridiculous method. We corrected him and got credit, but he warned us of the dangers of our methods (integration/differentiation) because he wasn't sure if they'd hold up for other situations, and that he wouldn't give partial credit if it lead to wrong answers in the future. There is a reason you get it wrong if you do it the incorrect way. You will run into problems that require you to do it the way he's trying to teach you, and if you refuse to practice that way you won't learn it. So you get it wrong. It's only fair. Ugh those were the days. My Calc 3 professor marked a whole question wrong because instead of doing 1-2-3 i did 2-1-3. But get the same result because it doesn't matter if u do 1 or 2 first. Nope question is wrong and the question is worth 15%. His like I taught you to do 1-2-3 not 2-1-3 In 4th grade I accidentally discovered some form of trigonometric identity using geometry and trial and error. Obviously we weren't doing Trigonometry, just drawing and graphing stuff to get a feel for geometry and shapes. The problem was that the assignment was to do such and such, and I showed I never needed to do all these other drawing to get the final drawing because it was always going to some factor of a number in this equation I wrote up. At first she was incredulous (she didn't know the rule I was describing). She did, however, talk to actual Math teachers about it and show them my work, rather than simply making me do the assignment their way. This little bit of extra effort propelled me into some advanced math program where I was done with enough math to nearly give me a college minor in it in Junior year of high school (of course I had to do 9 credits at the school, Diff Eq and two other random courses, I chose linear algebra and a stats course my CS degree require, but everything else transfered in) If she had just said "oh that's neat, but we're doing it this way," I maintain that I'd not have been propelled forward in my math interests and been given those opportunities. TL;DR - Good Girl 4th grade teacher saw potential in something I'd done in math, moved it up the ladder enough to find out I'd accidentally derived a trigonometric identity years ahead of what we were studying. This little bit of effort on her part propelled me forward in my opportunities :D It's ironic. I tell high schoolers all the time that they better adapt quickly to the fact that when high school ends virtually no one will give two shits about them or their life. They will have to demonstrate value to someone for some reason. Teachers are the last people, in a sense, who will ever care about the success of your endeavors in spite of your attitude and behavior. My favourite teacher was the exact in-your-face type you get in the real world, except he genuinely wants you to succeed. If you fuck up, he'd be hard on you, but he encouraged me to at least try. Also, he was one sarcastic motherfucker. My school never gave two fucks about its students... Everytime I hear about "good guy teacher this" or "My teacher didn't pass me because he wanted me to understand that" I always feel seriously jealous. My teachers came, collected their paychecks, and told their students to fuck off because we weren't worth their time. :/ (there may have been good teachers, I only know the shit ones though...) This is entirely false. In university you won't be chased down, but the advisors and professors (and even higher up faculty) definitely care about your success. Advisors and professors will spend hours helping you if you come to them with a specific issue. I am a teacher. Some teachers are good, some are bad. Those of us who try to do good work and help our students end up working very long hours and often feel that we are just spinning our wheels, because the payoff (the students "getting it") typically happens so slowly that we don't really see it. Most of us are, at one point or another, also taking classes ourselves (to maintain our certification) while we are teaching, which can be very stressful. It is not the hardest job in the world, or the most important, or the most underpaid. But it is a difficult job, it is an important job, and we often accept pay cuts and other fiscal difficulties because we want to keep doing something we care about. I don't think this meme is saying that all teachers are saints, because really none of us are (we are people). But I do think it is important not to villainize the profession because of the bad teachers we all had. Take comfort knowing you've made a difference, even if you don't immediately see it. It's the good teachers kids thank when they're older. Heck, my mom still gets phone calls from her old students because she left such a good impression on their lives. Keep doing what you do, because you're right, it is an important job. I'm sure if it was a good girl teacher meme then it would consist of something along the lines of "makes you stay after class; gives you a blowjob." Of course the teacher would look like a pornstar as well. I will assume they would have taken their specialist course for the subject they teach, and have at least have 8+ years of teaching experience. The increase of salary is nice, but few people understand that they have to stay with school boards for a long period of time to actually earn that much, and take additional qualification courses to get into the appropriate pay-grid. But yes, teachers here probably have it better in the states. If only this whole strike thing didn't have to be so bothersome.... They're both 25+ years experience, but not necessarily experts. My mom works herself to death to teach her kids (she stays 6 hours after school closes sometimes for no extra money), but not my dad. It's just what the saleries are here. I'm very glad that the "Salary Comfort Index" is on there. Otherwise people might think that the $60k average salary in California is better than the $50k in Ohio. It's much more affordable to live in IL, DE, GA, MI, MA, and PA. In all reality, California is in the bottom 10 for comfort while being in the top ten in pay. I am in California, so my decent pay is a direct result of the the high cost of living here. I have a family of four, we live in a good area, but cant afford to save for a down payment for a house. My salary is a ten month salary, I have to make it last 12 months though. Someone is going to say get a summer job, fuck that, I spend my summers with my kids, my year and a half old daughter loves playing with her dad. I agreed with a lot of the things, except the low pay. I mean it's a decent pay. You're not going to get a mansion on your salary, but I hate when people exaggerate. My mom used to live off a salary of $15,000 a year. A teacher's salary is alright. I can understand why you believe the pay isn't low. I also assume we're talking about the U.S. here, but things may be different in your country if not. The work you do as a teacher is unlike any other profession. You can't compare someone else's salary to a teacher's and say "look, they're making it work." People have this misconception that teaching must be easy work because you get summers off, you get out at 3, and you get to tell kids what to do. None of this is true, at least for GOOD teachers. Good teachers don't fly out the door at 3 and forget about everything over the summer. If you want teachers who do - who don't give a shit - then by all means keep lowering salaries and coming up for excuses as to how 30k a year is enough to live on nowadays. The only comparison I can make for someone who works in an office is this - brace yourself, a wall is coming: Imagine every day going into work, you have to make 6 different hour-long presentations, usually about completely different topics, to everyone in the office. You can't go over or under time - it has to be exactly 1 hour for each presentation. The majority of the office doesn't want to be there, too, but you have to make whatever you're presenting about interesting. You have to watch every word you say, to make sure what you're saying makes sense and isn't offensive to anyone. You have to constantly keep reminding a group of coworkers to pay attention, usually every 3-5 minutes. If it wasn't interesting enough and nobody remembered anything from your presentation, then some of your coworkers wives and husbands will come in and start complaining to you, and you have to smile and nod even though you'd love to just tell them that your coworkers aren't remembering anything because they don't pay attention or don't care. Every second of those 6 hours that you present each day, you're "on stage" - you have no breaks, not even a second to catch a breather. You have to be there to continue presenting and fielding questions; you are constantly doing something work related. And even though you only work for 6 hours, you probably have to spend 3-4 hours extra just to prepare for tomorrow. If you call in sick, someone will cover you, but you'll be behind in your presentation schedule and you might end up having to cut some important ones. Then tomorrow, and every workday after that, everything repeats. You get up at 4:30 or 5 AM in order to get to work at 6:30 so you have time to set up the conference room for 6 more presentations. And every couple months or so, you have to organize a field trip for your entire office to go on. You have to organize an itinerary, explain to your boss why this trip is necessary, get their approval, raise money for the trip, secure buses to take you, convince three of the spouses to come to help out, and hire a registered nurse to come with you - for each of the 6 presentation groups. Oh, and someone will probably poop themselves. You get asked to coach the office softball team. Now you're in charge of logistics for this group as well - organizing buses, schedules, trainers, etc. At least you get paid extra for this, but now this cuts into your prep time for the next day and you start getting home around 9PM. Then summer comes. You spend your time preparing lessons for next year and getting a head start on those field trips, organizing schedules, and so on. You also need to get a job teaching SAT classes, or tutoring, or something, so that you can afford to pay off whatever debts you still have going to college, since your job requires you to have a bachelors. You also start saving up for grad school, since the company you work for requires you to have a masters after 5 years. Very few teachers really take a summer off. My summer was filled with professional development classes and going up to the school to work on curriculum, etc. The reason why so many teachers are lining up for jobs is bc of cutbacks. Just in the DFW area last year and the year before thousands of teachers were laid off due to budget cuts. During the school year my usual time at the school is 6am to about 4 on a great day. We have to stay after to tutor, we have to go up on nights for evening events, we have to bring a lot of work home to either prepare or grade. I have not yet met a lazy teacher who took this job because it is easy. It is underpaid. Most school districts require you have a master's, if not when they hire you, within a certain amount of time. We have to spend our own money on a lot of things for our classrooms and our students. If you look at the hours we work and what we have to do, we are underpaid. I love my job and I am thankful to have it, but teachers do not make great money. I doubt you will find many teachers stating otherwise. I don't expect to change your mind. Just my two cents. I didn't mean to exaggerate, but my deal with this argument is that compared to other jobs where much less is at stake, teachers are underpaid. Honestly, there is not fucking reason people in sports should be paid millions of dollars a year, while the people who will mold the future get paid, relatively, in crumbs. The whole system itself is messed up, the people lining up to get the job are honestly just there for the benefits, not for the career. Honestly, there is not fucking reason people in sports should be paid millions of dollars a year You're looking at the top 1% of people involved in sports. You know how many people try to make it there and get nothing, MOST PEOPLE. This comparison does not make any sense! How many top earning athletes are there in comparison to the amount of teachers? Think about that. The whole system itself is messed up, the people lining up to get the job are honestly just there for the benefits, not for the career. Hmm.. You're being a bit vague here, but I will agree something is up with the system. We're leaking tons of money somewhere and I'm not sure where. America spends the most per student in the world, and we don't get much for it. I don't think it's teachers that suck up that money either. Actually in Michigan there is a shortage of teachers lining up for jobs..."Qualified teachers" to be precise, because all those things are great, but you need a college degree to be a teacher most places and most jobs that require a five year degree program will give you those same benefits and/or much better pay. Depending on the area, it can be a bit more than "alright". There are teachers in my local school district with nothing extra behind them aside from 10 years of experience who are pulling + 75,000/yr. Oddly enough, the IT staff of the same school district pays about 50-55,000 for their systems administrators. It's almost as if we know our education is broken, so we all collectively said "Lets just throw a shit-ton of money at the problem and hope it goes away". "with nothing extra behind them" This is an ignorant statement. Teachers need to get their Master's degree (6+ yrs) and constantly take staff development courses. Not to mention 10 years of experience is amazing as a teacher if you look at the numbers. That teacher most likely taught more than 1000 students. I think this is easily a good/deserved living wage. No, it's not an ignorant statement. I used the phrase "With Nothing extra behind them" because I'm speaking of instances where they have no masters degree, and teach standard students. "constantly take staff development courses" I work in the backend part of IT where everyone has to keep mutliple certs current from multiple vendors. Tell me all about the hardships of having to get the extra teacher's certification and the extra staff courses. It must be aweful. I dunno, I think it should be at least compatible to what the private sector pays for that level of education. All to often the government sector tends to vastly overpay it's employees. I know some like social workers make diddly squat, but when I was working construction in my younger days we used to have our pay go upwards of 40$/hr on government jobs. Why? Because employees who work on government project have to make government wages and benefits. So I was paid $40/hr which could come out close to $80,000/yr to do something that you could train any able body to do in a few weeks of on the job training. Depends entirely on the cost of living for the area. If property taxes are high - for instance if you live in an affluent area - then expect school quality to be better and teachers to be better as well. The real problem is complaining that some teachers make 75k and do less work to earn their wage than the school IT staff. Or otherwise complaining how teachers are overpaid and the education system sucks, simultaneously. That one ALWAYS gets me. Thanks for the caveat. It's annoying that "teacher pay" is treated as such a monolithic topic. For instance: the starting salary for a teacher in NY is $45k. In my state (NC), you don't make that until you've taught for 24 years. Assuming they ever unfreeze our salaries... Ehh, I had a history teacher who drove a perpetually new Audi because it turned out he was a mob enforcer on the side. He ended up getting arrested and now is serving a very long sentence in prison. That being said, he was a fucking awesome AP European History teacher. Good Lady Middle School Teacher: (Just as a note my teacher was about 70 at the time) Substitute teacher one day. Everyone was talking crud about the teacher but me and a couple people that decided to do our work. Next day: "Students I'm back from the Doctors!" (Continued noise and ruckus from students) (Teacher sits down and sighs) My friend and I went up to her desk and asked if she was okay. "I've been diagnosed with cancer." Skip forward a couple weeks Another Sub? That's bad. Next day: "Why weren't you here yesterday, Ma'am?" Teacher- "I have also now been told I have a blood clot in my leg." She couldn't drink anything to cold or to hot or she would be in extreme pain. She was on a bazillion different meds and some didn't mix well together. She had to have chemo and couldn't stand up without getting dizzy. But I saw her at school every day for my entire 7th grade year. She even came to all of the track meets that year for our team, and cheered them on. (That year we broke the county record for fastest relay, which she had coached at the beginning of the year.) I found out in 8th grade year that a week into summer, she died due to the cancer. As an aspiring teacher, I've always wanted to do something different. That's not a materialistic- natuerd 9 'tll 5 job. I've always envisioned myself doing something special, being in a job of responsibility and purpose. I want to do something where I can inspire others, help, and possibly change people's lives. That, is the reason why I want to be a teacher. As a teacher, it's because many teachers are idiots who don't even understand the subject they're teaching, let alone have a passion for learning. A coworker of mine told me recently, "Kinbensha, you really like learning. I hate learning. That's why I became a high school teacher instead of working at university. After I started teaching, I didn't have to read or study anymore." Really, for serious academics, going into teaching is mental suicide. Becoming a researcher or university professor is far less likely to end in you strangling someone. Edit: This isn't even mentioning teachers who hate their students. There are plenty of teachers who actively seek to distance themselves from students in any way possible, don't want to help them, don't want to spend time teaching them. As another teacher...I hear this way too often. Too many of my co-workers (I teach HS) will freely admit that they're there for the great healthcare, weekends and holidays off, and the opportunity to "lord over" their kids. Many of our Reading teachers cannot read. I've had an English teacher tell me she wanted to burn books. Upon returning from the summer break, I asked someone in the science department what they thought of the Mars Curiosity landing. She said - "The what?" It's enough to make you want to cry. Those of us teaching HS that are actually knowledgeable in our field and care about our students are becoming the exceptions rather than the rule. I think this changes a lot depending on where you are teaching. The overwhelming attitude of most teachers I work with is that they like their students and truly want to help them. We have a handful who might say otherwise, but they are a minority and usually don't stay in the building long. It's hard to build a positive school culture among teachers, but it can be done. The people who hate it really need to stop teaching, since they give the rest of us a bad name. Where do you work? I am in Detroit, at a charter school, but the vast majority of my coworkers are there because they truly want to make a difference and have a positive impact on education. I believe it happens, but maybe I haven't been teaching long enough for this to be my experience... But seriously, Detroit! I just hate hearing teachers bash the field by saying 'yeah, all those damn awful teachers except me!' I know, I am just trying to stay in denial... Yup. I ranted earlier in the thread, but basically we pay teachers so little in most states that they can do other occupations and have a much higher quality of life. I'm certified to teach accounting and IT, and both of them would pay better than what I'm making. Hold on a second I can have a job as a researcher? I'm planning in going into physics to become a professor. How could I get a job as a researcher? Would I be a researcher at a university? Also why do some teachers decide to become high school teachers instead of spending the extra two years in college to become professors? "Kinbensha, you really like learning. I hate learning. That's why I became a high school teacher instead of working at university. After I started teaching, I didn't have to read or study anymore." I love to learn and would love to teach at a university, but a PhD is four more years of school for the same pay that I'd receive now. Aside from less teaching time, I can't see the benefit. (Canadian) People don't deal with truck drivers as often as students deal with teachers. When youre a student you have to deal with the same person an hour a day for 5 days a week for a year, plus complete their assignments at home, all while they have complete control over your grade for the whole year. It explains why bad teachers are bashed. There are certainly some names that come up from my past teachers all the time as terrible teachers among my peers, while others are hailed as awesome, and some are just never mentioned. So, of course, the bad ones are bashed because they're terrible AND you have to deal with them so much. Truck drivers hardly deal with anyone long enough to get to know them. Mike seemed to be suggesting as to why a teacher should be bashed for being a teacher. In which case, my answer is that they are just people and vulnerable to insecurities, and can be bashed just like anyone else. I doubt he was primarily inquiring as to why teachers are bashed more often than others. Right, but I wasn't responding to this Mike fellow, I was responding to the individual who responded to Mike (I'm assuming). This is just getting redundant. If you are responding to Mike, then respond directly to him, otherwise he won't get notification that you responded at all. Everyone had good teachers, everyone had bad teachers. The flip side of the image is that someone who is underpaid and gets no respect is a terrible person to spend a year with, especially if they hate their job and the kids they teach. Some teachers are genuinely bad though, it's worst in small towns. I went to high school in a small town and one teacher was a raging alcoholic who never washed, threw accusations at students for no reason and called us names. They never fired him because they couldn't replace him. There were many other teachers like that at my school, one of whom backed me into a corner then tried to say I "physically intimidated her" and brushed off the many witnesses as liars because "they're probably her friend". She was kept on as well, despite having just had a mental break down and clearly being unfit to work. All the while these teachers moan about the "work" they do and the low pay. Teachers are well paid in Scotland and in my opinion a lot of my teachers were lazy, malicious fucks. I'd have guessed we went to the same school if the sexes weren't switched. Alcoholic HE teacher and my music teacher was the one who had a breakdown. He tried making every class sit in silence while he typed gibberish on an ancient typewriter; took one and a half lessons until we realised he couldn't keep us all there and walked out. He was kept on for nearly a year without teaching a single lesson. My partner's a teacher who really does care and he can confirm that the same is true of shitty teachers here in the UK. Our problems came when we started employing teachers straight out of university. Not for the subject they're teaching but for teaching itself. They know all the regs but can rarely command respect from a class full of kids. I want your life. I teach High School English, 4 preps (one of which is an AP class), run 2 clubs, and coach o make 43k a year with a masters. Americans really do not pay their teachers very well at all. Because all kids think when they grow up they're going to be multimillionaire professional ball players, famous musicians and awesome reality tv celebrities. Little do they know, "16 and pregnant" doesn't count... I believe that everyone always seems to have an opinion on teachers because everyone has had to interact with a lot of them over their entire adolescent life. Because of this, mostly everyone has had a negative experience with a teacher one time or another, so they think that the entire profession is a farce. But, like any other profession, we will always have exceptional, good, or bad people in the field. But people rarely complain about the incompetent bankers, or horrible lawyers, mainly because they have had no forms of personal interactions with multiple members of that field, or have needed their experience for 15+ years. for those complaining that teachers are overpaid, i invite you to see why this isn't actually true, at least if you think like an economist and realize that teaching isn't the only option for many of the people who end up being the best teachers. Short sighted is right. There are school districts that extend furloughs because they can't pay teachers. The kids might get longer vacations but that sure as shit won't help them 5 years down the line. ...wouldn't that depend on the particular circumstances under which a particular union is striking and their demands? This makes about as much sense as saying "I support politicians running for re-election." Some strikes are worth supporting, others are not. I'm going into teaching, and I think the "we want more money" sort of strikes are embarrassing and irresponsible. Sure, I can understand teachers wanting more money. In many places, they are underpaid. However, they do get enough to live on, and it's just irresponsible and selfish to hurt the kids' education over it. There are better ways to handle things like that. There are other situations though, in which I think the teachers are completely right. Class sizes in some places are getting enormous. I did a field study in a Milwaukee middle school last year, and there were 40-50 kids in every class. That is insane. They didn't learn anything because the class was constantly out of control, and it wasn't the fault of the teacher either. If she spent one minute helping a kid with a problem, the rest of the class would go completely nuts. The kids who wanted to learn and do their work were at a huge disadvantage because at least half of the class period was spent disciplining kids or getting the class to settle down. In cases like this, I think strikes are justified. It's a condition that is hurting the kids just as much or more than the teachers, and it's a huge problem that is getting in the way of education. I can't believe how low people's opinion of teachers is. There's alot of bad ones of course, but if you're even half decent you sure have to work A LOT. What was the suckiest thing about highschool? After sitting there working all day long, when you came home you STILL couldn't relax, because you had a shitload of homework to do and tests to prepare. That's EXACTLY what being a teacher is like. Except that the hourse spent working during the day are way more intense and stressful than when you go to school, of course, and the workload after hours is tremendous. All that combined to a huge pile of bullshit that you have to put up with, from horrible classes to horrible parents to ridiculous administrative shit to low pay and so on. I quit because I'm not dealing with so much work after hours and stress during the day, and all the bullshit that comes with the job just to get paid very little and get no respect at all from anyone. Yeah, I know. The only teachers they like are the ones that truly don't care and are just there to collect a check. For example, there is a middle school teacher that just jokes around and doesn't really care if students turn in completed work or not. Sure, the kids love him, but they are getting nothing from him. Then, there is a truly, caring teacher who wants to inspire kids. She is made fun of, puts in crazy hours, and tries to instill in the kids responsibility. The kids just torment her. It's horrible. Remember that being an "English teacher" isn't about English, or books, or essays, or poetry. An English teacher is a teacher first and foremost, and it's a rare day indeed when a student actually finds the beauty and awe within the subject area. Case in point: in my middle school class last year I spent more time dealing with disciplinary issues than anything else, and I was told point-blank by a parent that it wasn't my job to help her kids learn how to love literature. The kicker: she's a teacher, too. With my two degrees, level of content knowledge, and skills in education I think should be around 60-70k a year. I teach AP Lit, which potentially saves my STUDENTS collectively around 35k a year. If you look at what people pay just for daycare, not someone with an advanced skill set and content knowledge, they are averaging about 200 a week. Assuming I have a full class of 35 (my classes are sometimes larger) all day I should be compensated around 7k a week, pre-tax, for what I do. Just off of daycare cost. Instead of just watching them I manage to teach them how to read, write, understand subtext in a variety of media, apply social and philosophical theories to a text, and exist as a social being. I do not know many day care places that can also teach your child to apply Marxist or deconstructive theory to a work. Do you feel you're underpaid? Yes, I just answered why. I make 42k a year as a teacher with 5 years in my district and a masters degree. MBA ranges about 100k average. MBA simply makes money for other people, I instruct, shape, guide, and inform the future of our nation and world. Could you make more money with your skill set outside of academics? Yes. I could write ad copy, go in to technical writing, professional editing, content creation for web enterprise, and so on. All of these options would pay more than teaching. However, they can be very competitive positions. Do you feel teachers spend more time on the job as compared to other salary workers? Depends on the field and job. I work about 50-55 hours a week on average. I will not lie and say "Oh I put in so many more hours...blah blah blah." The private sector works very hard, and also puts in many hours. I would not say I work more than the average, mid-leveled, corporate job. I know guys that work those jobs that put in just as many hours as I do. However, there is a misnomer that teachers work only half the year or whatnot. This year I will spend 210 in an actual school building (according to contract), and that is not including professional development, workshops, planning at home on my own, grading, and whatnot. well, let's see: my skill set is worth? don't know. I never really thought of that. I have two Bachelor's degrees (special ed and general ed), so there's that. underpaid? Yeah, very much so. I make about $870 every two weeks. Could I make more money outside of academics? I can apply for other jobs in offices or whatever. It is hard to "move up" as a teacher in the profession. In time, I can become a mentor teacher, a reading or math coach, or other type of facilitator (for a slight increase in pay). The big bucks are in administration. Time on the job? My "workday" is supposed to be from 8:00 - 3:40, at least what my contract says. However, I usually get to work at around 7 am and leave home until 6. People don't know that grading, planning, and preparing is usually done on our own, unpaid time. I put in LOTS of unpaid overtime. Grade level - I teach 4th grade, but I can teach up to middle school. Rural/densely populated - Desert southwest and very rural. Cost of living - I would say it's low. Housing is expensive, though. Major forms of employment - There is lots of farmland, so agriculture is big here. We do have two military bases. Our town is about 100,000. There are officies, major retail stores, and so on. Arizona - very Red, both because of the politics and the scorching sun! I'll answer these questions as a teacher. 1. My skill set is probably worth somewhere between 40k - 50k US 2. I feel I'm underpaid and will continue to be underpaid. My district pays me $34,000 as a 4th year teacher. It will take me 16 more years to break 40k. With a masters degree it will take me 7 more years to break 40k. 3. I'm a band director and outside of academics I could make more but it's extremely difficult to find jobs. 4. I work from 7:30 a.m. to 5:00 p.m. at school. I come home, eat dinner, bathe my son and get back to work. I usually work on average about 2 hours a night on grading, paperwork, databases, music, etc. Skill set: I could go into industry instead of education with my degree and probably make 40k a year. That said, I only have a minor in the industry side of my degree. I'd like to think a full degree is worth more than a minor, so probably somewhere on the low side of 50-60k in my current market. Underpaid: I make just over 30k, so considering my answer to skillset, I do feel underpaid. Question 3: yes I do. I could do IT or accounting, or marketing and make more than I do currently. Time spent: Time on the job? Not necessarily. Time spent working on things for work outside of the job itself? Definitely spend more. Then again, with summers its probably about even. I had a teacher in High School who was the most lazy teacher I had ever even heard of. He didn't teach us jack shit (and I am convinced he didn't know jack shit because he did get his information wrong a lot) and spent his entire class time on the computer. He would assign us homework and then just give us all credit, even if we didn't do it...so you can imagine that no one did it once we wised up to that. He was a history teacher and then went on to be my AP US History teacher. You know what his advice for the AP test was? "There won't be any pre-revolutionary war era stuff on the test so we won't even cover it." Guess what about 50% of that test was on? Yup, colonial America. I just about flipped my shit. Needless to say I didn't pass that test. Every question I got right was about parts of history I had really taken an interest in and learned about before I had even taken that class. I really loved WWII history so I know I did pretty good on those questions. Unfortunately there weren't many. And you know what? I couldn't even complain. Everyone loved the guy. He was friends with all the students AND the soccer coach that lead our school on a pretty big undefeated streak that year. The guy was pretty cool as a person, but he was an absolutely horribly lazy teacher. He got paid too much for how much work he did. Meanwhile I had a drama teacher who worked her ass off each and every day and really cared about us who was threatened with a pink slip every year because of the budget. The lazy history teacher was basically assured his job because he was head of the history department and they weren't going to cut a teacher that taught something like history when they could just cut the hard working drama teacher. Thankfully she never got cut and still works there. Unfortunately I believe Mr. Lazy still works there too. TL;DR Had a really lazy teacher who didn't teach anything and never got fired, school was going to fire the hard working teacher who cared. As a full time teacher I arrive at 7:45am, prepare my classroom for the day (I might have 20 minutes yard duty). I teach 6 50 minute classes (~32 students each), including monitoring the hallways in between each class. I get a 35 minute duty-free lunch (which sometimes I eat with the other staff and sometimes I work in my room). After school ends I monitor the halls again and then attend meetings (parent, staff development, district, etc.) or grade papers, plan lessons, clean my classroom) until 4:00pm. Then I pick up my own children and mommy until their bedtime at 8pm. After the kids are in bed I often continue to process papers, return emails and plan until 10pm. Low pay? Bitch please. The average salary for a teacher in the district I grew up in is about $75k. What's that? Suburban schools have inflated salaries? Well, the average salary for a Chicago Public Schools teacher is actually about $76k. And remember this is for only 9 months of work. It's equivalent to 100k/year. Of course this doesn't include the all of the perks and benefits like the pension you're entitled to collect for the rest of your life after you've worked for only 30 years. What do I mean only 30 years? Well think about it this way, if you start working at 25 you're eligible for pension benefits at 55. According to the CDC, the life expectancy for someone who's made it to 25 is 58 years. So, 25 + 58 = 83. So let's say you work for your 30 years and then retire. You can be expected to live to 83, another 28 years, all while receiving pension benefits. You don't need a degree in math to realize that's not feasible. Anyways, I'm not saying teachers don't work hard or that they deserve anything less than the absolute best society has to offer. But this idea that teachers are broke is completely bullshit. You're taking a small sample (CPS) which has a notably high teacher pay average; and if you've ever been in a CPS school, you'd know that even these teachers are underpaid for the shit they put up with. You're also talking about the average, which includes teachers who have been teaching for 30 (or more) years and have obtained advanced degrees that balloon their pay. Some teachers do make plenty of money to make a great living; however there are also many younger teachers, especially at the elementary level, make less than $25k/year, even in Chicago suburbs. These folks are also facing pay freezes at the beginning of their careers and contracts that severely limit their pensions because of a current backlash toward teachers' unions. I'm not saying that no teachers make a good living, but there are a lot of woefully underpaid professionals in education who continue the grind because they care about what they're doing. And yet compared to police officers in those same towns, the can make much less. In my area cops make an easy 100k with almost 0 crime beyond high school kids being hooligans. I understand that, as a cop, you put your life on the line. However America focuses far too much on policing, then they do on educating. Which I'd bet would prevent much more crime then the massive amounts of security/military/police spending we put into our country would. Nah I live in the tri state area though. In the suburbs similar to where I live, cops make out like bandits. I know someone who became a cop, I think he got promoted to a county or state position now and has been working for 8 years. Makes 2 grand a week. You seem aware of the emotional and mental stress placed on teachers, addressing the fact that we work hard, so I won't go into all of that. In fact, I'd argue that the profession is substantially more rewarding and fun than it is stressful. I enjoy it a lot more than the jobs I had in retail, food service, and architecture. However, I would like to break things down mathematically for you. I have a Masters in Middle Grade Education, this is my second year teaching, and I work in a suburban district in North Carolina - $36k a year. (Not that the number of years teachers in NC have been working really matters, as there has been a pay freeze in effect for the past five years.) I work at the school from around 7:30-6:30 Monday through Friday, sometimes slightly earlier and sometimes slightly later. I am required to tutor students until 5:30 at least two days a week, every week for no additional pay. I am required to sponsor at least one club/group for no additional pay. I am required to work at least three weekend fundraising events, such as dances, car washes, or doughnut sales, for no additional pay. I am required to represent the 7th Grade in at least one committee, which usually meets twice a month and requires my footwork the rest of the month, for no additional pay. I am required to work through my 25 minute lunch time, chaperoning the students in the cafeteria, for no additional pay. I am required to meet with parents, EC teachers, ESL teachers, principals, assistant principals, Central Office representatives, textbook representatives, curriculum facilitators, technology coordinators, and students during my planning period, for no additional pay, so planning and grading almost never happen during this time. I am required to attend and sometimes present at meetings and training sessions during non-student days, for no additional pay. I am required to attend one PTO meeting a month, from 7pm-9pm, for no additional pay. I spend an average of 12 hours a week writing lesson plans and grading work at home, for no additional pay. I am required to work at least 5 athletic events a year, which last until 7:30 then I have to help clean up, for no additional pay. You mention that teachers only work 9 months a year, but people are paid by the hour, not the day. Go ahead and add all of that time up. If you only count the time I spend at the school during the normal work week, and none of the additional work I do at home and at the school on weekends, that's an average of 11 hours a day 180 days a year, 8 hours a day 5 teacher workdays a year, and 8 hours a day 5 training days a year, for a grand total of 2060 hours a year. An average full-time job is 2000 hours a year. Teachers also spend an ungodly amount of their own money on their students and classrooms. Although the school does not require me to do so, I buy academic supplies, art supplies, and books for my students because they can't afford them, but also can't work efficiently without them. So far I've spent over $700 this year cleaning, painting, and repairing my classroom because the district didn't have the money to bring in maintenance. I sponsor students who can't afford to go on field trips, play sports, or play instruments. I buy prizes for the school store. I pay for any incentives I offer my students. I send food home with students on the weekends. I send presents home with parents for Christmas. I send clothes home with students at the change of every season. Not to mention the sheer volume of stuff I've bought because our kids were having one of those wrapping paper/cookie dough/window decal/discount card fundraisers. Last year, I had documentation of around $5,600 in work-related expenditures, but I spent more than I documented, and I only got back an additional $23 on my tax return for itemizing it anyway. Finally, I'm not sure if you knew this, but teachers do pay into state pensions. It's not like we're just getting a pension for nothing. I pay 6% of my gross income every month toward the fund. Teachers working today pay the pensions of retired teachers today, so that teachers working in 30 years will pay their pensions when they retire. Inflation actually helps the system because pensions are based on how much you were paid when you were working. Just because YOUR district pays $75k doesn't mean every district does. I make $32k which INCLUDES extra days and a stipend. I have around 200 students. It works out to about six cents per student per hour. I think 30k a year is pretty fucking low for anybody who has spent at least 4 years in college. My education degree just got changed from a 5 year to 4 year program as well, so many teachers have gone through 5. My husband and I both grew up under the poverty line. I lived with my single mom and little brother. I know what it is to be poor. I paid for college all on my own through scholarships and working a food service job. I had zero loans or debt. I graduated top of my class from a major university and it took a year and a half to get this $30k job. In the mean time, my husband and I lived on retail+internship wages. We lived in a crappy apartment, worked extra jobs here and there, lived within our means, and we did just fine. Jumping up to $30k is going to make us feel like millionaires. I'm not saying I don't make enough. I do feel like I've put in enough work through high school, college, and post-college to earn a fair salary. But doctors are different in the sense that they're a pay for service industry. A doctor that sees 100 patients will make more money than a doctor that see 50 patients (assuming all other factors are the same). That's not the case for teachers with many students versus teachers with few students. What? There aren't unlimited awesome jobs in the world, and people often have to settle for less due to other limitations such as housing market, family, transportation and all manner of things rendering this discussion pointless? People always say this, and forget that teaching is one of those jobs where you can't always just quit when you go home at 5. There's often still grading, lecture prep, writing homework assignments and tests, and administrate work to do. My mom was a full time music teacher. She certainly wasn't struggling to say afloat, but it was a constant struggle to keep her job in a district that was regularly making cuts to everything. We had a decent condo and cable tv, but when I lived there from middle school to high school (roughly 1999-2006), we had the slowest dialup available, an awful computer that was almost impossible to do projects on, and had to eat off-brand "cheeri-q's!" and crap, and this was in a good state for teachers (Michigan, until recently). Every moment she wasn't giving to me and my brothers she gave to the district, and that was a LOT of time and a lot of work. The benefits were amazing, though. I've always had the best health insurance available. Even with that, I've never known a teacher to say "it's worth it, for the money." The ones who feel it's worth it do so for the satisfaction of helping their students. I make 32k per year before taxes. I teach in the lowest of the low socio-economic areas in my state. It's not unusual (in the 12th grade) for 60% or more of my students to be unable to read on grade level. Many of them read at a seventh grade level. I come early each day, I stay late to tutor and get no pay for it, and I still love my job. Your comments are about one of the highest paying school districts in America. And don't give me that "ITS ACTUALLY 100K" bullshit. No its not. It's actually 75K, you lunatic. And teachers work so much outside of school that it more than makes up for the time we get in the summer, which we usually have to spend taking college classes and getting re-certified. My girlfriend and I, both teachers, made about 35K between the two of us, including when we both coached Track and Field and I coached football. Teachers in the district we are currently in can make, maximum, 68K, which is after 105 Masters credits taken and 18 years in the district. Your example is FAR from typical, while our situation is very typical. And pension benefits and retirement are increasingly going away. So you don't know what you're talking about. I have a friend who was an Assistant Principal at a school in Fulton County (Atlanta), GA. They had a large spreadsheet that cross-referenced years of service, education level, teaching field and position (teacher/counselor/principal/commissioner) and each had a clearly defined salary. Like many/most jobs starting pay was a bit low but with some experience and good education you can make a very, very good living. Assistant Principal at the time I saw (2005'ish)( this was around $85K and a principal was over 6 figures. EDIT: Actually found the 2011-2012 pay schedule for Fulton County teachers online. Not sure what "step" refers to but it looks like salaries range from about $40k to over $80k. Also found the Assistant Principal / Principal pay schedules. Looks like the pay range is $66k - $128k. Not bad at all. As public employees it's all public info. Just google "(school district) salary" and you can find out what your HS teachers make. There are two types of salary increases, step and lane. I can never keep them straight, but one is for each year of experience and the other is for education. So if you go back and get your master's or things like that.
Browse Results Modify Your Results This undergraduate textbook is intended primarily for a transition course into higher mathematics, although it is written with a broader audience in mind. The heart and soul of this book is problem solving, where each problem is carefully chosen to clarify a concept, demonstrate a technique, or to enthuse. The exercises require relatively extensive arguments, creative approaches, or both, thus providing motivation for the reader. With a unified approach to a diverse collection of topics, this text points out connections, similarities, and differences among subjects whenever possible. This book shows students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics, by mentioning current activity in the mathematical community, and by discussing many famous and less well-known questions that remain open for future mathematicians. Ideally, this text should be used for a two semester course, where the first course has no prerequisites and the second is a more challenging course for math majors; yet, the flexible structure of the book allows it to be used in a variety of settings, including as a source of various independent-study and research
Product Description A welcome addition to Saxon's curriculum line, Saxon Geometry is the perfect solution for students and parents who prefer a dedicated geometry course...yet want Saxon's proven methods! Presented in the familiar Saxon approach of incremental development and continual review, topics are continually kept fresh in students' minds. Covering triangle congruence, postulates and theorems, surface area and volume, two-column proofs, vector addition, and slopes and equations of lines, Saxon features all the topics covered in a standard high school geometry course. Two-tone illustrations help students really "see" the geometric concepts, while sidebars provide additional notes, hints, and topics to think about. Parents will be able to easily help their students with the solutions manual, which includes step-by-step solutions to each problem in the student book; and quickly assess performance with the test book (test answers included). Tests are designed to be administered after every five lessons after the first ten. Please Note: This is the third printing and errors to date have been corrected. Product Reviews Saxon Geometry Homeschool Kit 3.9 5 18 18 Let me out of here! We began using this text for the first time this year at our co-op after using Saxon for Alg. I and Alg. II. Of nine students, five have dropped within the first five weeks due to frustrations with the text. This is not the Saxon I've loved and heartily endorsed. The questions are confusing and take forever to wade through (yes, we're using the corrected version). I would work through the lesson with my daughter and then she'd spend an hour doing the problems, miss 50-75% of them, and then we'd spend an hour correcting the lesson. It was with great relief we fled to a different curriculum that she is flying through. BTW, the Saxon tests are a breeze, so she obviously learned something, but we don't have 2+ hours to spend on math each day. September 25, 2013 Saxon Geometry is the best next step This was the best next step for my high-schooler. We've always used Saxon math and wanted to continue into high school. It's right on target and meets our needs perfectly. CBD was also the obvious choice of where to buy it as not only are they competitive on pricing, but the customer service is excellent. Thanks CBD for helping with all our homeschool needs. May 24, 2013 Poorly written textbook I have used Saxon for about 20 years so when they came out with a Geometry text I was pleased because this was lacking in the other texts. I've been highly disappointed. Number one, I was assured that I had the new solutions manual when I ordered but sadly there are many errors, as well as leaving out crucial steps in order to follow their solution. I have found solutions with concepts that have not even been taught. Number two, the text is just poorly written in general. The problem set questions are hard to understand what they are looking for. The explanations in the text are vague at best. I have had to rely on my engineering son to help me solve some problems for his youngest brother. This text does not hold up to the standard of the older textbooks. Don't buy it. You will be frustrated. April 10, 2013 It's Saxon, what else can you say My daughters were doing Switched on Schoolhouse Algebra. Both were lost. The combo of the DIVE videos and Teacher Videos did the trick. Saxon rocks. December 27, 2011
and the physical world. Stimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non ...Show synopsisStimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non-Euclidean geometries. Also describes how math is used in optics, astronomy, and other phenomenaA good copy, Unmarked, Solid, Cover shows wear, A good copy...A good copy, Unmarked, Solid, Cover shows wear, A good copy overall. We take great pride in accurately describing the condition of our books, ship within 48 hours and offer a 100% money back guarantee. Description:Good+ Spine and covers are heavily tanned, however text is only...Good+ Spine and covers are heavily tanned, however text is only lightly and evenly tanned. Edges and spine folds are worn. But, don't judge this book just by the cover-because the binding is tight and the text is clean-no names, inscriptions, underlining or highlighting. It's a decent copy.; "A history of mathematics, man's greatest invention for the investigation of the physical world. Morris Kline, professor of mathematics at New York University, shows the simultaneous growth of mathematics and the physical sciences, and the interaction of one domain with the other from the time the Greeks first recognized the mathematical design of nature up to modern science which, according to Kline, is a collection of mathematical theories adorned with physical facts."; 546
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Student Resources Math Useful Links Dave's Math Tables Virtually every table needed at any level of mathematics covering general math, geometry, discrete math, trigonometry, statistics, calculus and advanced math. Other good resources are also listed here. GoMath An on-line math help site. Topics covered includes algebra, geometry, graphing, formulas, and SAT math. There are interactive tutorials, quizzes, and math games available.
Who cares about the real world? I'm in pure math. Even then, a TI89, or whatever you call it, won't be of much use to you in most applied areas other than accounting. You will need to use things like maple, mathematica, or matlab because the math becomes too complicated. Knowing how to plug and chug is ultimately useless. Programming on maple, mathematica, or matlab at least requires you to know the mathematics behind what you're doing.
Mathematical theory in basic courses usually involves deterministic phenomena; however, in practice, the input to a linear system may contain a "random" quantity that yields uncertainty about the output. Probability theory and random process theory have become indispensable tools when analyzing these systems. This SPIE Field Guide discusses basic probability... more... The text is designed for use in a forty-lecture introductory course covering linear algebra, multivariable differential calculus, and an introduction to real analysis. The core material of the book is arranged to allow for the main introductory material on linear algebra, including basic vector space theory in Euclidean space and the initial theory... more... Karl Gustafson is the creater of the theory of antieigenvalue analysis. Its applications spread through fields as diverse as numerical analysis, wavelets, statistics, quantum mechanics, and finance. Antieigenvalue analysis, with its operator trigonometry, is a unifying language which enables new and deeper geometrical understanding of essentially every... more... A central area of study in Differential Geometry is the examination of the relationship between the purely algebraic properties of the Riemann curvature tensor and the underlying geometric properties of the manifold. In this book, the findings of numerous investigations in this field of study are reviewed and presented in a clear, coherent form, including... more... This comprehensive textbook is intended for a two-semester sequence in analysis. The first four chapters present a practical introduction to analysis by using the tools and concepts of calculus. The last five chapters present a first course in analysis. The presentation is clear and concise, allowing students to master the calculus tools that are crucial... more... Inverse limits with set-valued functions are quickly becoming a popular topic of research due to their potential applications in dynamical systems and economics. This brief provides a concise introduction dedicated specifically to such inverse limits. The theory is presented along with detailed examples which form the distinguishing feature of this... more... This book contains survey papers based on the lectures presented at the 3rd International Winter School "Modern Problems of Mathematics and Mechanics" held in January 2010 at the Belarusian State University, Minsk. These lectures are devoted to different problems of modern analysis and its applications. An extended presentation of modern... more... This book is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do... more...
Welcome to CPM Educational Program, an educational non-profit organization dedicated to improving grades 6-12 mathematics instruction. CPM offers professional development and curriculum materials. We invite you to learn more about the CPM mathematics program by clicking the "Learn about CPM" link at left. The other sections offer support materials for teachers, parents and students. Headlines The statistics and probability lessons from the CPM Core Connections series have been posted here. This series of Core Connections Statistics Supplements replaces the Statistics and Probability Resource that was previously available. These files are available for download free of charge. For a class not using Common Core State Standards (CCSS)–aligned curriculum materials, these lessons supplement the statistics and probability content to meet the statistics and probability standards of the CCSS. Even if CCSS-aligned curriculum is being used in the classroom, in the transition period to CCSS, it is possible, even likely, that a class will not cover all of the statistics and probability content standards. This content will have to be taught in a subsequent course. These CPM Statistics Supplements provide the materials for any delayed statistics content. Conference Handouts and Resources News New curriculum and technical support available Three CPM mentor teachers are now available to help parents, students and teachers who have questions about the CPM program. This support is primarily for questions about using the program, its technology and the website. To use CPM's support service, go to to see the available services. Follow the prompts to get help. Note that this service is not a "homework helpline." That support is at CPM now offers a new series of textbooks to meet the grade 6-8 and high school CCSS content and practice standards: Core Connections, Courses 1 - 3 and Core Connections Algebra 1 & 2 and Geometry. Learn how this series as well as the original Connections series of CPM textbooks are fully aligned with the CCSS Content and Mathematical Practice Standards. CPM can also provide professional development centered around embedding the eight CCSS Mathematical Practices into your current lessons and current textbook from any publisher. Start moving on the path to the CCSS today! Parent e-book licenses are now available! Parents may purchase a one-year e-book license of their student's book for $10 by calling CPM and using a credit card. See order form for a complete list of available one-year licenses. Contact CPM at (209) 745-2055. Sample Problem Core Connections Algebra 2: 9.2.1: How can an experiment show cause and effect? 9-45. Testing Cause and Effect with Experiments Consider this question: "Does a new type of motor oil improve gas mileage?" Assuming this new motor oil was already available to the public, you could survey some car owners who use the oil and some who do not and compare their mean gas mileages. a.) What problems might arise from using this method? What are some possible lurking variables that could affect the outcome of the study? [answer] b.) After reading the Math Notes box in this lesson, design an experiment to help decide whether the new motor oil improves gas mileage. [answer] c.) Sometimes a group gets no treatment. If people are involved, they may get a placebo (a treatment with no effect) so the psychological effects of participating in an experiment are the same among the groups. If you have not already done so, include a no-treatment or placebo group in your experiment. [answer]
Probability Theory - 07 edition Summary: This introductory text features well-chosen problems and exercises that illustrate algebras of events, discrete random variables, characteristic functions, and limit theorems. Prerequisites include knowledge of differential and integral calculus as well as the theory of real and complex functions. An extensive appendix introduces information theory. 1970
Encyclopedia of Algebraic Thinking A searchable database of students' algebraic thinking and misconceptions Over 800 studies research and discuss students' struggles in algebra and potential strategies to help students overcome conceptual obstacles. Here we distill those articles into encyclopedia entries for teachers to use. This Encyclopedia is part of the Center for Algebraic Thinking. Here you will find 64 entries based on five domains and linked to the Common Core State Standards (CCSS). Each entry in the encyclopedia contains answers to the below questions when available from the research. You can search by the five domains below, Common Core State Standard, Learning Progression, or by keyword. An advanced search allows you to search the entire Center site, including the Encyclopedia, Formative Assessments, and Technologies. At the end of each entry is an opportunity for you to include your experience with students' thinking on that topic. Please add your experience! DOMAINS 1) Variables and Expressions 2) Algebraic Relations (Equations and Inequalities) 3) Analysis of Change (Graphing) 4) Patterns and Functions 5) Modeling (Word Problems) GUIDING QUESTIONS FOR ENTRIES: 1. What Common Core State Standard(s) does this research address? 2. What is the symbolic representation of thinking with the idea? (What does it look like?) 3. How do students think about the algebraic idea? (What does it sound like?) 4. What are the underlying mathematical issues involved? 5. What research-based strategies/tools could a teacher use to help students understand? You need to log in with an account if you want to change the pages or add your experience. We encourage you to do so. Unfortunately, you need an account because spammers target open wikis. We don't do anything with your information. Thanks for understanding!
From differentiation to integration - solve problems with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or, worse yet, not know where to begin? Have no fear! This hands-on guide focuses on helping you solve the many types of calculus problems you encounter in a focused, step-by-step manner. With... more... Boost Your grades with this illustrated quick-study guide. You will use it from high school all the way to graduate school and beyond. Includes both Calculus I and II. Clear and concise explanations. Difficult concepts are explained in simple terms. Illustrated with graphs and diagrams. Search for the words or phrases. Access the guide anytime, anywhere... more... Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner,... more... Students can gain a thorough understanding of differential and integral calculus with this powerful study tool. They'll also find the related analytic geometry much easier. The clear review of algebra and geometry in this edition will make calculus easier for students who wish to strengthen their knowledge in these areas. Updated to meet the emphasis... more... In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation... more... This book project was initiated at The Tribute Workshop in Honour of Gunnar Sparr and the follow-up workshop Inequalities, Interpolation, Non-commutative, Analysis, Non-commutative Geometry and Applications INANGA08, held at the Centre for Mathematical Sciences, Lund University in May and November of 2008. The resulting book is dedicated in celebration
This lesson helps students further their understanding of linear functions by applying the material to a real-world example. The class will use data on an airline flight including travel time, ground speed, time... Based at the University of Plymouth, the Centre for Innovation in Mathematics Teaching has developed many instructional materials designed to help both novice and experienced math teachers. This particular area of the... This lesson from Illuminations asks students to solve a system of linear equations using a practical math problem. The lesson involves question for students; participants are asked to give a short presentation to the...
Basic Math Skills is an appealing full-color text for students and adults who need additional instruction on basic mathematical concepts and operations. Written at a third- to fourth-grade reading level, Basic Math Skills provides extensive examples and practice activities--all presented as real-life situations, so students see the relevance of math in their day-to-day lives. ...show lessEdition/Copyright: 03 Cover: Hardcover Publisher: American Guidance Service Published: 01/01/2003 International: No List Price: $71.00 Used Currently Sold Out New Currently Sold Out Rental $27.99 Due back 08/01/2014 Save $43.01 (61%) Free return shipping In stock 21-day satisfaction guarantee CDs or access codes may not be included Marketplace sellers starting at $14.95 More prices and sellers below. Additional Sellers for Basic Math Skills14.99 +$3.99 s/h Good ChattanoogaBookServices CHATTANOOGA, TN 2006-01-30 Hardcover Good *MULTIPLE COPIES AVAILABLE2003 COPYRIGHT USED GOOD CONDITION-HAS SOME COVER WEAR-MAY BE SHOWING SLIGHT CARDBOARD ON CORNERS-MAY HAVE SOME WRITING OR HIGHLIGHTING-GREA...show moreT USABLE BOOKDELIVERY CONFIRMATION ON EVERY ORDER! * ...show less $17.01 +$3.99 s/h VeryGood Georgia Book Company Chickamauga, GA Hardcover Very Good 0785429522 MULTIPLE COPIES AVAILABLE 2003 copyright USED VERY GOOD CONDITION MAY HAVE SLIGHT CORNER WEAR-MAY HAVE MINIMAL WRITING-EXPERIENCE AMAZING CUSTOMER SERVICE-WE SHIP DAI...show moreLY. ...show less Hardcover New 0785429522 New book with very minor shelf wear. Never used. Nice gift. Best buy. Shipped promptly and packaged carefully. $38.95 +$3.99 s/h Good One Stop Text Books Store Sherman Oaks, CA 2006-01-30 Hardcover Good Expedited shipping is available for this item! $41.39 +$3.99 s/h New TextbookBarn Woodland Hills, CA 0785429522
California Mathematics enables students understand large numbers and addition, subtraction, multiplication, and division of whole numbers. Also describe and compare simple fractions and decimal, understand the properties of, and the relationships between, plane geometric figures and to collect, represent, and analyze data to answer questions. Prentice Hall Science Explorer, the nation's leading middle school science program, is the perfect fit for today's classroom. The book provides students with more options so they can deliver lessons aligned to their standards and preferences. This high-interest, relevant program teaches students the reading, writing, listening, and speaking skills they need to achieve success at school and in the workplace. Realistic work scenarios demonstrate the importance of communication skills in the book. This textbook is organized to support students learning of the California Science Content Standards, understanding this organization can help students master the standards provided in the book. Every chapter begins with a Focus on the Big Idea question that is linked to a California Science Standard. Focus on the Big Idea poses a question for students to think about as they study the chapters. They will discover the answers to the question as they readThe updated Magruder's American Government meets the changing needs of today's high school students and teachers. The program's engaging narrative is enhanced with numerous primary sources, political cartoons, charts, graphs, and photos, making the structure and principles of government accessible and motivating to students of all abilitiesStudents of grades 10 and 11 will take the test called the Texas Assessment of Knowledge and Skills (TAKS) in science. The objectives are based on the learning requirements listed in the Texas Essential Knowledge and Skills (TEKS). Students can take steps to improve their performance and increase the chances of doing well on the test, and this biology textbook will help them take these steps. Each chapter has many features that prepare for the TAKS, followed by an overview of the Biology TEKS. Oklahoma's Priority Academic Student Skills (PASS) objectives list the inquiry skills and science content you are expected to learn in biology. The goal of the PASS standards is for all students to have scientific literacy. Prentice Hall Literature Missouri edition is divided into four strands: Reading; Writing; Listening and Speaking; and Information Literacy. The MAP Reading Skills Review in the book provides six short tests with questions like the ones students will find on the MAP. Prentice Hall Mathematics Course 2: A structured approach to a variety of topics such as ratios, percents, equations, inequalities, geometry, graphing and probability.Test Taking Strategies provide a guide to problem solving strategies that are necessary for success on standardized tests. Checkpoint Quizzes assess student understanding after every few lessons. Daily Guided Problem Solving in the text is supported by the Guided Problem Solving worksheet expanding the problem, guiding the student through the problem solving process and providing extra practice
NCR 9583 CBEST Math Workshop Description: This course is designed for students seeking math review. It covers all of the arithmetic, algebra, measurement, and geometry concepts and skills covered in the CBEST exam. Test-taking strategies, techniques for solving word problems, and alternative problem-solving methods also are discussed.
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Intermediate Algebra 9780321233868 ISBN: 0321233867 Edition: 7 Pub Date: 2006 Publisher: Addison-Wesley Summary: The goal of Intermediate Algebra: Concepts and Applications, 7e is to help today's students learn and retain mathematical concepts by preparing them for the transition from "skills-oriented" intermediate algebra courses to more "concept-oriented" college-level mathematics courses, as well as to make the transition from "skill" to "application." This edition continues to bring your students a best-selling text that in...corporates the five-step problem-solving process Bittinger, Marvin A. is the author of Intermediate Algebra, published 2006 under ISBN 9780321233868 and 0321233867. One hundred forty three Intermediate Algebra textbooks are available for sale on ValoreBooks.com, one hundred forty two used from the cheapest price of $0.75, or buy new starting at $34
Pages to are hidden for "Computer Lab #1 Introduction to Matlab" Please download to view full document 2680414330086 Computer Lab #1: Introduction to Matlab M161, Fall 2006 Section 1 Introduction: In M160 calculators were used when technology was required. The calculators are great tools---all the way from the TI83 that will evaluate certain functions, plot and integrate numerically to the TI89 (or the TI92 or Voyager) that will perform just about every calculation that you learned to do in M160. The good news about the calculators is that they are portable and the bad news is that they have bad keyboards and screens. There are three different (at least three) computer algebra systems (CAS"s) for PC"s and Mac"s that will do all of the things that the calculators will do and more. Matlab, Mathematica and Maple are complete programming environments. The packages will perform both numerical and symbolic operations. These packages are apt to be the packages that you will use someday on the job. We feel that it is imperative that you learn something about the capabilities of the CAS"s. We have chosen to use Matlab in M161 because Matlab is used in a variety of upper division mathematics and engineering classes. We will emphasize the symbolic capability of Matlab but will use some of the numerical capabilities when it is appropriate. Laboratory reports will be required for the labs. We will give you information about writing the reports at the end of this lab. Purpose: To introduce the student to some basic Matlab facilities. Symbolic expressions and plotting are emphasized. The student should read through the lab and follow along with Matlab. After the lab is completed, the student can complete the exercises given at the end of the lab. Commands Reviewed: sym, pi, format, ezplot, axis, hold, subs, vpa, limit, diff, int Symbolic Expressions One of the capabilities of Matlab is that it can perform abstract operations on expressions. Usually these expressions are formulas---just like the formulas you have used in algebra and calculus---involving unknown variables like x, y, and z. Open up a Matlab session by clicking on the Matlab icon and type the following commands in the command window (where the cursor >> appears). Try 1/4+1/6 and then compare this to sym(1/4+1/6) The first command performed a numerical evaluation while the second performed a symbolic evaluation. When the first operation was done, the computer stored all the entries in decimal form (essentially forgetting that the numbers were fractions) and performed the operation of addition on these approximations. For the second command, the computer understood that these fractions were not to be stored in decimal form. The computer stores the numbers as fractions and operated on these numbers as fractions. Notice that for the second command, matlab even reduced its final answer from 10/24 to 5/12. Both the numerical and symbolic evaluation of these expressions is useful. It is nice to have both evaluations at our disposal. Let"s consider another example of the distinction between numerical and symbolic evaluations in matlab. If you type pi The computer returns the numerical value of  with the number of digits of accuracy, which you have chosen. You can change the displayed accuracy with the command format long (or format short) Now try typing pi again. Matlab interprets pi as a real number which has either four or fourteen places after the decimal point (depending on which format you are using). This is an approximation and is clear when we try cos(pi/2) cos(pi/6) The answer to be the first command should in fact be 0. Now Matlab"s given answer is terribly small and for all practical purposes 0. It shows that the evaluation of cos was performed numerically. To tell Matlab to think of pi as the object  (half of the circumference of a circle of radius 1), we need to make pi symbolic. Try the following commands spi=sym("pi") cos(spi/2) cos(spi/6) Notice that although here we have Matlab perform the same calculations as we did earlier, we now get the exact answers. Let"s examine one more example of Matlab"s ability to manipulate symbolic expressions. Recall from class that we have the cancellation equations for sin and arcsin x=arcsin(sin(x)) for x  [-,] x=sin(arcsin(x)) for x  [-1,1] Let"s verify if these identities really work in Matlab. Using the well known fact that sin(/4) = 2/2 we will evaluate the right hand side of the identities given above sin(asin(1/22^(1/2))) asin(sin(pi/4)) It works but the final answer is not in the same form as the original input. If we want Matlab to give the exact answers, not just a numerical value, we need to make both terms symbolic as follows. spi=sym("pi") sinpiover4=sin(spi/4) Then we get sin(asin(sinpiover4)) asin(sin(spi/4)) By telling Matlab that an object is symbolic, it performs the operations using basic operations and identities, instead of doing the computation numerically. We briefly mention that in this last set of commands, we could have skipped the line spi=sym("pi") since we had typed it earlier. Also, the variable sinpiover4 is symbolic because when we take the sine of a symbolic variable spi/4, the result will be symbolic. Graphing Symbolic Expressions Matlab has a variety of plotting capabilities and commands. In this course, all of your plotting will be done with the ezplot command. Let"s consider the plot of arcsin. The first method is to try ezplot("asin") or ezplot("asin(x)") In this form, we can also choose the axis, the previous command had a default range of [-1,1]. Try ezplot("asin(x)",[-1,0]) If we make x into a symbolic variable then the apostrophes in the previous commands can be omitted. x=sym("x") ezplot(asin(x)) The y-axis for the graph can also be changed with the axis command. To change the axis on the last graph it suffices to type axis([-1,1,-2,2]) The first pair defines the range on the x-axis and the second pair defines the range on the y-axis. For example, if you wanted to change the range on the x-axis to [0,1] and the y-axis to [-3,3], you would write axis([0,1,-3,3]) or you could just write ezplot(asin(x),[-1,1,-3,3]) Being able to control the axes is very useful since often graphs won"t look very nice the first time you graph them. For example, let"s plot two curves one on top of each other f= x^3-x^2-9x+9 ezplot(f) hold on ezplot(f+1) shows two curves almost on top of each other but if we rescale the graph with axis([-4,4,-30,20]) the plots look much better. Let us explain some of the syntax that we used above. To save some typing, the symbolic expression x^3-x^2-9x+9 was given a name, here f. Then f was plotted. The hold on command told Matlab that all future curves would be plotted on the same graph (until we negate the hold on command with a hold off command). The final ezplot command graphed f+1, namely x^3-x^2-9x+10. If you want to make another plot and not have it add the new plot to the previous, you use the command hold off. Then the now plot will a new window. It should be clear that the above notation is similar to the common function notation of mathematics. If we wanted to evaluate f at x=2, we would like to write f(2). Try it---it doesn't work. To evaluate the function f at x=2 in Matlab, we must use the subs command. It's not pretty but it works. We write subs(f,x,2) This command should be read as "substitute in f for x, 2." Calculus Operations on Symbolic Expressions The most important operations in calculus can be performed symbolically in Matlab. Among these we review computing limits, derivatives, and integrals. If it is at all convenient, we encourage you to double check your homework solutions with Matlab. To compute the limit as x approaches 3 of sinh(x), type limit(sinh(x),x,3) Notice that this was computed symbolically. To obtain a numerical answer try vpa(limit(sinh(x),x,3)) In general, it is best to begin by defining a function f and then to perform your operations directly on f. For example: f=sin(x)/x vpa(limit(f,x,0)) or when computing the first and third derivatives diff(f) diff(f,3) or even indefinite and definite integrals g=1/(1+4*x^2); int(g) int(g,-pi/2,pi) If we do a calculation and we might want to use the result again, we can give our result a name such as fp=diff(f) Then if we wanted to compute the second and the third derivative, we could type fpp=diff(fp) fppp=diff(fpp) Or if we wanted to evaluate f '(2), we could write fpo2=subs(fp,x,2) We could play the same game with the integral by writing intg=int(g) and check our answer by writing intgp=diff(intg) And finally, we suggest that you type help limit help diff help int to see all the options for these commands. If you take some time to look at the "help" facility, you will find that Matlab has a very good and useful help. Often, you may find it more useful to click on the question mark on the tool bar at the top of the window. Then click on "Index" and type the command you would like to look up. Once you find the command that you want to use, it is usually enough skim to the bottom to see the examples that are included to find what you want to do. It should be made clear that we have just touched on some of the capabilities of Matlab---some of the useful capabilities and some of the calculus capabilities. The symbolic part of Matlab can perform the full range of algebraic manipulations, such as solving equations, simplifying equations, expanding equations, etc. Homework Give your answers in the form of a clearly written report. You should begin with some sort of Introduction or Purpose of the lab. You should end with some sort of Summary. You should answer the questions with complete sentences. Do not list the Matlab commands you used unless it relevant to your answer. Include the Matlab graphs into your text (not at the end in some obscure appendix) by exporting the picture as a jpeg file and then pasting it into your word processor. You will be graded on the presentation. 1) Make x into a symbolic variable and study the expressions sin(asin(x)) asin(sin(x)) Does Matlab give correct results? Does Matlab give the results that you would have guessed it should give---given that you have studied these equations in the book? Why do you think Matlab gives the answers that they give? 2) Plot the relation "tan(y)=x" using ezplot over the domain [-10,10,-2,2]. Then use ezplot to plot "atan" on the same range. Compare and contrast these two plots. lim cos(x) lim cosh(x) 3) Compare the limits x3 and x3 . Graph both cos(x) and cosh(x). Does there seem to be any connection between these two functions? sin(h  x)  sin( x) lim 4) Compute the limit h0 h and interpret the result. 5) Compute the following derivatives dg x2  3 (0) g ( x)  3 (a) dx for x 2 2 d g (0) for g ( x)  e cos(x  1) 2 x2 2 (b) dx 6) Compute the following integrals 3 x e 4 3x dx (a) 0 1 x 2 sin(4 x)dx (b) 1 I hope you realize that you have been given just a small introduction to some of the things that Matlab can do for you. We next try to give you some information about writing the report. We told you earlier what we want you to put in the report. In general use your imagination but write a nice report. One tool within Word that might help you is the Symbol window. By clicking Insert and then Symbol, you get a window that will allow you to insert an assortment of symbols in your Word document. If you do not have any of the equation packages, it's about the only way that you can easily include mathematics in your document. Another very important tool that you need is how to import your plots into your Word document. This can be a very useful skill for other classes, other than M161. When you use ezplot to plot a function, your plot comes up in a Figure window. Before you do anything with it, you might experiment with the tools in this window that will allow you to edit your plot. You can insert text, arrows and lines. The all can be very useful. When you have the plot that you like, you click on File and then on Export. An export window will pop up. In the File Name slot, you see that the default file type is .emf. I don't know what the .emf file type is good for, but I'm pretty sure you don't want it here. In the slot below, labeled "Save as Type", it tells you what the .emf is a provides you with a down arrow to give you some choices for the file type. Click on the arrow and choose .jpg (read jpeg). Then name the file and save it, preferably on a floppy that you brought with you. When it is time to insert your plot in your Word document, you click on Insert, and File. Word will give you"re a browsing window that will let you find your file. You click on the file that you want inserted and it appears in your Word document where your cursor was situated. And finally, you should understand that by clicking on the plot in your Word document, Word will allow you to move the plot around and to resize it. Good luck
Elementary Statistics - With CD (High School) - 2nd edition Summary: For algebra-based Introductory Statistics courses. Elementary Statistics teams the proven authorship and pedagogical expertise of Larson with Farber's 30 years of statistics-teaching experience. It will appeal to today's visually oriented and more technologically savvy students. Highlights Graphical Approach that incorporates the graphical display of data throughout. Flexible tec...show morehnology--Introduces each new technique with hand calculations before a worked-out Technology Example is presented. More than 1,700 exercises--Includes a wide variety in each section that moves from basic concepts and skill development to more challenging problems. Titled examples paired with unique Try It Yourself problems--Illustrates every concept in the text with step-by-step examples numbered and titled for easy reference, Immediately followed with a similar problem. "Real Statistics, Real Decisions" challenges students to make decisions about which techniques to use3.99 +$3.99 s/h Acceptable Borgasorus Books, Inc. MO Wentzville, MO Hardcover Fair 01304888526.05 32E $6.35 32E $6.49 +$3.99 s/h Acceptable AlphaBookWorks Alpharetta, GA 0130488852
2003 Paperback New Book New and in stock. 4/11John Bird's approach to mathematics, based on numerous worked examples and interactive problems, is ideal for level 2 and 3 vocational courses including the BTEC National specifications. Theory is kept to a minimum, with the emphasis firmly placed on problem-solving skills, making this a thoroughly practical introduction to the mathematics that students need to master. Now in its seventh edition, Engineering Mathematics has helped thousands of students to succeed in their exams. The new edition includes a section at the start of each chapter that explains and how it relates to real life engineering projects. It is supported by a fully updated companion website with resources for both students and lecturers. It has 1000 worked problems, 238 multiple-choice questions, and full solutions to all 1800 further questions contained in the 237 practice exercises. All 525 illustrations used in the text can also be downloaded for use in the classroom. Audience: Students following vocational engineering courses / first year undergraduates. Suitable for all Level 3 engineering programmes, and core units at Level 3. Matched to New BTEC National specifications: Mathematics for Technicians; Further Mathematics for Technicians; AVCE: Applied Mathematics for Engineering; Further Mathematics for Engineering. Related Subjects Meet the Author John Bird is the former Head of Applied Electronics in the Faculty of Technology at Highbury College, Portsmouth, UK. More recently, he has combined freelance lecturing at the University of Portsmouth, with examiner responsibilities for Advanced Mathematics with City and Guilds, and examining for the International Baccalaureate Organisation. He is the author of over 125 textbooks on engineering and mathematical subjects, with worldwide sales of one million copies. He is currently a Senior Training Provider at the Defence School of Marine Engineering in the Defence College of Technical Training at HMS Sultan, Gosport, Hampshire
Larson's market-leading text, PRECALCULUS is known for delivering sound, consistently structured explanations and exercises of mathematical concepts to expertly prepare students for the study of calculus. With the ninth edition, the author continues to revolutionize the way students learn material by incorporating more real-world applications, ongoing review, and innovative technology. How Do You See It? exercises give students practice applying the concepts, and new Summarize features, Checkpoint problems, and a Companion Website reinforce understanding of the skill sets to help students better prepare for tests.
Reflecting the latest New York State curriculum change, this brand-new addition to Barron's Let's Review series covers all topics prescribed by the New York State Board of Regents for the new Integrated Algebra Regents exam, which replaces the Math A Regents exam. This book stresses rapid learning, using many step-by-step demonstration examples, helpful diagrams, enlightening "Math Fact" summaries, and graphing calculator approaches. Fourteen chapters review the following topics: sets, operations, and algebraic language; linear equations and formulas; problem solving and technology; ratios, rates, and proportions; polynomials and factoring; rational expressions and equations; radicals and right triangles; area and volume; linear equations and graphing; functions, graphs, and models; systems of linear equations and inequalities; quadratic and exponential functions; statistics and visual representations of data; and counting and probability of compound events. Exercise sections within each chapter feature a large sampling of Regents-type multiple-choice and extended response questions, with answers at the back of the book. Students will find this book helpful when they need additional explanation and practice on a troublesome topic, or when they want to review specific topics before taking a classroom test or the Regents exam. Teachers will value it as a lesson-planning aid, and as a source of classroom exercises, homework problems, and test questions.
Saxon Algebra 1 Program Algebra 1 is made up of five instructional components: Introduction of the New Increment, Examples with Complete Solutions, Practice of the Increment, Daily Problem Sets, and Cumulative Tests. Algebra 1 covers topics typically treated in a first-year algebra course. Algebra 1 includes such topics as: arithmetic and evaluation of expressions involving signed numbers, exponents, and roots solution of quadratic equations by factoring, completing the square, and quadratic formula direct and inverse variation exponential growth computation of the perimeter and area of two-dimensional regions computation of the surface area and volume of a wide variety of geometric solids statistics probability Algebra 1 Homeschool Kit Contents Saxon's secondary mathematics programs have been carefully planned and packaged. All you need to successfully teach each program is included in the homeschool kit. The Algebra 1 homeschool kit includes a textbook, an answer key for textbook problem sets, and tests with solutions/answers. Solutions Manual The Solutions Manual provides step-by-step solutions for each problem in the textbook problem sets. Solutions to test problems included with answer key.
2734152 / ISBN-13: 9780962734151 Doing Simple Math in Your Head Almost all adults suffer a little math anxiety, especially when it comes to everyday problems they think they should be able to figure out in their ...Show synopsisAlmost all adults suffer a little math anxiety, especially when it comes to everyday problems they think they should be able to figure out in their heads. Want to figure the six percent sales tax on a $34.50 item? A 15 percent tip for a $13.75 check? The carpeting needed for a 121/2-by-17-foot room? No one learns how to do these mental calculations in school, where the emphasis is on paper-and-pencil techniques. With no math background required and no long list of rules to memorize, this book teaches average adults how to simplify their math problems, provides ample real-life practice problems and solutions, and gives grown-ups the necessary background in basic arithmetic to handle everyday problems quickly.Hide synopsis Easy-to-learn techniques for solving math problems quickly without pencil and paper or calculator. A Book- of-the-Month Club Selection.Easy-to-learn techniques for solving math problems quickly without pencil and paper or calculator. A Book- of-the-Month Club Selection962734152 Book is lightly used with little or no...Very Good. 0962734152
The first edition has gained a strong following among instructors who find Kaseberg's use of guided discovery and problem solving facilitates learning new concepts and strengthens skill retention. In the second edition, Kaseberg's informal, interactive style makes algebra accessible to students while maintaining rigorous mathematical accuracy. To reduce preparation time for course leaders, and facilitate use by adjuncts, the new Instructor's Resource Manual accompanies the second edition. The manual provides a structured lesson and group-activity worksheet for each section in the textbook; incorporates materials from the textbook with supplemental projects and activities; suggests core homework assignments; and furnishes guided-discussion questions to use in recitations. This resource serves to bridge the gap between traditional pedagogy and a reform approach. [via] Alice Kaseberg's respected Intermediate Algebra: Everyday Explorations, Fourth Edition, helps students build confidence in algebra. This text's popularity is attributable to the author's use of guided discovery, explorations, and problem solving, all of which help students learn new concepts and strengthen their skill retention. Known for an informal, interactive style that makes algebra more accessible to students while maintaining a high level of mathematical accuracy, Intermediate Algebra includes a host of teaching and learning tools that work together for maximum flexibility and a high student success rate. With the Fourth Edition, instructors have access to an Instructor's Annotated Edition that provides additional examples, as well as a robust Instructor's Resource Manual, algorithmic computerized testing, and an extensive online homework system-discussion questions. This resource serves to bridge the gap between traditional pedagogy and a reform approach. [via] If you've ever wondered, "What is algebra good for?" Alice Kaseberg will help you answer this age-old question with her respected text. INTRODUCTORY ALBEGRA, FOURTH EDITION, uses guided discovery, explorations, and problem solving to help you learn new concepts and strengthen the retention of new skills. Known for an informal, interactive style that makes algebra more accessible while maintaining mathematical accuracy, INTRODUCTORY ALGEBRA: EVERYDAY EXPLORATIONS, FOURTH EDITION, includes a host of learning tools that work together to help you succeed. A robust website and Enhanced WebAssign support you with practice problems, end-of-chapter problems that incorporate figures and examples, and quizzes that provide immediate feedback on your progress discussion questions. This resource serves to bridge the gap between traditional pedagogy and a reform approach. [via]
Introductory Algebra - 4th edition Summary: The latest book from Cengage Learning on Introductory Algebra Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's INTRODUCTORY ALGEBRA, Fourth Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math users, algebra is like a foreign language. They have difficulty translating the words, their meanings, and how they apply to problem solving. Emphasizing the ''language of algebra,'' the b...show moreook's fully integrated learning process is designed to expand users' reasoning abilities and teach them how to read, write, and think mathematically. It blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology47871 Item in good condition. Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!! $6.796882010 Paperback Good Domestic Media Mail shipping ONLY. 100% of this purchase will support literacy programs through a nonprofit organization! $2529.98 +$3.99 s/h Good One Stop Text Books Store Sherman Oaks, CA 2010-01-01 Paperback Good Expedited shipping is available for this item! $29.98 +$3.99 s/h Good One Stop Text Books Store Sherman Oaks
Section 01H This is a traditional lecture section that will cover Unit 6 only. Computer access will still be required for homework and quizzes. Class dates: August 21 through September 18, 2013 The student will learn to perform operations on exponential expressions and polynomials. Students will also learn techniques to factor polynomials and use these techniques to solve polynomial equations. Emphasis should be on learning all the different factoring methods, and solving application problems using polynomial equations. Credit is not applicable toward graduation. Prerequisite(s): MTE 5
Main menu Study Tools math.com: has good overview explanations for pre-algebra through calculus concepts and terms. It also has a section that gives study advice as well as strategies to overcome test anxiety. However, there is no step by step help available on this site. mathway.com: is easily navigated and has several options for what a student might be looking for from a given equation (solve for x, solve for y, find the 0, etc). The site costs $19.99/month. For no fee, it does show the answer to a given equation and it can also generate sample worksheets for any subject type (like arithmetic operations).
3383090 / ISBN-13: 9780073383095 Discrete Mathematics and Its Applications The goal of this text is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, ...Show synopsisThe goal of this text is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, expansive discussion, and detailed exercise sets. These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students. The fifth edition includes a more thorough and linear presentation of logic, proof types and proof writing, and mathematical reasoning. This enhanced coverage will provide students with a solid understanding of the material as it relates to their immediate field of study and other relevant subjects. The inclusion of applications and examples to key topics has been significantly addressed to add clarity to every subject. True to the fourth edition, the text specific web site supplements the subject matter in meaningful ways, offering additional material for students and instructors. Discrete math is an active subject with new discoveries made every year. The continual growth and updates to the web site reflect the active nature of the topics being discussed. This text is designed for a one- or two-term introductory discrete mathematics course to be taken by students in a wide variety of majors, including computer science, mathematics, and engineering. College Algebra is the only explicit prerequisite.Hide synopsis ...Show more!from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields3383090 Brand New International Edition. Guaranteed...New. 0073383090 Brand New International Edition. Guaranteed Super Fast Delivery. Same Contents as US Editions. Cover & ISBN of the book could be different from US EditionBeware of international editions. The one my son received did not have the same questions in it as the american one and therefore we had to buy the american edition as his instructor required these questions be anwsered as part of his assignment
Rosemead SATAs it turns out, this same chapter of math is essential when one constructs a numerical representation of a differential operator, or a system of partial differential equations. Linear Algebra is not only essential in defining numerical differentiation operators, but also it is a crucial tool i... ...When doing long term, repeat visits, I assign tasks or extra problems to help keep the student thinking. If the student is young I try to establish a relationship with the parents, or communicate with the teacher, so I can assist with these good study habits I mentioned, and track the students p...
Let's Review: Geometry (Barron's Review Course) Book Description: This classroom text presents a detailed review of all topics prescribed as part of the high school curriculum. Separate chapters analyze and explain: the language of geometry; parallel lines and polygons; congruent triangles and inequalities; special quadrilaterals and coordinates; similarity (including ratio and proportion, and proving products equal); right triangles and trigonometry; circles and angle measurement; transformation geometry; locus and coordinates; and working in space (an introduction to solid geometry). Each chapter includes practice exercises with answers provided at the back of the book
The math level needed involves basic algebra formulas and arithmetic used to solve appropriate word problems. Topics include: the properties, classification and structure of matter, the use of the periodic table and the laws and facts for using it as a reference, chemical bonding, chemical react... ...To me problem solving is like to solve a puzzle. You list the information you know and use variables for unknown information. Then you find the connection between them to form one or more equations.
0132757451 9780132757454 Introductory Algebra for College Students:For a one-semester undergraduate introductory algebra course. The goal of this text is to provide students with a strong foundation in Basic Algebra skills; to develop students' critical thinking and problem-solving capabilities and prepare students for Intermediate Algebra and some "service" math courses. Topics are presented in an interesting and inviting format incorporating real world sourced data modeling. Back to top Rent Introductory Algebra for College Students 2nd edition today, or search our site for Robert textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Prentice Hall.
Many of the documents for this module will be available on the web in two formats: ps (postscript) or pdf (portable document format). Which format is easiest to view will probably depend on the computer that you are using. Corrections, room changes etc. will be announced on the Student Portal Course Homepage for G1BCOF. To find this, log in to choose the My Teaching tab, and click on the View your modules button. You should check this page regularly for announcements and to read the module Message Board (see below). Module Message Board: The module message board is available from the Student Portal Course Homepage for G1BCOF. To find this, log in to choose the My Teaching tab, and click on the View your modules button. You should regularly check the Message Board topic Discussion, questions and answers. Feel free to post questions, answers and suggestions concerning the module there. I will keep an eye on this, and may well contribute my own answers if appropriate. Answers to Frequently Asked Questions appear in the G1BCOF FAQ document, available from the module web page at Summary of content: This module provides an introduction to the theory and applications of functions of a complex variable, using an approach oriented towards methods and applications. The elegant theory of complex functions is developed and then used to evaluate certain real integrals. Topics to be covered will include: analytic functions and singularities; series expansions; contour integrals and the calculation of residues; applications of contour integration. Prerequisites: Knowledge of elementary analysis, calculus of real functions, and complex numbers, as provided by the modules G11CAL, G11ACF and G11LMA. Corequisites: None Module aims: This module forms part of both the Pure Mathematics strand and one of the Applied Mathematics strands. The theory of functions of a complex variable is very important for applications as well as leading to more advanced study in the level 4 module G14COA. Learning outcomes: A student who completes this module successfully will develop a variety of intellectual, professional and transferable skills. Such a student should also gain the knowledge and understanding to be able to: identify analytic functions and their singularities; calculate Taylor and Laurent series; calculate residues of functions and compute contour integrals; evaluate real definite integrals using residues. Assessment: Assessment will be entirely by one 2-hour written examination. There will be five questions, and your best four answers will count. If you answer four of the questions perfectly, then you will obtain full marks. Please note that no calculators will be permitted in the examination. Should a resit examination be required in August/September it will take the same form as above (five questions, best four answers count, no calculators). You may find it useful to look at the Spring 2003-4 exam paper for this module. (See below for solutions to that exam paper and comments on the students' answers to it.) You may also find it useful to look at past exam papers for Professor Langley's module G12CAN Complex Analysis: solutions to the 2002/2003 paper are available from the short loan section of the George Green Library. Note, however, that the G12CAN examinations had a different rubric. Coursework (problems/exercises) Coursework is due in at the end of the Friday lecture in weeks 2, 4, 6, 8 and 10. It does not form part of the assessment, but should give you useful feedback, and its completion is strongly advised in order to master the techniques of the module. If you have any queries about the marking of your work you should see Dr Feinstein. Problem classes take place in weeks 3, 5, 7, 9 and 11 and there will also be a special examples class in teaching week 2. Solutions to all of these questions will be made available on the web as the module progresses. Coursework and problem class questions: ps, pdf These questions were originally compiled by Professor Langley for his module G12CAN Complex Analysis 2002-3. Dr Feinstein has made very minor alterations to the questions and solutions.
...Show more facilitate students understanding. "Progress Checks" follow most examples and enable students to assess their comprehension. Exceptional explanation in Chapter 2 (Sections 2.2 and 2.4) on how to solve word problems help students master this important skill. "Warnings" reinforce good mathematical habits by pointing out incorrect practices most commonly found in homework and on exams
"A good textbook." ― Mathematical Gazette. This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. Each chapter begins with an optional commentary on the history of geometry. Contents in... read more Customers who bought this book also bought: Our Editors also recommend: The Gentle Art of Mathematics by Dan Pedoe This lighthearted work uses a variety of practical applications and puzzles to take a look at today's mathematical trends. In nine chapters, Professor Pedoe covers mathematical games, chance and choice, automatic thinking, and more. Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene F. Krause Fascinating, accessible introduction to unusual mathematical system in which distance is not measured by straight lines. Illustrated topics include applications to urban geography and comparisons to Euclidean geometry. Selected answers to problemsProblems and Solutions in Euclidean Geometry by M. N. Aref, William Wernick Based on classical principles, this book is intended for a second course in Euclidean geometry and can be used as a refresher. More than 200 problems include hints and solutions. 1968 editionProduct Description: "A good textbook." ― Mathematical Gazette. This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. Each chapter begins with an optional commentary on the history of geometry. Contents include modern elementary geometry, isometries and similarities in the plane, vectors and complex numbers in geometry, inversion, and isometries in space. Numerous exercises appear throughout the text, many of which have corresponding answers and hints at the back of the book. Prerequisites for this text, which is suitable for undergraduate courses, include high school algebra, geometry, and elementary trigonometry. 1972
Pages to are hidden for "Elements of Integral Calculus using SAGE" Please download to view full document 483780635008038 Elements of Integral Calculus using SAGE (preliminary version) Dale Hoffman, William Stein, David Joyner 4-2-2008 Contents vi Contents 0 Preface ix 1 The Integral 1 1.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Some applications of area . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Total Accumulation as "Area" . . . . . . . . . . . . . . . 8 1.2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Sigma notation and Riemann sums . . . . . . . . . . . . . . . . . 10 1.3.1 Sums of areas of rectangles . . . . . . . . . . . . . . . . . 12 1.3.2 Area under a curve Riemann sums . . . . . . . . . . . . . 14 1.3.3 Two special Riemann sums: lower and upper sums . . . . 19 1.3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 The definite integral . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.1 The Fundamental Theorem of Calculus . . . . . . . . . . 24 1.4.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.3 Properties of the definite integral . . . . . . . . . . . . . . 28 1.4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5 Areas, integrals, and anti-derivatives . . . . . . . . . . . . . . . . 32 1.5.1 Integrals, Antiderivatives, and Applications . . . . . . . . 34 1.6 Indefinite Integrals and Change . . . . . . . . . . . . . . . . . . . 35 1.6.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . 35 1.6.2 Physical Intuition . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Substitution and Symmetry . . . . . . . . . . . . . . . . . . . . . 38 1.7.1 The Substitution Rule . . . . . . . . . . . . . . . . . . . . 38 1.7.2 Changing the variable and definite integrals . . . . . . . . 41 1.7.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 Applications of the integral 45 2.0.5 Using integration to determine areas . . . . . . . . . . . . 45 vii Chapter 0 Preface This is a preface. ix Chapter 1 The Integral The subject of Differential Calculus starts with the "simple" geometrical idea of the slope of a tangent line to a curve, develops it into a combination of theory about derivatives and their properties, techniques for calculating derivatives, and applications of derivatives. This book begins the development of Integral Calculus and starts with the "simple" geometric idea of area. This idea will be developed into another combination of theory, techniques, and applications. The integral will be introduced in two (completely different) way: as a limit of "Riemann sums" and as an "inverse" of differentiation ("anti-derivative"). Conceptually, one is geometric, or numerical, and the other is somewhat more algebraic. One of the most important results in mathematics, The Fundamental Theo- rem of Calculus, appears in this chapter. It connects these two notions of the integral and also provides a relationship between differential and integral calcu- lus. Historically, this theorem marked the beginning of modern mathematics, and it provided important tools for the growth and development of the sciences. The chapter begins with a look at area, some geometric properties of areas, and some applications. First we will see ways of approximating the areas of regions such as tree leaves that are bounded by curved edges and the areas of regions bounded by graphs of functions. Then we will find ways to calculate the areas of some of these regions exactly. Finally, we will explore more of the rich variety of uses of "areas". The primary purpose of this introductory section is to help develop your intuition about areas and your ability to reason using geometric arguments about area. This type of reasoning will appear often in the rest of this book and is very helpful for applying the ideas of calculus. 1.1 Area We know from previous experience how to compute the areas of simple geomet- rical shapes, like triangles and circles and rectangles. Formulas for these have been known since the days of the ancient Greeks. But, how do you find the area 1 1.1. AREA under a "more complicated" curve, such as y = x2 , −1 < x < 1? First, let's graph it. For this, we can use SAGE as follows1 : SAGE sage: a = -1; b = 1 sage: f = lambda x: xˆ2 sage: Lb = [[b,f(b)],[b,0],[a,0],[a,f(a)]] sage: Lf = [[i/20,f(i/20)] for i in range(20a,20b+1)] sage: P = polygon(Lf+Lb,rgbcolor=(0.2,0.8,0)) sage: Q = plot(f(x),x,a-0.5,b+0.5) sage: show(P+Q) Here is the plot: Figure 1.1: Plot using SAGE of y = x2 . The rough, general idea introduced in this section is the following. To compute the area of the "complicated" shaded region in Figure 1.1, we break it up into lots of "simpler" subregions, whose areas are easy to compute, then add them up to get the total area. We shall return to this example later. The basic shape we will use is the rectangle; the area of a rectangle is (base)×(height). If the units for each side of the rectangle are "meters," then the area will have the units ("meters")×("meters") = "square meters" = m2 . The only other area formulas needed for this section are for triangles, area = bh/2, and for circles, area = πr2 . Three other familiar properties of area are assumed and will be used: • Addition Property: The total area of a region is the sum of the areas of the nonoverlapping pieces which comprise the region. (Figure 1.2) • Inclusion Property: If region B is on or inside region A, then the area of region B is less than or equal to the area of region A. (Figure 1.3) 1 Feel free to try this yourself, changing a, b and x2 to something else if you like. 2 1.1. AREA • Location-Independence Property: The area of a region does not depend on its location. (Figure 1.4) Figure 1.2: Addition of areas. Figure 1.3: Estimating areas using rectangles. Figure 1.4: Independence of area under translations and rotations. Example 1.1.1. Determine the area of the region in Figure 1.5(a). Solution: The region can easily be broken into two rectangles, Figure 1.5(b), with areas 35 square inches and 3 square inches respectively, so the area of the original region is 38 square inches. We can use the three properties of area to get information about areas that are difficult to calculate exactly. For instance, let A be the region bounded by the graph of f (x) = 1/x, the x–axis, and vertical lines at x = 1 and x = 3. 3 1.1. AREA Figure 1.5: Figure for Example 1.1.1. Since the two rectangles in Figure 1.6 are inside the region A and do not overlap each, the area of the rectangles, 1/2 + 1/3 = 5/6, is less than the area of region A. Figure 1.6: The area under of y = 1/x, 1 ≤ x ≤ 3. Practice 1.1.1. Build two rectangles, each with base 1 unit, outside the shaded region in Figure 1.6 and use their areas to make a valid statement about the area of region A. (Ans: Outside rectangular area = 1.5.) Practice 1.1.2. What can be said about the area of region A in Figure 1.6 if we use both inside and outside rectangles with base 1/2 unit? (Ans: The area of the region is between 0.95 and 1.2.) Example 1.1.2. In Figure 1.7, there are 32 dark squares, 1 centimeter on a side, and 31 lighter squares of the same size. We can be sure that the area of the leaf is smaller than what number? Solution: The area of the leaf is smaller than 32 + 31 = 63 cm2 . Practice 1.1.3. We can be sure that the area of the leaf is at least how large? 4 1.1. AREA Figure 1.7: The area of a "leaf". Functions can be defined in terms of areas. For the constant function f (t) = 2, define A(x) to be the area of the rectangular region bounded by the graph of f , the t-axis, and the vertical lines at t = 1 and t = x (Figure 1.8(a)). Then A(2) is the area of the shaded region in Figure 1.8(b), and A(2) = 2. Similarly, A(3) = 4 and A(4) = 6. In general, A(x) = (base)×(height) = (x − 1)(2) = 2x − 2, for any x ≥ 1. The graph of y = A(x) is shown in Figure 1.8(c), and A′ (x) = 2 for every value of x > 1. Figure 1.8: The area as a function. Sometimes it is useful to move regions around. The area of a parallelogram is obvious if we move the triangular region from one side of the parallelogram to fill the region on the other side and ending up with a rectangle (Figure 1.9). Figure 1.9: The area of a parallelogram. 5 1.2. SOME APPLICATIONS OF AREA At first glance, it is difficult to estimate the total area of the shaded regions in Figure 1.10(a). However, if we slide all of them into a single column (Figure 1.10(b)), then it is easy to determine that the shaded area is less than the area of the enclosing rectangle = (base)×(height) = (1)(2) = 2. Figure 1.10: An irregular area. 1.2 Some applications of area One reason "areas" are so useful is that they can represent quantities other than simple geometric shapes. For example, if the units of the base of a rectangle are "hours" and the units of the height are "miles/hour", then the units of the "area" of the rectangle are (hours)×(miles/hour) = miles, a measure of distance. Similarly, if the base units are centimeters and the height units are grams, then the "area" units are gram×centimeters, a measure of work. Example 1.2.1. Distance as an "area:" In Figure 1.11, f (t) is the velocity of a car in "miles per hour," and t is the time in "hours." Then the shaded "area" will be (base)×(height) = (3 hours)×(20 miles/hour ) = 60 miles, the distance traveled by the car in the 3 hours from 1 o'clock until 4 o'clock. Figure 1.11: Distance as "area". Here is the general statement of the idea illustrated in the example above. 6 1.2. SOME APPLICATIONS OF AREA Theorem 1.2.1. ("Area" as Distance) If f (t) is the (positive) forward velocity of an object at time t, then the "area" between the graph of f and the t-axis and the vertical lines at times t = a and t = b will be the distance that the object has moved forward between times a and b. This "area as distance" fact can make some difficult distance problems much easier. Example 1.2.2. A car starts from rest (velocity = 0) and steadily speeds up so that 20 seconds later it's speed is 88 feet per second (60 miles per hour). How far did the car travel during those 20 seconds? Solution: If "steadily speeds up" means that the velocity increases linearly, then the idea of "area as distance" is applicable. The "area" of the triangular region (Figure 1.12) represents the distance traveled, so distance = 1 (base) × (height) 2 1 = 2 (20 seconds) × (88 feet/second) = 880 feet. Figure 1.12: Distance a car travels as "area". Practice 1.2.1. A train traveling at 45 miles per hour (66 feet/second) takes 60 seconds to come to a complete stop. If the train slowed down at a steady rate (the velocity decreased linearly), how many feet did the train travel while coming to a stop? Practice 1.2.2. You and a friend start off at noon and walk in the same di- rection along the same path at the rates shown in Figure 1.13. • Who is walking faster at 2 pm? Who is ahead at 2 pm? • Who is walking faster at 3 pm? Who is ahead at 3 pm? • When will you and your friend be together? (Answer in words.) 7 1.2. SOME APPLICATIONS OF AREA Figure 1.13: Illustration for Practice 1.2.2. 1.2.1 Total Accumulation as "Area" In the previous examples, the function represented a rate of travel (miles per hour), and the area represented the total distance traveled. For functions rep- resenting other rates such as the production of a factory (bicycles per day), or the flow of water in a river (gallons per minute) or traffic over a bridge (cars per minute), or the spread of a disease (newly sick people per week), the area will still represent the total amount of something. Theorem 1.2.2. ("Area" as Total Accumulation) If f (t) represents a positive rate (in units per time interval) at time t, then the "area" between the graph of f and the t-axis and the vertical lines at times t = a and t = b will be the total units which accumulate between times a and b. Practice 1.2.3. Figure 1.14 shows the number of telephone calls made per hour on a Tuesday. Approximately how many calls were made between 9 am and 11 am? Figure 1.14: Illustration for Practice 1.2.3. 1.2.2 Problems 1. (a) Calculate the sum of the rectangular areas in Figure 1.15(a). (b) From part (a), what can we say about the area of the shaded region in Figure 1.15(b)? 8 1.2. SOME APPLICATIONS OF AREA 2. (a) Calculate the sum of the areas of the shaded regions in Figure 1.15(c). (b) From part (a), what can we say about the area of the shaded region in Figure 1.15(b)? Figure 1.15: Estimating areas. 3. Let A(x) represent the area bounded by the graph and the horizontal axis and vertical lines at t = 0 and t = x for the graph in Fig. 25. Evaluate A(x) for x = 1, 2, 3, 4, and 5. Figure 1.16: Computing areas. 4. Police chase: A speeder traveling 45 miles per hour (in a 25 mph zone) passes a stopped police car which immediately takes off after the speeder. If the police car speeds up steadily to 60 miles/hour in 20 seconds and then travels at a steady 60 miles/hour, how long and how far before the police car catches the speeder who continued traveling at 45 miles/hour? (Figure 1.17) Figure 1.17: Computing areas. 9 1.3. SIGMA NOTATION AND RIEMANN SUMS 5. What are the units for the "area" of a rectangle with the given base and height units? Base units Height units "Area" units miles per second seconds hours dollars per hour square feet feet kilowatts hours houses people per house meals meals 1.3 Sigma notation and Riemann sums One strategy for calculating the area of a region is to cut the region into simple shapes, calculate the area of each simple shape, and then add these smaller areas together to get the area of the whole region. We will use that approach, but it is useful to have a notation for adding a lot of values together: the sigma (Σ) notation. The function to the right of the sigma is called the summand, and the num- bers below and above the sigma are called the lower and upper limits of the summation. (Figure 1.18) Figure 1.18: Summation notation. 10 1.3. SIGMA NOTATION AND RIEMANN SUMS x f (x) g(x) h(x) 1 2 4 3 2 3 1 3 3 1 2 3 4 0 3 3 5 3 5 3 Figure 1.19: Table for Example 1.3.1. Summation A way to read Sigma notation the sigma notation notation 5 2 2 1 + 2 + 32 + 42 + 52 the sum of k squared k=1 k 2 for k equals 1 to k equals 5 1 1 1 1 1 7 3 + 4 + 5 + 6 + 7 the sum of 1 over k k=3 k −1 for k equals 3 to k equals 7 5 20 + 21 + 22 + 23 + 24 + 25 the sum of 2 to the j-th power j=0 2j for j equals 0 to j equals 5 7 a2 + a3 + a4 + a5 + a6 + a7 the sum of a sub i i=2 ai for i equals 2 to i equals 7 The variable (typically i, j, or k) used in the summation is called the counter or index variable. Practice 1.3.1. Write the summation denoted by each of the following: 5 7 j1 4 (a) k=1 k3 , (b) j=2 (−1) j , (c) m=0 (2m + 1). In practice, the sigma notation is frequently used with the standard function notation: 3 f (k + 2) = f (1 + 2) + f (2 + 2) + f (3 + 2) = f (3) + f (4) + f (5) k=1 and 4 f (xi ) = f (x1 ) + f (x2 ) + f (x3 ) + f (x4 ). k=1 5 Example 1.3.1. Use the values in Table 1.19 to evaluate k=2 2f (k) and 5 j=3 (5 + f (j − 2)). 5 Solution: k=2 2f (k) = 2f (2) + 2f (3) + 2f (4) + 2f (5) = 2(3) + 2(1) + 2(0) + 5 2(3) = 14. j=3 (5 + f (j − 2)) = (5 + f (32)) + (5 + f (42)) + (5 + f (52)) = (5 + f (1)) + (5 + f (2)) + (5 + f (3)) = (5 + 2) + (5 + 3) + (5 + 1) = 21. 11 1.3. SIGMA NOTATION AND RIEMANN SUMS Practice 1.3.2. Use the values of f , g and h in Table 1.19 to evaluate the following: 5 4 5 (a) g(k), (b) h(j), (c) [f (i − 1) + g(i)]. k=2 j=1 i=3 Since the sigma notation is simply a notation for addition, it has all of the familiar properties of addition. Theorem 1.3.1. (Summation Properties) n • Sum of Constants: k=1 C = C + C + C + · · · + C (n terms) = nC. n n n • Addition: k=1 (ak + bk ) = k=1 ak + k=1 bk . n n n • Subtraction: k=1 (ak − bk ) = k=1 ak − k=1 bk . n n • Constant Multiple: k=1 Cak = C k=1 ak . n n • Preserves positivity: if bk ≥ ak for all k then k=1 bk ≥ k=1 ak . In n particular, if ak ≥ 0 for all k then k=1 ak ≥ 0. m n n • Additivity of ranges: if 1 ≤ m ≤ n then k=1 ak + k=m+1 ak = k=1 ak . 1.3.1 Sums of areas of rectangles Later, we will approximate the areas under curves by building rectangles as high as the curve, calculating the area of each rectangle, and then adding the rectangular areas together. Example 1.3.2. Evaluate the sum of the rectangular areas in Figure 1.20, and write the sum using the sigma notation. Solution: We have sum of the rectangular areas = sum of (base) × (height) for each rectangle = (1)(1/3) + (1)(1/4) + (1)(1/5) = 47/60. Using the sigma notation, 3 1 (1)(1/3) + (1)(1/4) + (1)(1/5) = . k k=1 Practice 1.3.3. Evaluate the sum of the rectangular areas in Figure 1.21, and write the sum using the sigma notation. Example 1.3.3. Write the sum of the areas of the rectangles in Figure 1.22 using the sigma notation. Solution: The area of each rectangle is (base)×(height). 12 1.3. SIGMA NOTATION AND RIEMANN SUMS Figure 1.20: Area and summation notation. Figure 1.21: Area and summation notation. rectangle base height area 1 x1 − x0 f (x1 ) (x1 − x0 )f (x1 ) 2 x2 − x1 f (x2 ) (x2 − x1 )f (x2 ) 3 x3 − x2 f (x3 ) (x3 − x2 )f (x3 ) The area of the k-th rectangle is (xk − xk−1 )f (xk ), and the total area of the 3 rectangles is the sum k=1 (xk − xk−1 )f (xk ). Figure 1.22: Area and summation notation. 13 1.3. SIGMA NOTATION AND RIEMANN SUMS 1.3.2 Area under a curve Riemann sums Suppose we want to calculate the area between the graph of a positive function f and the interval [a, b] on the x–axis (Fig. 7). The Riemann Sum method is to build several rectangles with y = f (x) bases on the interval [a, b] and sides that reach up to the graph of f (Fig. 8). Then the areas of the rectangles can be calculated and added together to get a number called a Riemann sum of f on [a, b]. The area of the region formed by the rectangles is an approximation of the area we want. Example 1.3.4. Approximate the area in Figure 1.23(a) between the graph of f and the interval [2, 5] on the x–axis by summing the areas of the rectangles in Figure 1.23(b). Solution: The total area of rectangles is (2)(3) + (1)(5) = 11 square units. Figure 1.23: Illustration for Example 1.3.4. In order to effectively describe this process, some new vocabulary is helpful: a "partition" of an interval and the mesh of the partition. A partition P of a closed interval [a, b] into n subintervals is a set of n + 1 points {x0 = a, x1 , x2 , x3 , ..., xn−1 , xn = b} in increasing order, a = x0 < x1 < x2 < x3 < ... < xn = b. (A partition is a collection of points on the axis and it does not depend on the function in any way.) The points of the partition P divide the interval into n subintervals (Figure 1.24): [x0 , x1 ], [x1 , x2 ], [x2 , x3 ], . . . , and [xn−1 , xn ] with lengths ∆x1 = x1 − x0 , ∆x2 = x2 − x1 , . . . , ∆xn = xn − xn−1 . The points xk of the partition P are the locations of the vertical lines for the sides of the rectangles, and the bases of the rectangles have lengths ∆xk for k = 1, 2, 3, ..., n. The mesh or norm of the partition is the length of the longest of the subin- tervals [xk−1 , xk ], or, equivalently, the maximum of the ∆xk for k = 1, 2, 3, ..., n. For example, the set P = {2, 3, 4.6, 5.1, 6} is a partition of the interval [2, 6] and divides it into 4 subintervals with lengths ∆x1 = 1, ∆x2 = 1.6, ∆x3 = 0.5 and ∆x4 = 0.9. The mesh of this partition is 1.6, the maximum of the lengths of 14 1.3. SIGMA NOTATION AND RIEMANN SUMS Figure 1.24: Partition of the interval [a, b]. the subintervals. (If the mesh of a partition is "small," then the length of each one of the subintervals is the same or smaller.) A function, a partition, and a point in each subinterval determine a Rie- mann sum. Suppose f is a positive function on the interval [a, b], P = {x0 = a, x1 , x2 , x3 , ..., xn−1 , xn = b} is a partition of [a, b], and ck is an xvalue in the k- th subinterval [xk−1 , xk ] : xk−1 ≤ ck ≤ xk . Then the area of the k-th rectangle is f (ck ) · (xk − xk−1 ) = f (ck )∆xk . (Figure 1.25) Figure 1.25: Part of a Riemann sum. n Definition 1.3.1. A summation of the form k=1 f (ck )∆xk is called a Rie- mann sum of f for the partition P . This Riemann sum is the total of the areas of the rectangular regions and is an approximation of the area between the graph of f and the x–axis. Example 1.3.5. Find the Riemann sum for f (x) = 1/x and the partition {1, 4, 5} using the values c1 = 2 and c2 = 5. Solution: The two subintervals are [1, 4] and [4, 5] so ∆x1 = 3 and ∆x2 = 1. Then the Riemann sum for this partition is n 1 1 f (ck )∆xk = f (c1 )∆x1 + f (c2 )∆x2 = f (2)(3) + f (5)(1) = (3) + (1) = 1.7. 2 5 k=1 15 1.3. SIGMA NOTATION AND RIEMANN SUMS Practice 1.3.4. Calculate the Riemann sum for f (x) = 1/x on the partition {1, 4, 5} using the values c1 = 3, c2 = 4. Practice 1.3.5. What is the smallest value a Riemann sum for f (x) = 1/x and the partition {1, 4, 5} can have? (You will need to select values for c1 and c2 .) What is the largest value a Riemann sum can have for this function and partition? Here is a SAGE example. Example 1.3.6. Using SAGE, we construct the Riemann sum of the function y = x2 using a partition of 6 equally spaced points, where the ck 's are taken to be the midpoints. SAGE sage: f1(x) = xˆ2 sage: f = Piecewise([[(-1,1),f1]]) sage: g = f.riemann_sum(6,mode="midpoint") sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: Q = g.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40) sage: L = add([line([[pf[0][0],0],[pf[0][0],pf[1](x=pf[0][0])]],\ rgbcolor=(0.7,0.6,0.6)) for pf in g.list()]) sage: show(P+Q+L) Here is the plot: Figure 1.26: Plot using SAGE of a Riemann sum for y = x2 . At the end of this section is a Python2 program listing for calculating Riemann sums of f (x) = 1/x on the interval [1, 5] using 100 subintervals. It can be 2 Python is a cross-platform, free and open source computer language. It is widely used in industry and academicia, and is available for download at Alternatively, you can download the mathematical software system SAGE from It comes with Python pre-installed. 16 1.3. SIGMA NOTATION AND RIEMANN SUMS modified easily to work for different functions, different endpoints, and different numbers of subintervals. Table 1.27 shows the results of running the program with different numbers of subintervals and different ways of selecting the points ci in each subinterval. When the mesh of the partition is small (and the number of subintervals large), all of the ways of selecting the ci lead to approximately the same number for the Riemann sums. 17 1.3. SIGMA NOTATION AND RIEMANN SUMS Here is a Python program to calculate Riemann sums of f (x) = 1/x on [1, 5] using 100 equal length subintervals, based on the "lefthand" endpoints. Python f = lambda x: 1/x # define the function a = 1.0 # left endpoint of integral b = 5.0 # right endpoint of integral n = 100 # number of subintervals Dx = (b-a)/n # width of each subinterval rsum = sum([f(a+iDx)Dx for i in range(n)]) # compute the Riemann sum print rsum # print the Riemann sum Other Riemann sums can be calculated by replacing the "rsum" line with one of: rsum = sum([f(a+(i+0.5)Dx)Dx for i in range(n)]) "midpoint" rsum = sum([f(a+(i+1)Dx)Dx for i in range(n)]) "right-hand" Written as Python "functions", these three are written as below3 Python def rsum_lh(n): f = lambda x: 1/x a = 1.0 b = 5.0 Dx = (b-a)/n return sum([f(a+iDx)Dx for i in range(n)]) def rsum_mid(n): f = lambda x: 1/x a = 1.0 b = 5.0 Dx = (b-a)/n return sum([f(a+(i+0.5)Dx)Dx for i in range(n)]) def rsum_rh(n): f = lambda x: 1/x a = 1.0 b = 5.0 Dx = (b-a)/n return sum([f(a+(i+1)Dx)Dx for i in range(n)]) The command sizes = [5, 10, 20, 100, 1000] table = [[n, (b-a)/n, rsum_lh(n),rsum_mid(n), rsum_rh(n)] for n in sizes] yields the following data: In fact, the exact value is log(5) = 1.609437..., so these last few lines yielded pretty good approximations. 3 If you have an electronic copy of this file, and "copy-and-paste" these into Python, keep in mind indenting is very important in Python syntax. 18 1.3. SIGMA NOTATION AND RIEMANN SUMS left–hand midpoint right–hand n ∆xi Riemann sum Riemann sum Riemann sum 5 0.8 1.9779070602600015 1.5861709609993364 1.3379070602600014 10 0.4 1.7820390106296689 1.6032106782106783 1.462039010629669 20 0.2 1.6926248444201737 1.6078493243021688 1.5326248444201738 100 0.04 1.6255658911511259 1.6093739310551827 1.5935658911511259 1000 0.004 1.6110391924319691 1.6094372724359669 1.607839192431969 Figure 1.27: Table for Python example. Practice 1.3.6. Replace 1/x by x2 and [a, b] = [1, 5] by [a, b] = [−1, 1] in the Python code above and find the Riemann sum for the new function and n = 100. Use the midpoint approximation. (You may use SAGE or Python, whichever you prefer.) Example 1.3.7. Find the Riemann sum for the function f (x) = sin(x) on the interval [0, π] using the partition {0, π/4, π/2, π} with c1 = π/4, c2 = π/2, c3 = 3π/4. Solution: The 3 subintervals are [0, π/4], [π/4, π/2], and [π/2, π] so ∆x1 = π/4, ∆x2 = π/4 and ∆x3 = π/2. The Riemann sum for this partition is 3 k=1 f (ck )∆xk = sin(π/4)(π/4) + sin(π/2)(π/4) + sin(3π/4)(π/2) = √2 π + 1 · π + √2 π 1 4 4 1 2 = 2.45148... . Practice 1.3.7. Find the Riemann sum for the function and partition in the previous example, but use c1 = 0, c2 = π/2, c3 = π/2. 1.3.3 Two special Riemann sums: lower and upper sums Two particular Riemann sums are of special interest because they represent the extreme possibilities for Riemann sums for a given partition. Definition 1.3.2. Suppose f is a positive function on [a, b], and P is a partition of [a, b]. Let mk be the xvalue in the k-th subinterval so that f (mk ) is the minimum value of f in that interval, and let Mk be the xvalue in the k-th subinterval so that f (Mk ) is the maximum value of f in that interval. n lower sum: LS = k=1 f (mk )∆xk . n upper sum: U S = k=1 f (Mk )∆xk . Geometrically, the lower sum comes from building rectangles under the graph of f (Figure 1.28(a)), and the lower sum (every lower sum) is less than or equal to the exact area A: LS ≤ A for every partition P . The upper sum comes from building rectangles over the graph of f (Figure 1.28(b)), and the upper sum (every upper sum) is greater than or equal to the exact area A: U S ≥ A for 19 1.3. SIGMA NOTATION AND RIEMANN SUMS Figure 1.28: Lower and upper Riemann sums. every partition P . The lower and upper sums provide bounds on the size of the exact area: LS ≤ A ≤ U S. Unfortunately, finding minimums and maximums can be a timeconsuming business, and it is usually not practical to determine lower and upper sums for "arbitrary" functions. If f is monotonic, however, then it is easy to find the values for mk and Mk , and sometimes we can explicitly calculate the limits of the lower and upper sums. For a monotonic bounded function we can guarantee that a Riemann sum is within a certain distance of the exact value of the area it is approximating. Theorem 1.3.2. If f is a positive, montonically increasing, bounded function on [a, b], then for any partition P and any Riemann sum for P , distance between the Riemann sum and the exact area ≤ distance between the upper sum (US) and the lower sum (LS) ≤ (f (b) − f (a)) · (mesh of P ). Proof: The Riemann sum and the exact area are both between the upper and lower sums so the distance between the Riemann sum and the exact area is less than or equal to the distance between the upper and lower sums. Since f is monotonically increasing, the areas representing the difference of the upper and lower sums can be slid into a rectangle whose height equals f (b)−f (a) and whose base equals the mesh of P . Then the total difference of the upper and lower sums is less than or equal to the area of the rectangle, (f (b)−f (a))·(mesh of P ). 1.3.4 Problems For problems the next four problems, sketch the function and find the smallest possible value and the largest possible value for a Riemann sum of the given function and partition. 1. f (x) = 1 + x2 (a) P = {1, 2, 4, 5} (b) P = {1, 2, 3, 4, 5} (c) P = {1, 1.5, 2, 3, 4, 5} 20 1.4. THE DEFINITE INTEGRAL 2. f (x) = 7 − 2x (a) P = {0, 2, 3} (b) P = {0, 1, 2, 3} (c) P = {0, .5, 1, 1.5, 2, 3} 3. f (x) = sin(x) (a) P = {0, π/2, π} (b) P = {0, π/4, π/2, π} (c) P = {0, π/4, 3π/4, π}. 4. f (x) = x2 − 2x + 3 (a) P = {0, 2, 3} (b) P = {0, 1, 2, 3} (c) P = {0, .5, 1, 2, 2.5, 3}. 5. Suppose we divide the interval [1, 4] into 100 equally wide subintervals and calculate a Riemann sum for f (x) = 1 + x2 by randomly selecting a point ci in each subinterval. (a) We can be certain that the value of the Riemann sum is within what distance of the exact value of the area between the graph of f and the interval [1, 4] ? (b) What if we take 200 equally long subintervals? 6. If f is monotonic decreasing on [a, b] and we divide the interval [a, b] into n equally wide subintervals, then we can be certain that the Riemann sum is within what distance of the exact value of the area between f and the interval [a, b]? 7. Suppose LS = 7.362 and U S = 7.402 for a positive function f and a partition P of the interval [1, 5]. (a) We can be certain that every Riemann sum for the partition P is within what distance of the exact value of the area under the graph of f over the interval [1, 5]? (b) What if LS = 7.372 and U S = 7.390? 1.4 The definite integral Each particular Riemann sum depends on several things: the function f , the interval [a, b], the partition P of the interval, and the values chosen for ck in each subinterval. Fortunately, for most of the functions needed for applications, as the approximating rectangles get thinner (as the mesh of P approaches 0 and the number of subintervals gets bigger) the values of the Riemann sums approach the same value independently of the particular partition P and the 21 1.4. THE DEFINITE INTEGRAL points ck . For these functions, the limit (as the mesh approaches 0) of the Riemann sums is the same number no matter how the ck 's are chosen. This limit of the Riemann sums is the next big topic in calculus, the definite integral. Integrals arise throughout the rest of this book and in applications in almost every field that uses mathematics. n Definition 1.4.1. If limmesh(P )→0 k=1 f (ck )∆xk equals a finite number I then f is said to be (Riemann) integrable on the interval [a, b]. The number I is called the definite integral of f over [a, b] and is written b a f (x) dx. b The symbol a f (x) dx is read "the integral from a to b of f of x dee x"or "the integral from a to b of f (x) with respect to x." The lower limit is a, upper limit is b, the integrand is f (x), and x is sometimes called the dummy b variable. Note that a f (u) du numerically means exactly the same thing, but b with a different dummy variable. The value of a definite integral a f (x) dx depends only on the function f being integrated and on the endpoints a and b. The following integrals each represent the integral of the function f on the interval [a, b], and they are all equal: b b b b f (x) dx = f (t) dt = f (u) du = f (z) dz. a a a a Also, note that when the upper limit and the lower limit are the same then the integral is always 0: a f (x)dx = 0. a There are many other properties, as we will see later. Example 1.4.1. (Relation between velocity and area) Suppose you're reading a car magazine and there is an article about a new sports car that has this table in it: Time (seconds) 0 1 2 3 4 5 6 Speed (mph) 0 5 15 25 40 50 60 They claim the car drove 1/8th of a mile after 6 seconds, but this just "feels" wrong... Hmmm... Let's estimate the distance driven using the formula distance = rate × time. We overestimate by assuming the velocity is a constant equal to the max on each interval: 195 estimate = 5 · 1 + 15 · 1 + 25 · 1 + 40 · 1 + 50 · 1 + 60 · 1 = miles = 0.054... 3600 22 1.4. THE DEFINITE INTEGRAL (Note: there are 3600 seconds in an hour.) But 1/8 ∼ 0.125, so the article is inconsistent. (Doesn't this sort of thing just bug you? By learning calculus you'll be able to double-check things like this much more easily.) Insight! The formula for the estimate of distance traveled above looks exactly like an approximation for the area under the graph of the speed of the car! In fact, if an object has velocity v(t) at time t, then the net change in position from time a to b is b v(t)dt. a If f is a velocity, then the integrals on the intervals where f is positive measure the distances moved forward; the integrals on the intervals where f is negative measure the distances moved backward; and the integral over the whole time interval is the total (net) change in position, the distance moved forward minus the distance moved backward. Practice 1.4.1. A bug starts at the location x = 12 on the x–axis at 1 pm and walks along the axis in the positive direction with the velocity shown in Figure 1.29. How far does the bug travel between 1 pm and 3 pm, and where is the bug at 3 pm? Figure 1.29: Velocity of a bug on the x–axis. Practice 1.4.2. A car is driven with the velocity west shown in Figure 1.30. (a) Between noon and 6 pm how far does the car travel? (b) At 6 pm, where is the car relative to its starting point (its position at noon)? Figure 1.30: Velocity of a car on the x–axis. Units For the Definite Integral We have already seen that the "area" under a graph can represent quantities whose units are not the usual geometric units 23 1.4. THE DEFINITE INTEGRAL of square meters or square feet. In general, the units for the definite integral b a f (x)dx are (units for f (x))×(units for x). A quick check of the units can help avoid errors in setting up an applied problem. For example, if x is a measure of time in seconds and f (x) is a velocity with units feet/second, then ∆x has the units seconds and f (x)∆x has the units (feet/second)(seconds) = feet, a measure of distance. Since each Riemann sum f (xk )∆xk is a sum of feet and the definite integral is the limit of the Riemann b sums, the definite integral a f (x)dx, has the same units, feet. b If f (x) is a force in grams, and x is a distance in centimeters, then a f (x)dx is a number with the units "gram·centimeters," a measure of work. 1.4.1 The Fundamental Theorem of Calculus Example 1.4.2. For the function f (t) = 2, define A(x) to be the area of the region bounded by the graph of f , the t–axis, and vertical lines at t = 1 and t = x. (a) Evaluate A(1), A(2), A(3), A(4). (b) Find an algebraic formula for A(x), for x ≥ 1. d (c) Calculate dx A(x). (d) Describe A(x) as a definite integral. Solution : (a) A(1) = 0, A(2) = 2, A(3) = 4, A(4) = 6. (b) A(x) = area of a d d rectangle = (base)×(height) = (x−1)·(2) = 2x−2. (c) dx A(x) = dx (2x−2) = 2. x (d) A(x) = 1 2 dt. Practice 1.4.3. Answer the questions in the previous Example for f (t) = 3. A curious "coincidence" appeared in this Example and Practice problem: the derivative of the function defined by the integral was the same as the integrand, the function "inside" the integral. Stated another way, the function defined by the integral was an "antiderivative" of the function "inside" the integral. We will see that this is no coincidence: it is an important property called The Fundamental Theorem of Calculus. Let f be a continuous function on the interval [a, b]. Theorem 1.4.1. ("Fundamental Theorem of Calculus") If F (x) is any differ- entiable function on [a, b] such that F ′ (x) = f (x), then b f (x)dx = F (b) − F (a). a The above theorem is incredibly useful in mathematics, physics, biology, etc. One reason this is amazing, is because it says that the area under the entire curve is completely determined by the values of a ("magic") auxiliary function 24 1.4. THE DEFINITE INTEGRAL at only 2 points. It's hard to believe. It reduces computing (1.4.1) to finding a single function F , which one can often do algebraically, in practice. Whether or not one should use this theorem to evaluate an integral depends a lot on the application at hand, of course. One can also use a partial limit via a computer for certain applications (numerical integration). Example 1.4.3. I've always wondered exactly what the area is under a "hump" of the graph of sin. Let's figure it out, using F (x) = − cos(x). π sin(x)dx = − cos(π) − (− cos(0)) = −(−1) − (−1) = 2. 0 In SAGE, you can do this both "algebraically" and "numerically" as follows. SAGE sage: f = lambda x: sin(x) sage: integral(f(x),x,0,pi) 2 sage: numerical_integral(f(x),0,pi) (1.9999999999999998, 2.2204460492503128e-14) For the "algebraic" computation, SAGE knows how to integrate sin(x) exactly, π so can compute 0 sin(x)dx = 2 using its integral command4 . On the last line of output, the first entry is the approximation, and the second is the error bound. For the "numerical" computation, SAGE obtains5 the approximation π 0 sin(x)dx ≈ 1.99999... by taking enough terms in a Riemann sum to achieve a very small error. (A lot of theory of numerical integration goes into why numerical_integral works correctly, but that would take us too far afield to explain here.) Example 1.4.4. Let [...] denote the "greatest integer" (or "floor") function, so 3/2 [1/2] = [0.5] = 0 and [3/2] = [1.5] = 1. Evaluate 1/2 [x] dx. (The function of y = [x] is sometimes called the "staircase function" because of the look of its discontinuous graph, Figure 1.31.) Solution: f (x) = [x] is not continuous at x = 1 in the interval [1/2.3/2] so the Fundamental Theorem of Calculus can not be used. We can, however, use 3/2 our understanding of the meaning of an integral as an area to get 1/2 [x] dx = (areaundery=0between0.5and1)+(areaundery=1between1and1.5) = 0+1/2 = 1/2. Now, let's try something illegal - using the Fundamental Theorem of Calculus to evaluate this. Pretend for the moment that the Fundamental Theorem of Calculus is valid for discontinuous functions too. Let 4 In fact, SAGE includes Maxima ( and calls Maxima to compute this integral. 5 In fact, SAGE includes the GNU Scientific Library ( and calls it to approximate this integral. 25 1.4. THE DEFINITE INTEGRAL Figure 1.31: Plot of the "greatest integer" function. 1, 1/2 ≤ x ≤ 1, F (x) = x, 1 < x ≤ 3/2. This function F is continuous and satisfies F ′ (x) = [x] for all x in [1/2, 3/2] except x = 1 (where f (x) = [x] is discontinuous), so this F could be called an 3/2 "antiderivative" of f . If we use it to evaluate the integral we get 1/2 [x] dx = 3/2 F (x)\|1/2 = 3/2 − 1 = 1/2. This is correct. (Surprised?) Let's try another antiderivative. Let 2, 1/2 ≤ x ≤ 1, F (x) = x, 1 < x ≤ 3/2. This function F also satisfies F ′ (x) = [x] for all x in [1/2, 3/2] except x = 1. If 3/2 3/2 we use it to evaluate the integral we get 1/2 [x] dx = F (x)\|1/2 = 3/2−2 = −1/2. This doesn't even have the right sign (the integral of a non-negative function must be non-negative!), so it must be wrong. Moral of the story: In general, the Fundamental Theorem of Calculus is false for discontinuous functions. But does such an F as in the fundamental theorem of calculus (Theorem 1.4.1) always exist? The surprising answer is "yes". x Theorem 1.4.2. Let F (x) = a f (t)dt. Then F ′ (x) = f (x) for all x ∈ [a, b]. Note that a "nice formula" for F can be hard to find or even provably non- existent. The proof of Theorem 1.4.2 is somewhat complicated but is given in complete detail in many calculus books, and you should definitely (no pun intended) read and understand it. 26 1.4. THE DEFINITE INTEGRAL Proof: [Sketch of Proof] We use the definition of derivative. F (x + h) − F (x) F ′ (x) = lim h→0 h x+h x = lim f (t)dt − f (t)dt /h h→0 a a x+h = lim f (t)dt /h h→0 x x+h Intuitively, for h sufficiently small f is essentially constant, so x f (t)dt ∼ hf (x) (this can be made precise using the extreme value theorem). Thus x+h lim f (t)dt /h = f (x), h→0 x which proves the theorem. 1.4.2 Problems In problems 1 – 4 , rewrite the limit of each Riemann sum as a definite integral. n 1. limmesh(P )→0 k=1 (2 + 3ck )∆xk on the interval [0, 4]. n 2. limmesh(P )→0 k=1 cos(5ck )∆xk on the interval [0, 11]. n 3 3. limmesh(P )→0 k=1 ck ∆xk on the interval [2, 5]. n 4. limmesh(P )→0 k=1 ck ∆xk on the interval [2, 5]. 5. Write as a definite integral (don't evaluate it though): The region bounded by y = x3 , the x–axis, the line x = 1, and x = 5. 6. Write as a definite integral (don't evaluate it though): The region bounded √ by y = x, the x–axis, and the line x = 9. 7. Write as a definite integral (do evaluate it, using geometry formulas): The region bounded by y = 2x, the x–axis, the line x = 1, and x = 3. 8. Write as a definite integral (do evaluate it, using geometry formulas): The region bounded by y = \|x\|, the x–axis, and the line x = −1. 9. For f (x) = 3 + x, partition the interval [0, 2] into n equally wide subinter- vals of length ∆x = 2/n. (a) Write the lower sum for this function and partition, and calculate the limit of the lower sum as n → ∞. (b) Write the upper sum for this function and partition and find the limit of the upper sum as n → ∞. 27 1.4. THE DEFINITE INTEGRAL 10. For f (x) = x3 , partition the interval [0, 2] into n equally wide subintervals of length ∆x = 2/n. (a) Write the lower sum for this function and partition, and calculate the limit of the lower sum as n → ∞. (b) Write the upper sum for this function and partition and find the limit of the upper sum as n → ∞. 1.4.3 Properties of the definite integral Definite integrals are defined as limits of Riemann sums, and they can be inter- preted as "areas" of geometric regions. This section continues to emphasize this geometric view of definite integrals and presents several properties of definite integrals. These properties are justified using the properties of summations and the definition of a definite integral as a Riemann sum, but they also have natural interpretations as properties of areas of regions. These properties are used in this section to help understand functions that are defined by integrals. They will be used in future sections to help calculate the values of definite integrals. Since integrals are a lot like sums (they are, after all, limits of them), their properties are similar too. Here is the integral analog of Theorem 1.3.1. Theorem 1.4.3. (Integral Properties) b • Integral of a constant function: a c dx = c · (b − a). b b b • Addition: a (f (x) + g(x)) dx = a f (x) dx + a g(x) dx. b b b • Subtraction: a (f (x) − g(x)) dx = a f (x) dx − a g(x) dx. b b • Constant Multiple: a c · f (x) dx = c a f (x) dx. • Preserves positivity: If f (x) ≥ g(x) on for all x ∈ [a, b], then b b f (x) dx ≥ g(x) dx. a a In particular, if f (x) ≥ 0 on for all x ∈ [a, b], then b f (x) dx ≥ 0. a b c c • Additivity of ranges: a f (x) dx + b f (x) dx = a f (x) dx. Here are some other properties. Theorem 1.4.4. b (b − a) · ( min f (x)) ≤ f (x) dx ≤ (b − a) · ( min f (x)). x∈[a,b] a x∈[a,b] 28 1.4. THE DEFINITE INTEGRAL Figure 1.32: Plot illustrating Theorem 1.4.4. Which Functions Are Integrable? This important question was finally an- swered in the 1850s by Georg Riemann, a name that should be familiar by now. Riemann proved that a function must be badly discontinuous to not be integrable. Theorem 1.4.5. Every continuous function is integrable. If f is continuous on n the interval [a, b], then limmesh(P )→0 ( k=1 f (ck )∆xk ) is always the same finite b number, namely, a f (x) dx, so f is integrable on [a, b]. In fact, a function can even have any finite number of breaks and still be integrable. Theorem 1.4.6. Every bounded, piecewise continuous function is integrable. If f is defined and bounded ( for all x in [a, b], M ≤ f (x) ≤ M for some M > 0), and continuous except at a finite number of points in [a, b], then n b limmesh(P )→0 ( k=1 f (ck )∆xk ) is always the same finite number, namely, a f (x) dx, so f is integrable on [a, b]. Example 1.4.5. (A Nonintegrable Function) Though rarely encountered in "everyday practice", there are functions for which the limit of the Riemann sums does not exist, and those functions are not integrable. A nonintegrable function: The function 1, if x is a rational number, f (x) = 0, if x is an irrational number is not integrable on [0, 1]. Proof: For any partition P , suppose that you, a very rational (pun intended) person, always select values of ck which are rational numbers. (Every subinterval contains rational numbers and irrational numbers, so you can always pick ck to be a rational number.) Then f (ck ) = 1, and your Riemann sum is always n n YP = f (ck )∆xk = ∆xk = xn − x0 = 1. k=1 k=1 Suppose your friend, however, always selects values of ck which are irrational numbers. Then f (ck ) = 0, and your friend's Riemann sum is always 29 1.4. THE DEFINITE INTEGRAL n n FP = f (ck )∆xk = 0 · ∆xk = 0. k=1 k=1 Now, take finer and finer partitions P so that mesh(P ) → 0. Keep in mind that, no matter how you refine P , you can always make "rational choices" for ck and your friend can always make "irrational choices". We have limmesh(P )→0 YP = 1 and limmesh(P )→0 FP = 0, so the limit of the Riemann sums doesn't have a unique value. Therefore the limit n lim ( f (ck )∆xk ) mesh(P )→0 k=1 does not exist, so f is not integrable. 1.4.4 Problems Problems 1 – 20 refer to the graph of f in Figure 1.33. Use the graph to determine the values of the definite integrals. (The bold numbers represent the area of each region.) Figure 1.33: Plot for problems. 3 1. 0 f (x) dx 5 2. 3 f (x) dx 2 3. 2 f (x) dx 7 4. 6 f (x) dx 5 5. 0 f (x) dx 7 6. 0 f (x) dx 6 7. 3 f (x) dx 7 8. 5 f (x) dx 30 1.4. THE DEFINITE INTEGRAL 0 9. 3 f (x) dx 3 10. 5 f (x) dx 0 11. 6 f (x) dx 3 12. 0 2f (x) dx 4 13. 4 f (x)2 dx 3 14. 0 1 + f (t) dt 3 15. 0 x + f (x) dx 5 16. 3 3 + f (x) dx 5 17. 0 2 + f (x) dx 5 18. 3 \|f (x)\| dx 3 19. 7 1 + \|f (x)\| dx For problems 21–28, sketch the graph of the integrand function and use it to help evaluate the integral. (\|...\| denotes the absolute value and [...] denotes the integer part.) 4 21. 0 \|x\| dx, 4 22. 0 1 + \|x\| dx, 2 23. −1 \|x\| dx, 2 24. 1 \|x\| − 1 dx, 3 25. 1 [x] dx, 3.5 26. 1 [x] dx, 3 27. 1 2 + [x] dx, 1 28. 3 [x] dx. 31 1.5. AREAS, INTEGRALS, AND ANTI-DERIVATIVES 1.5 Areas, integrals, and anti-derivatives This section explores properties of functions defined as areas and examines some of the connections among areas, integrals and antiderivatives. In order to focus on the geometric meaning and connections, all of the functions in this section are nonnegative, but the results are generalized in the next section and proved true for all continuous functions. This section also introduces examples to illustrate how areas, integrals and antiderivatives can be used. When f is a continuous, x nonnegative function, then the "area function" A(x) = a f (t) dt represents the area between the graph of f , the t–axis, and between the vertical lines at t = a and t = x (Figure 1.34), and the derivative of A(x) represents the rate of change (growth) of A(x). Figure 1.34: Plot of an "area function". Let F (x) be a differentiable function. Call F (x) an antiderivative of f (x) if d dx F (x) = f (x). We have seen examples which showed that, at least for some functions f , the derivative of A(x) was equal to f so A(x) was an antiderivative of f . The next theorem says the result is true for every continuous, nonnegative function f . Theorem 1.5.1. ("The Area Function is an Antiderivative") If f is a contin- x d x uous nonnegative function, x ≥ a, and A(x) = a f (t) dt then dx a f (t) dt = d dx A(x) = f (x), so A(x) is an antiderivative of f (x). This result relating integrals and antiderivatives is a special case (for non- negative functions f ) of the Fundamental Theorem of Calculus. This result is important for two reasons: • it says that a large collection of functions have antiderivatives, and • it leads to an easy way of exactly evaluating definite integrals. x d x Example 1.5.1. Let G(x) = dx 0 cos(t)dt. Evaluate G(x) for x = π/4, π/2, and 3π/4. x Solution: It is not hard to plot the graph of A(x) = 0 cos(t)dt = sin(x) (Figure 1.35). By the theorem, A′ (x) = G(x) = cos(x) so A′ (π/4) = cos(π/4) = .707..., A′ (π/2) = cos(π/2) = 0, and A′ (3π/4) = cos(3/4) = −0.707... . 32 1.5. AREAS, INTEGRALS, AND ANTI-DERIVATIVES x Figure 1.35: Plot of y = 0 G(t) dt and y = G(x). Here is the plot of y = A(x) and y = G(x): Incidentally, this was created using the following SAGE commands. SAGE sage: P = plot(cos(x),x,0,2pi,linestyle="--") sage: Q = plot(sin(x),x,0,2pi) sage: R = text("$y=A(x) = \sin(x)$",(3.1,1)) sage: S = text("$y=G(x) = \cos(x)$",(6.8,0.7)) sage: show(P+Q+R+S) Theorem 1.5.2. ("Antiderivatives and Definite Integrals") If f is a continu- ous, nonnegative function and F is any antiderivative of f (F ′ (x) = f (x)) on the interval [a, b], then area bounded between the graph b of f and the x–axis and = a f (x) dx = F (b) − F (a). vertical lines at x = a and x = b The problem of finding the exact value of a definite integral reduces to finding some (any) antiderivative F of the integrand and then evaluating F (b)F (a). Even finding one antiderivative can be difficult, and, for now, we will stick to functions which have easy antiderivatives. Later we will explore some methods for finding antiderivatives of more difficult functions. The evaluation F (b) − F (a) is represented by the symbol F (x)\|b . a 3 Example 1.5.2. Evaluate 1 x dx in two ways: (a) By sketching the graph of y = x and geometrically finding the area. (b) By finding an antiderivative of F (x) of f and evaluating F (3) − F (1). Solution: (a) The graph of y = x is a straight line, so the area is a triangle 1 which geometrical formulas (area= 2 bh) tell us has area 4. d (b) One antiderivative of x is F (x) = 1 x2 (check that dx ( 1 x2 ) = x), and 2 2 33 1.5. AREAS, INTEGRALS, AND ANTI-DERIVATIVES 1 2 1 2 F (x)\|3 = F (3) − F (1) = 1 3 − 1 = 4, 2 2 which agrees with (a). Suppose someone chose another antiderivative of x, say 1 d F (x) = 2 x2 + 7 (check that dx ( 1 x2 + 7) = x), then 2 1 1 F (x)\|3 = F (3) − F (1) = ( 32 + 7) − ( 12 + 7) = 4. 1 2 2 No matter which antiderivative F is chosen, F (3) − F (1) equals 4. 3 Practice 1.5.1. Evaluate 1 (x − 1) dx in the two ways of the previous example. Practice 1.5.2. Find the area between the graph of y = 3x2 and the horizontal axis for x between 1 and 2. 1.5.1 Integrals, Antiderivatives, and Applications The antiderivative method of evaluating definite integrals can also be used when we need to find an "area," and it is useful for solving applied problems. Example 1.5.3. Suppose that t minutes after putting 1000 bacteria on a petri plate the rate of growth of the population is 6t bacteria per minute. (a) How many new bacteria are added to the population during the first 7 minutes? (b) What is the total population after 7 minutes? (c) When will the total population be 2200 bacteria? Solution: (a) The number of new bacteria is the area under the rate of growth d graph, and one antiderivative of 6t is 3t2 (check that dx (3t2 ) = 6t) so new 7 bacteria = 0 6t dt = 3t2 \|7 = 147. 0 (b) The new population = (old population) + (new bacteria) = 1000 + 147 = 1147 bacteria. (c) If the total population is 2200 bacteria, then there are 2200 − 1000 = 1200 new bacteria, and we need to find the time T needed for that many new bacteria T to occur. 1200 new bacteria = 0 6t dt = 3t2 \|T = 3(T )2 − 3(0)2 = 3T 2 so 0 2 T = 400 and T = 20 minutes. After 20 minutes, the total bacteria population will be 1000 + 1200 = 2200. Practice 1.5.3. A robot has been programmed so that when it starts to move, its velocity after t seconds will be 3t2 feet/second. (a) How far will the robot travel during its first 4 seconds of movement? (b) How far will the robot travel during its next 4 seconds of movement? (c) How many seconds before the robot is 729 feet from its starting place? 34 1.6. INDEFINITE INTEGRALS AND CHANGE (Hint: an antiderivative of 3t2 is t3 .) Practice 1.5.4. The velocity of a car after t seconds is 2t feet per second. (a) How far does the car travel during its first 10 seconds? (b) How many seconds does it take the car to travel half the distance in part (a)? 1.6 Indefinite Integrals and Change 1.6.1 Indefinite Integrals The notation f (x)dx = F (x) means that F ′ (x) = f (x) on some (usually speci- fied) domain of definition of f (x). Recall, we call such an F (x) an antiderivative of f (x). Proposition 1.6.1. Suppose f is a continuous function on an interval (a, b). Then any two antiderivatives differ by a constant. Proof: If F1 (x) and F2 (x) are both antiderivatives of a function f (x), then ′ ′ (F1 (x) − F2 (x))′ = F1 (x) − F2 (x) = f (x) − f (x) = 0. Thus F1 (x) − F2 (x) = c from some constant c (since only constant functions have slope 0 everywhere). Thus F1 (x) = F2 (x) + c as claimed. We thus often write f (x)dx = F (x) + C, where C is an (unspecified fixed) constant. Note that the proposition need not be true if f is not defined on a whole interval. For example, f (x) = 1/x is not defined at 0. For any pair of constants c1 , c2 , the function ln(\|x\|) + c1 x < 0, F (x) = ln(x) + c2 x > 0, satisfies F ′ (x) = f (x) for all x = 0. We often still just write 1/x = ln(\|x\|) + c anyways, meaning that this formula is supposed to hold only on one of the intervals on which 1/x is defined (e.g., on (−∞, 0) or (0, ∞)). We pause to emphasize the notation difference between definite and indefinite integration. b f (x)dx = a specific number a f (x)dx = a (family of) functions 35 1.6. INDEFINITE INTEGRALS AND CHANGE There are no small families in the world of antiderivatives: if f has one an- tiderivative F (as it always does, unless f is a really unusual function), then f has an infinite number of antiderivatives and every one of them has the form F (x) + C. Example 1.6.1. There are many ways to write a particular indefinite inte- gral and some of them may look very different. You can check that F (x) = sin(x)2 , G(x) = − cos(x)2 , and H(x) = 2 sin(x)2 + cos(x)2 all have the same derivative f (x) = 2 sin(x) cos(x), so the indefinite integral of 2 sin(x) cos(x), 2 sin(x) cos(x) dx, can be written in several ways: sin(x)2 +C, or − cos(x)2 +C, or 2 sin(x)2 + cos(x)2 + C. One of the main goals of this course is to help you to get really good at computing f (x)dx for various functions f (x). It is useful to memorize a table of examples (see, e.g., page 406 of Stewart), since often the trick to integration is to relate a given integral to a known one. Integration is like solving a puzzle or playing a game, and often you win by moving into a position where you know how to defeat your opponent, e.g., relating your integral to integrals that you already know how to do. If you know how to do a basic collection of integrals, it will be easier for you to see how to get to a known integral from an unknown one. Whenever you successfully compute F (x) = f (x)dx, then you've constructed b a mathematical gadget that allows you to very quickly compute a f (x)dx for any a, b (in the interval of definition of f (x)). The gadget is F (b) − F (a). This is really powerful. Example 1.6.2. 1 1 x2 + 1 + dx = x2 dx + 1dx + dx x2 + 1 x2 + 1 1 2 = x + x + tan−1 (x) + c. 3 Example 1.6.3. 5 √ −1/2 √ dx = 5x dx = 2 5x1/2 + c. x Example 1.6.4. sin(2x) 2 sin(x) cos(x) dx = = 2 cos(x) = 2 sin(x) + c sin(x) sin(x) Particular Antiderivatives: You can verify the following yourself. • Constant Function: k dx = kx + C xn+1 • Powers of x: xn dx = n+1 + C, n = −1. 36 1.6. INDEFINITE INTEGRALS AND CHANGE x−1 dx = ln(x) + C. Common special cases: √ 2 x dx = 3 x3/2 + C. 1 √ x dx = 2x1/2 + C. • Trigonometric Functions: cos(x) dx = sin(x) + C. sin(x) dx = − cos(x) + C. sec(x)2 dx = tan(x) + C. csc(x)2 dx = − cot(x) + C. sec(x) tan(x) dx = sec(x) + C. csc(x) cot(x) dx = − csc(x) + C. 1.6.2 Physical Intuition In the previous lecture we mentioned a relation between velocity, distance, and the meaning of integration, which gave you a physical way of thinking about integration. In this section we generalize our previous observation. The following is a restatement of the fundamental theorem of calculus. Theorem 1.6.1. (Net Change Theorem) The definite integral of the rate of change F ′ (x) of some quantity F (x) is the net change in that quantity: b f ′ (x)dx = f (b) − f (a). a For example, if p(t) is the population of students at UCSD at time t, then p′ (t) is the rate of change. Lately p′ (t) has been positive since p(t) is growing (rapidly!). The net change interpretation of integration is that t2 p′ (t)dt = p(t2 ) − p(t1 ) = change in number of students from time t1 to t2 . t1 Another very common example you'll seen in problems involves water flow into or out of something. If the volume of water in your bathtub is V (t) gallons at time t (in seconds), then the rate at which your tub is draining is V ′ (t) gallons per second. If you have the geekiest drain imaginable, it prints out the drainage rate V ′ (t). You can use that printout to determine how much water drained out from time t1 to t2 : t2 V ′ (t)dt = water that drained out from time t1 to t2 t1 Some problems will try to confuse you with different notions of change. A standard example is that if a car has velocity v(t), and you drive forward, then 37 1.7. SUBSTITUTION AND SYMMETRY slam it in reverse and drive backward to where you start (say 10 seconds total elapse), then v(t) is positive some of the time and negative some of the time. 10 The integral 0 v(t)dt is not the total distance registered on your odometer, since v(t) is partly positive and partly negative. If you want to express how far 10 you actually drove going back and forth, compute 0 \|v(t)\|dt. The following example emphasizes this distinction: Example 1.6.5. A bug is pacing back and forth, and has velocity v(t) = t2 − 2t − 8. Find (1) the displacement of the bug from time t = 1 until time t = 6 (i.e., how far the bug is at time 6 from where it was at time 1), and (2) the total distance the bug paced from time t = 1 to t = 6. For (1), we compute 6 6 1 3 10 (t2 − 2t − 8)dt = t − t2 − 8t =− . 1 3 1 3 For (2), we compute the integral of \|v(t)\|: 6 4 6 1 3 1 3 44 98 \|t2 − 2t − 8\|dt = − t − t2 − 8t + t − t2 − 8t = 18 + = . 1 3 1 3 4 3 3 1.7 Substitution and Symmetry Remarks: t 1. The total distance traveled is t12 \|v(t)\|dt since \|v(t)\| is the rate of change of F (t) = distance traveled (your speedometer displays the rate of change of your odometer). b 2. How to compute a \|f (x)\|dx. (a) Find the zeros of f (x) on [a, b], and use these to break the interval up into subintervals on which f (x) is always ≥ 0 or always ≤ 0. (b) On the intervals where f (x) ≥ 0, compute the integral of f , and on the intervals where f (x) ≤ 0, compute the integral of −f . (c) The sum of the above integrals on intervals is \|f (x)\|dx. This section is primarly about a powerful technique for computing definite and indefinite integrals. 1.7.1 The Substitution Rule In first quarter calculus you learned numerous methods for computing deriva- tives of functions. For example, the power rule asserts that (xa )′ = a · xa−1 . 38 1.7. SUBSTITUTION AND SYMMETRY We can turn this into a way to compute certain integrals: 1 xa dx = xa+1 if a = −1. a+1 Just as with the power rule, many other rules and results that you already know yield techniques for integration. In general integration is potentially much trickier than differentiation, because it is often not obvious which technique to use, or even how to use it. Integration is a more exciting than differentiation! Recall the chain rule, which asserts that d f (g(x)) = f ′ (g(x))g ′ (x). dx We turn this into a technique for integration as follows: Proposition 1.7.1. (Substitution Rule) Let u = g(x), we have f (g(x))g ′ (x)dx = f (u)du, assuming that g(x) is a function that is differentiable and whose range is an interval on which f is continuous. Proof: Since f is continuous on the range of g, Theorem 1.4.2 (the funda- mental theorem of Calculus) implies that there is a function F such that F ′ = f . Then f (g(x))g ′ (x)dx = F ′ (g(x))g ′ (x)dx d = F (g(x)) dx dx = F (g(x)) + C = F (u) + C = F ′ (u)du = f (u)du. If u = g(x) then du = g ′ (x)dx, and the substitution rule simply says if you let u = g(x) formally in the integral everywhere, what you naturally would hope to be true based on the notation actually is true. The substitution rule illustrates how the notation Leibniz invented for Calculus is incredibly brilliant. It is said that Leibniz would often spend days just trying to find the right notation for a concept. He succeeded. As with all of Calculus, the best way to start to get your head around a new concept is to see severally clearly worked out examples. (And the best way to actually be able to use the new idea is to do lots of problems yourself!) In this section we present examples that illustrate how to apply the substituion rule to compute indefinite integrals. 39 1.7. SUBSTITUTION AND SYMMETRY Example 1.7.1. x2 (x3 + 5)9 dx Let u = x3 + 5. Then du = 3x2 dx, hence dx = du/(3x2 ). Now substitute it all in: 1 9 1 10 1 3 x2 (x3 + 5)9 dx = u = u = (x + 5)10 . 3 30 30 There's no point in expanding this out: "only simplify for a purpose!" Example 1.7.2. ex dx 1 + ex Substitute u = 1 + ex . Then du = ex dx, and the integral above becomes du = ln \|u\| = ln \|1 + ex \| = ln(1 + ex ). u Note that the absolute values are not needed, since 1 + ex > 0 for all x. Example 1.7.3. x2 √ dx 1−x Keeping in mind the power rule, we make the substitution u = 1 − x. Then du = −dx. Noting that x = 1 − u by solving for x in u = 1 − x, we see that the above integral becomes (1 − u)2 1 − 2u + u2 − √ du = − du u u1/2 =− u−1/2 − 2u1/2 + u3/2 du 4 2 = − 2u1/2 − u3/2 + u5/2 3 5 4 2 = −2(1 − x)1/2 + (1 − x)3/2 − (1 − x)5/2 . 3 5 The steps of the "change of variable" method can be summarized as 1. set a new variable, say u , equal to some function of the original variable x (usually u is set equal to some part of the original integrand function), 2. calculate the differential du as a function of dx, 3. rewrite the original integral in terms of u and du, 4. integrate the new integral to get an answer in terms of u, 5. replace the u in the answer to get an answer in terms of the original variable. 40 1.7. SUBSTITUTION AND SYMMETRY A "Rule of thumb" for changing the variable: If part of the integrand is a composition of functions, f (g(x)), then try setting u = g(x), the "inner" function. Example 1.7.4. elect a function for u for each integral and rewrite the integral in terms of u and du. 5ex (a) cos(3x) 2 + sin(3x) dx, (b) 2+ex dx, (c) ex sin(ex ) dx. Solution: (a) Put u = 2 + sin(3x). Then du = 3 cos(3x) dx, and the integral 1√ becomes 3 u du. 5 (b) Put u = 2 + ex . Then du = ex dx, and the integral becomes u du. x x (c) Put u = e . Then du = e dx, and the integral becomes sin(u) du. 1.7.2 Changing the variable and definite integrals Once an antiderivative in terms of u is found, we have a choice of methods. We can (a) rewrite our antiderivative in terms of the original variable x, and then evaluate the antiderivative at the integration endpoints and subtract, or (b) change the integration endpoints to values of u, and evaluate the an- tiderivative in terms of u before subtracting. If the original integral had endpoints x = a and x = b, and we make the sub- stitution u = g(x) and du = g ′ (x)dx, then the new integral will have endpoints u = g(a) and u = g(b): x=b u=g(b) (original integrand) dx becomes (new integrand) du. x=a u=g(a) Example 1.7.5. To evaluate 1 (3x − 1)4 dx, 0 we can, in line with the "Rule o thumb" above, use the substitution u = 3x − 1. d Then du = dx (3x − 1)dx = 3dx, so the indefinite integral (3x − 1)4 dx becomes 1 4 1 5 3 u du = 15 u + C. (a) Converting our antiderivative back to the variable x and evaluating with the original endpoints: 1 1 32 −1 11 (3x − 1)4 dx = ( (3x − 1)5 + C)\|1 = 0 − = = 2.2. 0 15 15 15 5 (b) Converting the integration endpoints to values of u : when x = 0, then u = 3x − 1 = 3 · 0 − 1 = −1, and when x = 1, then u = 3x − 1 = 3 · 1 − 1 = 2 so 1 2 1 4 1 11 (3x − 1)4 dx = u du = ( u5 + C)\|2 = −1 = 2.2. 0 −1 3 15 5 41 1.7. SUBSTITUTION AND SYMMETRY Both approaches typically involve about the same amount of work and calcula- tion. Of course, these approaches lead to the same numberical answer, by the "substitution rule" (Proposition 1.7.1). Here's how to do this using SAGE. Note that the area under the two curves plotted below, y = (3x − 1)4 , 0 < x < 1, and y = x4 /3, −1 < x < 2, are the same. SAGE sage: x,u = var("x,u") sage: integral((3x-1)ˆ4,x,0,1) 11/5 sage: integral(uˆ4/3,u,-1,2) 11/5 sage: P = plot((3x-1)ˆ4,x,0,1,rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: Q = plot(uˆ4/3,u,-1,2,linestyle=":") sage: R = text("$y=(3x-1)ˆ4$",(1.4,12)) sage: S = text("$y=xˆ4/3$",(2,2.5)) sage: plot(P+Q+R+S) Figure 1.36: Plots of y = (3x − 1)4 and y = x4 /3. 1.7.3 Symmetry An odd function is a function f (x) (defined for all reals) such that f (−x) = −f (x), and an even function one for which f (−x) = f (x). If f is an odd function, then for any a, a f (x)dx = 0. −a If f is an even function, then for any a, a a f (x)dx = 2 f (x)dx. −a 0 Both statements are clear if we view integrals as computing the signed area between the graph of f (x) and the x-axis. 42 1.7. SUBSTITUTION AND SYMMETRY Example 1.7.6. An even example, 1 1 1 1 3 2 x2 dx = 2 x2 dx = 2 x = , −1 0 3 0 3 and an odd example, 1 1 1 4 x3 dx = x = 0. −1 4 −1 These computations are consistent with the symmetry (or "anti-symmetry") of the graphs and what you know about the relationship between the integral and area. Figure 1.37: Plots of y = x2 and y = x3 . 1.7.4 Problems For the problems below, let f (x) = x2 and g(x) = x and verify that 1. f (x) · g(x) dx = f (x) dx · g(x) dx. 2. f (x)/g(x) dx = f (x) dx/ g(x) dx. 3. f (x) dx = f (x) dx. 1 1 4. R f (x) dx = f (x) dx. 43 1.7. SUBSTITUTION AND SYMMETRY 44 Chapter 2 Applications of the integral The development of calculus by Newton and Leibniz was a vital step in the advancement of pure mathematics, but Newton also advanced the applied sci- ences and mathematics. Not only did he discover theoretical results, but he immediately used those results to answer important applied questions about gravity and motion. The success of these applications of mathematics to the physical sciences helped establish what we now take for granted: mathematics can and should be used to answer questions about the world. Newton applied mathematics to the outstanding problems of his day, problems primarily in the field of physics. In the intervening 300 years, thousands of people have contin- ued these theoretical and applied traditions and have used mathematics to help develop our understanding of all of the physical and biological sciences as well as the behavioral sciences and business. Mathematics is still used to answer new questions in physics and engineering, but it is also important for modeling ecological processes, for understanding the behavior of DNA, for determining how the brain works, and even for devising strategies for voting effectively. The mathematics you are learning now can help you become part of this tradition, and you might even use it to add to our understanding of different areas of life. It is important to understand the successful applications of integration in case you need to use those particular applications. It is also important that you understand the process of building models with integrals so you can apply it to new problems. Conceptually, converting an applied problem to a Riemann sum is the most valuable and the most difficult step. 2.0.5 Using integration to determine areas This section is about how to compute the area of fairly general regions in the plane. Regions are often described as the area enclosed by the graphs of sev- eral curves. ("My land is the plot enclosed by that river, that fence, and the highway.") b Recall that the integral a f (x)dx has a geometric interpretation as the signed area between the graph of f (x) and the x-axis. We defined area by subdividing, 45 adding up approximate areas (use points in the intervals) as Riemann sums, and taking the limit. Thus we defined area as a limit of Riemann sums. The fundamental theorem of calculus asserts that we can compute areas exactly when we can finding antiderivatives. Figure 2.1: Area between y = f (x) and y = g(x). Instead of considering the area between the graph of f (x) and the x-axis, we consider more generally two graphs, y = f (x), y = g(x), and assume for simplicity that f (x) ≥ g(x) on an interval [a, b]. Again, we approximate the area between these two curves as before using Riemann sums. Each approximating rectangle has width (b − a)/n and height f (x) − g(x), so Area bounded by graphs ∼ [f (ci ) − g(ci )]∆x. Note that f (x) − g(x) ≥ 0, so the area is nonnegative. From the definition of integral we see that the exact area is b Area bounded by graphs = (f (x) − g(x))dx. (2.1) a Why did we make a big deal about approximations instead of just writing down (2.1)? Because having a sense of how this area comes directly from a Riemann sum is very important. But, what is the point of the Riemann sum if all we're going to do is write down the integral? The sum embodies the geometric manifestation of the integral. If you have this picture in your mind, then the Riemann sum has done its job. If you understand this, you're more likely to know what integral to write down; if you don't, then you might not. Remark 2.0.1. By the linearity property of integration, our sought for area is the difference b b f (x)dx − g(x)dx, a a of two signed areas. 46 Example 2.0.7. Find the area enclosed by y = x + 1, y = 9 − x2 , x = −1, x = 2. 2 Area = (9 − x2 ) − (x + 1) dx −1 We have reduced the problem to a computation: 2 2 2 1 1 39 [(9 − x2 ) − (x + 1)]dx = (8 − x − x2 )dx = 8x − x2 − x3 = . −1 −1 2 3 −1 2
What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area? By combining the mathematical history of extrema with contemporarySobolev Spaces presents an introduction to the theory of Sobolev Spaces and other related spaces of function, also to the imbedding characteristics of these spaces. This theory is widely used in pure and Applied Mathematics and in the Physical Sciences. This second edition of Adam's 'classic' reference text contains many additions and much modernizing
0471877 Algebra Fundamental to all areas of mathematics, algebra provides the cornerstone for the student's development. The concepts are often intuitive, but some can take years of study to fully absorb. For over twenty years, the author's classic three-volume set, Algebra, has been regarded by many as the most outstanding introductory work available. This work, Classic Algebra, combines a fully updated Volume 1 with the essential topics from Volumes 2 and 3, and provides a self-contained introduction
More About This Textbook Overview This innovative text, a product of an NSF funded calculus consortium based at Harvard University, was developed as part of the calculus reform movement. Functions are presented graphically, numerically and algebraically to give students the benefit of alternate interpretations. The text is problem driven and features exceptional exercises based on real-world applications. Technology is used as a tool to help readers learn to think. The Draft Version of this book was class tested at over 100 schools. Editorial Reviews Booknews A text for a one-semester course for students who have mastered a course in single variable calculus. Encourages students to think and write about the geometrical and numerical meaning of calculus, as well as the algebraic aspect, and builds on the authors' text on single variable calculus. Includes exercises and review problems, and tear- out pages with three-ring holes. This preliminary edition incorporates suggestions from users of the draft version, including rewritten and new
Intermediate Algebra : Connecting Concepts ThroughINTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master concepts, problem solving, and communication skills. It modifies the rule of four, integrating algebraic techniques, graphing, the use of data in tables, and writing sentences to communicate solutions to application problems. The authors have developed several key ideas to make concepts real and vivid for students. First, the authors integrate applications, drawing on real-world data to show students why they need to know and how to apply math. The applications help students develop the skills needed to explain the meaning of answers in the context of the application. Second, they emphasize strong algebra skills. These skills support the applications and enhance student comprehension. Third, the authors use an eyeball best-fit approach to modeling. Doing models by hand helps students focus on the characteristics of each function type. Fourth, the text underscores the importance of graphs and graphing. Students learn graphing by hand, while the graphing calculator is used to display real-life data problems. In short, INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS takes an application-driven approach to algebra, using appropriate calculator technology as students master algebraic concepts and skills.
131.Review of Adding, Subtracting, and Multiplying Monomials ...arkerboards, transparency of warm up, questions, worksheets, basketball hoop, ball Set: TTW review the do now and lead a discussion about what the most difficult subject is that we have been covering, in terms of monomials and binomials. After getting students to talk about what confuses them about polynomials and monomials, the teacher will announce that in the last class, we learned how to do multiply binomials using Punnett Squares and that, today, the objective is to review everything we've done in terms of adding, subtracting, and multiplying monomials and binomials. Prep: Name, date, objectives, agenda, and... User Rating: Grade Level: K-12 132.quadratic equation quiz Quadratic review and quiz July 18, 2008 Period 2 Mike Warner Algebra 1 50 minutes OBJECTIVES The students will review solving quadratic formula, making tables, making graphs and justifying why polynomial is prime. (DOK 2, Algebra 1, 2.i,j,k) MATERIALS pencil and paper dry erase markers transparencies and wet erase markers overhead quiz Do Now- have several systems quadratic functions on the board. Set- Hello class, last class we reviewed the concepts from earlier in the week with Mr. Sealand. These included q... 134.Quadratics intro and factoring 071608 ...e going to learn how to distinguish between quadratic equations and non-quadratic equations and you will learn how to solve them using factoring. D. PROCEDURE 1. TTW give examples and non-examples, showing the same format as the factoring lessons (standard form). [5 minutes] 2. TSW practice putting polynomial equations into standard form. [10 minutes] 3. TTW introduce the three types of quadratic equations. [5 minutes] 4. TTW model how to solve type I quadratic equations (*stressing that there are two solutions). [2 minutes] 5. TSW practice solving type I quadratic equations. [4 minutes] 6. TTW model how to solve... User Rating: Grade Level: K-12 135.Factoring Snowflake Method 071508 Justin Sealand Algebra I 7/15/08 Period 2 50 minutes A. OBJECTIVE 1. TSW use the "snowflake" method to factor polynomials. (DOK 2 - Algebra I, 2.h) B. MATERIALS Overhead Projector; Transparencies; Algebra I, Foster et al. Do Now: [5 minutes] What is the single most difficult part about factoring for you? C. SET How many of you all cook at home? If you don't cook, have you ever watched your mom do it? Have you realized that you... User Rating: Grade Level: K-12 136.Systems of Linear Equations 071408 ...slope-intercept form? You can graph it. That graph contains all of your answers to what x and y equal. Today we are going to learn how to solve linear equations using graphical and algebraic methods, and whether or not they even have a solution and how many they have. But before we can move on from polynomials and area and perimeter, we have to go over a few last things. D. PROCEDURE 1. TTW hand back all the tests and other grades that have not yet been returned. TTW explain the grading procedure for select questions from Friday's exam, and explain to the students what the teacher wants to see in a full, correct a... User Rating: Grade Level: K-12 137.review distance and midpoint ...CTIVES The students will review the Distance and Midpoint formulas and the pythagorean theorem (DOK 2, Algebra 1, 4.b) MATERIALS pencil and paper dry erase markers transparencies and wet erase markers overhead Do Now- problems involving exponent operations. Set- Hello class, last class we went over polynomial models with Mr. Sealand. This class were going to review some concepts from earlier on in the week. These will include exponents and radicals, Distance formula, Midpoint formula and Pythagorean theorem. We're going to review for the test next period so it would be wise to pay attention. Procedure: A. The stud... User Rating: Grade Level: K-12 138.4b and 4c exam LP 071108 ...ine segment on a Cartesian grid. (DOK 2 - Algebra I, 4.b) 3. TSW apply the Pythagorean theorem to find the hypotenuse, a missing side length, or to determine whether or not a triangle is a right triangle. (DOK 3 - Algebra I, 4.b) 4. TSW solve problems for area and perimeter using simple and complex polynomials. (DOK 3 - Algebra I, 4.c) B. MATERIALS Four (4) Exams Do Now: [5 minutes] Clear your desks of everything except for: Calculator Pencil (or pen) Booklets Spare paper C. SET When Reggie Bush walks on the field at the Super Dome in New Orleans, what do you think is going through his head? This is your chance to... User Rating: Grade Level: K-12 139.Pythagorean Theorem 071008 ...thagorean theorem. E. CLOSURE Today we re-used our knowledge of area and perimeter again, but this time for a figure for which we did not know all the side lengths. How did we find the missing side length? What is the Pythagorean theorem? For the remainder of this block, we are going to review some polynomial work. F. ASSESSMENT - TSW apply the Pythagorean theorem to triangles with missing lengths in order to solve perimeter and area questions. (DOK 2 - Algebra I, 4.b) Informal: TTW monitor student progress by verbally checking (M) for understanding of the Pythagorean theorem(C). Formal: Students will take a test... User Rating: Grade Level: K-12 140.Midpoint Formula 070908 ...bonus word problems. [any remaining time] E. CLOSURE Today I wanted you to be able to use the midpoint formula to determine the midpoint of a line segment. What do you remember about the formula? It's just the average of x values and the average of y values. Tomorrow we will be moving back to both polynomials and shapes with perimeter and area. F. ASSESSMENT - TSW apply the midpoint formula to determine the midpoint of a line segment. (DOK 2 - Algebra I, 4.b) Informal: TTW monitor student progress by verbally checking (M) for understanding of the proper application of the midpoint formula while the students work...
Description: This book gives a clear rational explanation of some basic mathematical concepts, dispelling the usual rote and how-to approaches that make too many people dislike mathematics. Just as the artist wishes people to enjoy his art, I wish people toMore... This book gives a clear rational explanation of some basic mathematical concepts, dispelling the usual rote and how-to approaches that make too many people dislike mathematics. Just as the artist wishes people to enjoy his art, I wish people to enjoy the beauty of rational thinking and mathematics. This very useful, easy-to-read book will help students and the public understand and appreciate
consists of the textbook plus an access kit for MyMathLab/MyStatLab. #xA0; The second book of the Mathematics in Actionseries, An Introduction to Algebraic, Graphical, and Numerical Problem Solving, Third Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and accompanying practice exercises. #xA0; This unique approach helps students increase their knowledge of mathematics, sharpen their problem-solving skills, and raise their overall confidence in their ability to learn. Technology integrated throughout the text helps students interpret real-life data algebraically, numerically, symbolically, and graphically. The active style of this book develops students#x19; mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines. #xA0; MyMathLabprovides a wide range of homework, tutorial, and assessment tools that make it easy to manage your course online. The Consortium for Foundation Mathematics is a team of fourteen co-authors, primarily from the State University of New York and the City University of New York systems. Using the AMATYC Crossroads standards, the team developed an activity-based approach to mathematics in an effort to reach the large population of college students who, for whatever reason, have not yet succeeded in learning mathematics. Chapter 1. Number Sense Cluster 1. Introduction to Problem Solving Activity 1.1 The Bookstore Objectives: 1. Practice communication skills. 2. Organize information. 3. Write a solution in sentences. 4. Develop problem-solving skills. Activity 1.2 The Classroom Objectives: 1. Organize information. 2. Develop problem-solving strategies. • Draw a picture. • Recognize a pattern. • So a simpler problem. 3. Communicate problem-solving ideas. Activity 1.3 Properties of Arithmetic Objectives: 1. Identify and use the commutative property in calculations. 2. Use the distributive property to evaluate arithmetic expressions. 3. Use the order of operations convention to evaluate arithmetic expressions. 4. Identify and use the properties of exponents in calculations 5. Covert numbers to and from scientific notation. 6. Identify, understand, and use formulas. 7. Use the order of operations convention in formulas involving whole numbers.
Matrix Computations The use of numerical methods continues to expand rapidly. At their heart lie matrix computations. Written in a clear, expository style, it allows ...Show synopsisThe use of numerical methods continues to expand rapidly. At their heart lie matrix computations. Written in a clear, expository style, it allows students and professionals to build confidence in themselves by putting the theory behind matrix computations into practice instantly. Algorithms that allow students to work examples and write programs introduce each chapter. The book then moves on to discuss more complicated theoretical material. Using a step-by-step approach, it introduces mathematical material only as it is needed. Exercises range from routine computations and verifications to extensive programming projects and challenging proofs significantly revised and improved introduction to a...New. A significantly revised and improved introduction to a critical aspect of scientific computationMatrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understan
More About This Textbook Overview This is a gentle introduction to the vocabulary and many of the highlights of elementary group theory. Written in an informal style, the material is divided into short sections, each of which deals with an important result or a new idea. Includes more than 300 exercises and approximately 60 illustrations. Editorial Reviews Booknews Textbook introduces the vocabulary and some highlights of elementary group theory. Emphasis is placed on concrete examples so that finite rotation groups and the 17 wallpaper groups are treated in detail alongside theoretical results. Over 300 exercises and some 60 illustrations. Annotation c. Book News, Inc., Portland, OR (booknews.com) From the Publisher M.A. Armstrong Groups and Symmetry "This book is a gentle introductory text on group theory and its application to the measurement of symmetry. It covers most of the material that one might expect to see in an undergraduate course . . . The theory is amplified, exemplified and properly related to what this part of algebra is really for by discussion of a wide variety of geometrical phenomena in which groups measure symmetry. Overall, the author's plan, to base his treatment on the premise that groups and symmetry go together, is a very good one, and the book deserves to succeed."—MATHEMATICAL REVIEW
The purpose of this letter is to acquaint you with the instruments used in math placement from grade eight to grade nine and to provide you with the ranges we use as guidelines. Math placement is not something that happens at the end of the year. Rather, it is an ongoing process with specific milestones throughout the year. The math department also feels it is important to include a general description of a student in each course. These characteristics are: College Prep: Algebra I Algebra students demonstrate emerging insights into algebraic thinking but need to deepen their understanding of algebraic concepts and skills. This course takes a traditional approach to algebra at a moderate pace with extensive review. Students in this course may or may not have taken algebra previously and would take any of the geometry classes listed below in their sophomore year. College Prep III: Algebra 1 This is a year long algebra course for students who have not taken algebra before. Topics are explored in less depth than CP Algebra I due to the slower pace of the class and the additional time for in-class practice and review. Students in this course typically take College Prep II or III Geometry in their sophomore year. College Prep II: Geometry with Algebra review Students can follow a model and reach an abstract level of understanding but do not consistently show mathematical insights in problem solving. They learn well from periods of direct instruction in combination with in-class guided practice. Given several models, they are able to select the one which works best for them and master it. This is a fast paced course which teaches paragraph style geometric proof and includes regular review of new material as well as a good deal of algebra review. College Prep I: Geometry Students understand and analyze complex situations with guidance. Able to understand and apply general principles of mathematics, they visualize concepts and relationships. They can assimilate and reproduce ideas in variations of problem solving situations already experienced. Given several models, students understand and use several. They recall previous skills and topics. This is a fast paced course which teaches traditional two column proof and expects proficiency with minimal review. Honors Geometry Students can understand concepts and relate those concepts to new situations. They have the ability to think abstractly, are creative problem posers and problem solvers, have insights into complex relationships, and see patterns beyond expectations. They execute skills reliably and demonstrate proficiency of previous topics and skills. Students are able to complete tests in the allotted time. The honors course has an accelerated pace with limited review. We use five formal instruments in determining placement. They are listed in order of importance. Students curretnly in Guided or Indepenent Algebra take the following: Test Average: weighted once The test averages for each term are considered as well as the overall test average. Calculators are allowed on most tests. Midyear Exam: weighted once The midyear exam includes material covered in the first half of the year. It is usually given the week prior to February vacation. American Mathematics Competition (AMC8): weighted once This 25 question multiple choice test is given in November during a 40 minute class period. Calculators are allowed. The items cover computation, geometry, interpreting graphs and problem solving. National Mathematics League Contests (NML): weighted one-half The contests require students to apply algebraic concepts which they have learned. Some questions are fairly direct applications while others require some integration of knowledge. The six question contest is given in January. No calculators are allowed. Students currently in Directed take the following: Iowa Algebra Prognosis (in place of Geometry Prognosis) The Iowa Algebra Prognosis Test is a 63 question multiple choice test which assesses algebra readiness skills. Calculators are not allowed. The four part test takes about 40 minutes and is given in January. Continental Math League (CML in place of NML) These contests are given monthly from November through February. Each consists of six non-routine problems which are to be solved in 25 minutes without the aid of a calculator, ruler, protractor or graph paper. Students may use any method they like and are required only to give an answer. The chart below provides the benchmarks used to place students in the ninth grade. Since no placement instrument is perfect, there are some overlaps in the ranges to allow teachers some flexibility for individual differences and special circumstances. CPIII Algebra CP Algebra CPII Geometry CPI Geometry Honors Geometry GeometryPrognosis (raw) up to 20 19-24 23 to 30 30 or above Iowa Prognosis (raw) up to 35 32 or above Test average (% ) and midyear (guided and independent) up to 75 70 to 85 80 to 95 92 or above Test average and midyear (%) directed up to 82 75 and above AMC8 (raw) up to 7 up to 7 6 to 11 10 to 15 15 or above NML (raw) up to 2 2 to 4 3 to 5 5 to 6 CML (%) up to 25% 20% or above We hope that having this information in the fall helps parents and students set appropriate goals. To further insure this, we would like to invite all CMS parents to the Math Placement meeting to answer any questions you may have regarding math placement. If possible, forward your questions to either office so that we may best address the needs of the group.
Free mathematics/graphing program that will allow you to develop and visually analyse mathematical expressions quickly and easily. Boasts a powerful editor/calculator and a unique graphing module that permits expressions with up to 8 parameters to be plotted directly, such as y=Asin(kx+b). An excellent tool for visualizing functions and their dependencies. Intuitive, simple to use, and suitable for everyone from students to theoretical physicists. ... Most popular algebra software downloads in Mathematics needs to solve problems ranging from simple elementary algebra to complex equations. Its underling implementation encompasses high precision, sturdiness and multi-functionality. MultiplexCalc also has the unlimited ability to extend itself by using user-defined variables. You can add your own variables to MultiplexCalc in order to convenience your work. Any instance ... Simple Solver is a free Windows application that can simplify computer logic systems, Boolean equations, and truth tables. The application includes six different tools:Logic Design Draw, Logic Simulation, Logic Design Auto, Boolean, Permutation and Random Number. These tools are built on years of engineering design experience and are intended for both ... ... needs to solve problems ranging from simple elementary algebra to complex equations. InnoCalculator also has the unlimited ability to extend itself by using user-defined variables. You can add your own variables to InnoCalculator in order to convenience your work. Any instance of a formula can be parsed once, and calculated ... PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a ... mathematical entities such as matrices, polynomials, power series, algebraic numbers etc., and a lot of transcendental functions. PARI ... The Math quizzes freeware a math quizzer designed to help students to improve there math skills by random math quizzes. The application is designed for all ages and have a number of difficult types. There is also a math game. The math quizzer is also connected to a web site in ... ... very intuitive approach. Most equations are supported, including algebraic equations, trigonometric equations, exponential equations, parametric equations. DeadLine solves equations graphically and numerically. It displays the graph of the function and a list of the real roots of the equation. You can evaluate the function and the first two derivatives, find ... ... way. The program covers the areas Analysis, Geometry, Algebra, Stochastics,Vector algebra.MathProf helps Junior High School students with problems in Geometry and Algebra. High School and College students, seeking to expand ...
... More About This Book that they understand, and with the expectation that their backgrounds may have some gaps. Smith/Minton provide exceptional, reality-based applications that appeal to students' interests and demonstrate the elegance of math in the world around us. New features include: • Many new exercises and examples (for a total of 7,000 exercises and 1000 examples throughout the book) provide a careful balance of routine, intermediate and challenging exercises • New exploratory exercises in every section that challenge students to make connections to previous introduced material. • New commentaries ("Beyond Formulas") that encourage students to think mathematically beyond the procedures they learn. • New counterpoints to the historical notes, "Today in Mathematics," stress the contemporary dynamism of mathematical research and applications, connecting past contributions to the present. • An enhanced discussion of differential equations and additional applications of vector calculus. • Exceptional Media Resources: Within MathZone, instructors and students have access to a series of unique Conceptual Videos that help students understand key Calculus concepts proven to be most difficult to comprehend, 248 Interactive Applets that help students master concepts and procedures and functions, 1600 algorithms , and 113 e-Professors
books.google.com - This book contains selected topics from the history of geometry, with 'modern' proofs of some of the results, as well as a fully modern treatment of selected basic issues in geometry. It is geared towards the needs of future mathematics teachers. One of my goals for this book is to open up for the dynamic...
the theory of numbers. An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory ...Show synopsisAn Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory. Updates include a chapter by J. H. Silverman on one of the most important developments in number theory -- modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader. The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists An introduction to the theory of numbers. The book has a lot of good information, but the style is a bit dated. It's a lot of the basic ideas of number theory - prime numbers, etc - but the format makes it a little difficult to follow, since proofs are not blocked out in the way a more modern math book would do it. Still, it's an excellent
Longmont Math extend your skills to thinking mathematically and further developing organized problem solving skills. Calculus is the pathway to solutions to problems you can solve in no other way. Without it, we wouldn't have many of the modern conveniences and technologies and other achievements we take for granted in today's world
Precalculus Functions And Graphs 9780495108375 ISBN: 0495108375 Edition: 11 Pub Date: 2007 Publisher: Thomson Learning Summary: Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text popular among students year after year. This latest edition of Swokowski and Cole's PRECALCULUS: FUNCTIONS AND GRAPHS retains these features. The problems have been consistently praised for being at just the right level for precalculus students like you. The book also p...rovides calculator examples, including specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics. Swokowski, Earl W. is the author of Precalculus Functions And Graphs, published 2007 under ISBN 9780495108375 and 0495108375. One hundred fifty six Precalculus Functions And Graphs textbooks are available for sale on ValoreBooks.com, thirty one used from the cheapest price of $9.29, or buy new starting at $164 readable copy. All pages and cover are intact. Dust jacket is torn or missing. The book has moderate to heavy wear. Covers have wear; Edges are yellowed and/or dirty; The c [more] A readable copy. All pages and cover are intact. Dust jacket is torn or missing. The book has moderate to heavy wear. Covers have wear; Edges are yellowed and/or dirty; The corners of the covers are bent and worn
We then move to: 1. exponential and logarithmic functions including the natural logarithm. 2. trigonometric and hyperbolic functions. 3. the derivative and applications involving the derivative 4. the integral and applications involving the integral. I use the student's own text book along with
Post a reply Topic review (newest first) irspow 2005-12-06 07:43:34 You can't learn physics without calculus. You would just be memorizing formulas otherwise. Then you would only be able to solve problems that have already been figured out. It would be like an "artist" who could only paint one picture again and again. ryos 2005-12-05 11:27:08 I'm in a college physics course that requires calculus. Ricky 2005-12-05 11:10:08 "ignore air resistance and friction, my eye. Teach me something I can actually use, and then I might be interested." Friction is related to speed. This makes things very complicated, because friction changes the speed and the speed change the friction. This lead to differential equations such as: 3y' + 5y = cos(x), solve for y, and it can't be in terms of y'. And that's a simple one! The problem with equations like this is it takes a full understanding of integrals and derivatives to solve (basic calculus). This is why they aren't taught at a high school level. But as soon as you get the advanced math, physics becomes frighteningly accurate (as well as difficult). ryos 2005-12-05 10:47:05 Physics uses math to model reality. Some models work better than others. In that sense both are abstract, and both are real: physics is abstract until it accurately describes reality, and math is abstract until it is used to model real situations. Don't get me started on Physics...(ignore air resistance and friction, my eye. Teach me something I can actually use, and then I might be interested.) math_phys_bio 2005-12-05 09:49:12 what could be the relation between these two? math is abstract and physics is part of the physical world.
This Teaching Notes and Solutions Manual accompanies Singapore Math's sold-separately Discovering Mathematics Textbook 7A, Common Core Edition. It contains one page of teaching notes for each chapter, with a paragraph for each concept and an overall suggested approach. Class Activities are included from the student text with answer and objectives. Fully-worked solutions for all questions and problems in the textbook are included. 174 pages, softcover. Click Here. Double your traffic. Get Vendio Gallery - Now FREE! You are buying One New Zimmer Orthopaedic Rush Pin Driver-Extractor for 1/4 pins. Please check out my other Zimmer Twist Drill Bits and Orthopaedic instruments Buy e-Study Guide for: Evolution: Making Sense of Life by Carl Zimmer, ISBN 9781936221363 by Cram101 Textbook Reviews and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones! $2.99 Kobo eBooksElliptic Functions - An Elementary Text-Book for Students of Mathematics eBook Free Shipping /elliptic-functions-an/v6VBGaw0NBXDOHU7ze29GQ==/info Kobo eBooks ( In stock ) Store not rated Free Shipping Buy Elliptic Functions - An Elementary Text-Book for Students of Mathematics by Arthur Baker and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones! Free Worldwide Delivery : Essential Mathematics for Cambridge IGCSE Student Book : Paperback : Oxford University Press : 9780199128747 : 019912874X : 02 Feb 2012 : Written by an examiner to exactly match the 0580 syllabus, this text gives you a sequential and logical teaching path through the full syllabus, addressing the four curriculum areas. A full set of solutions, exam practice and worked examples encourage active learning, and a CD offers dynamic material for every section of the book. hananasi Store ZIMMER DRIVER BENDER EXTRACTOR BONE PIN RUSH CAGE 67239 ZIMMER DRIVER-BENDER-EXTR ACTOR-BONE PIN, RUSH REF: 0693-00 71351200 NEW, SEE PICTURES FOR ADDITIONAL DETAILS PRICE PER UNIT 2 AVAILABLE QUESTIONS: EBAY Shipping Policy - All of the items I sell on eBay are in stock. Once payment is received,I will ship out the items same day or next. SALES ARE FINAL PLEASE DON'T ASK ABOUT OUR PRODUCTS AUTHENTICITY 100% AUTHENTIC All of our merchandise is marketed on numerous e-commerce sites Free Worldwide Delivery : About Teaching Mathematics : Paperback : Math Solutions Publications : 9780941355766 : 0941355764 : 04 Sep 2007 : Marilyn Burns' signature resource continues to be at the forefront of current trends and practices for improving the teaching and learning of mathematics. Supporting a student- centered, problem-solving approach proven by research to best develop students' thinking and reasoning of mathematics, Marilyn presents invaluable insights in a standards- driven, practical, and accessible manner. This essential resource includes more than 240 classroom-tested activities and presents the content and pedagogy... Free Worldwide Delivery : Teaching Mathematics : Paperback : Oxford University Press Australia : 9780195568455 : 0195568451 : 12 Jan 2012 : I was never any good at maths. I dropped it as soon as I could How many times have you heard your students make a similar statement when asked to study mathematics? Teaching Mathematics counters this with a rich set of practical pedagogy that will connect students to the bigger picture of mathematics and show them how to: Communicate mathematically Feel positive about mathematics and their role in teaching it Enter the classroom confident they are equipped with the practical knowledge, skills
Home of the Redhawks More Useful Links Pre-Algebra 2 - Grade 7 This course includes an introduction to algebra, integers and negative exponents, rational numbers, percentages, and strategies for solving equations and inequalities. Topics in geometry, graphing of linear equations, areas and volume, ratio and proportion, and the Pythagorean Theorem are also studied. My "office hours" are Monday and Friday mornings after 745 and Tuesday, Wednesday, and Thursday after school until 4pm.
hands to explore and build proficiency and eventually to replicate... I've previously taken regular calculus classes with engineers and won... This is not the same at all. We were solving real problems every day...Students work through problems using an online interactive textbook D...When teams become stuck on a problem Chiel or a teaching assistant m... hands, to explore and build proficiency, and, eventually, to replicate and build on recent math models used in the biological sciences. The course is cross-listed as both a biology and biomedical engineering class. "I've previously taken regular calculus classes with engineers and wondered what would the classes ever be useful for," said Kate Coyle, a biology major who completed the Dynamics class and graduated this semester. "Labs I've had in biology and physics show you the protocol and the expected result. "This is not the same, at all. We were solving real problems every day." Students work through problems using an online interactive textbook, Dynamics of Biological Systems: A Modeling Manual Chiel wrote and the computer programming language Mathematica, which scientists worldwide rely on to build mathematical models of complex systems. Chiel's book is available free to students as well as teachers who may want to use it as is or as a model for their own classes. When teams become stuck on a problem, ,Chiel or a teaching assistant makes suggestions, gives clues and tries to coax out the answer. After success, teachers quiz individuals about how they found the solution and what they'd learned. The class of 30 is spread out among hexagonal tables. Teams power up their laptops and go to work. Each day the teachers rotate to a different group of students, and after each class they compare notes on who has mastered the skills and who needs extra help, Gill said. When the second half of the semester begins, teams choose a mathematical model that was recently published in a scientific journal, begin reconstructing and analyzing it and then writing in detail what they learn. The students then extend the model to answer new questions that they ask themselves, and write up results as if they were writing for a scientific journal. Coyle and her teammates Valencia Williams and Joshua DeRivera focused on a pa
Beginning And Intermediate Algebra An Integrated Approach 9780495117933 ISBN: 0495117935 Edition: 5 Pub Date: 2007 Publisher: Thomson Learning Summary: Easy to understand, filled with relevant applications, and focused on helping students develop problem-solving skills, BEGINNING AND INTERMEDIATE ALGEBRA is unparalleled in its ability to engage students in mathematics and prepare them for higher-level courses. Gustafson and Frisk's accessible style combines with drill problems, detailed examples, and careful explanations to help students overcome any mathematics anx...iety. Their proven five-step problem-solving strategy helps break each problem down into manageable segments: analyze the problem, form an equation, solve the equation, state the conclusion, and check the result. Examples and problems use real-life data to make the text more relevant to students and to show how mathematics is used in a wide variety of vocations. Plus, the text features plentiful real-world application problems that help build the strong mathematical foundation necessary for students to feel confident in applying their newly acquired skills in further mathematics courses, at home or on the job. Gustafson, R. David is the author of Beginning And Intermediate Algebra An Integrated Approach, published 2007 under ISBN 9780495117933 and 0495117935. Two hundred forty two Beginning And Intermediate Algebra An Integrated Approach textbooks are available for sale on ValoreBooks.com, one hundred thirty eight used from the cheapest price of $14.50, or buy new starting at $109
Calculus is one of the greatest achievements of the human intellect. Inspired by problems in astronomy, Newton and Leibniz developed the ideas of calculus 300 years ago. Since then, each century has demonstrated the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the social and biological sciences. Calculus has been so successful both because its central theme—change—is pivotal to an analysis of the natural world and because of its extraordinary power to reduce complicated problems to simple procedures. This book offers a fresh approach to algebra that focuses on teaching readers how to truly understand the principles, rather than viewing them merely as tools for other forms of mathematics. It relies on a storyline to form the backbone of the chapters and make the material more engaging. Conceptual exercise sets are included to show how the information is applied in the real world. Using symbolic notation as a framework, business professionals will come away with a vastly improved skill set. In this long-awaited follow-up to the best-selling first edition of How to Draw Cars Like a Pro, renowned car designer Thom Taylor goes back to the drawing board to update his classic with all-new illustrations and to expand on such topics as the use of computers in design today. Taylor begins with advice on selecting the proper tools and equipment, then moves on to perspective and proportion, sketching and cartooning, various media, and light, shadow, reflection, color, and even interiors. Written to help enthusiasts at all artistic levels, his book also features more than 200 examples from many of today's top artists in the automotive field. Updated to include computerized illustration techniques
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
Featured Research from universities, journals, and other organizations The aftermath of calculator use in college classrooms Date: November 12, 2012 Source: University of Pittsburgh Summary: Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, experts say. They have proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students. Share This Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, says Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research & Development Center. King has proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students published in the British Journal of Educational Technology. "We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard." Together with Carol Robinson, coauthor and director of the Mathematics Education Centre at Loughborough University in England, King examined whether the inherent characteristics of the mathematics questions presented to students facilitated a deep or surface approach to learning. Using a limited sample size, they interviewed 10 second-year undergraduate students enrolled in a competitive engineering program. The students were given a number of mathematical questions related to sine waves -- a mathematical function that describes a smooth repetitive oscillation -- and were allowed to use calculators to answer them. More than half of the students adopted the option of using the calculators to solve the problem. "Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," said King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values." After completing the problems, the students were interviewed about their process. A student who had used a calculator noted that she struggled with the answer because she couldn't remember the "rules" regarding sine and it was "easier" to use a calculator. In contrast, a student who did not use a calculator was asked why someone might have a problem answering this question. The student said he didn't see a reason for a problem. However, he noted that one may have trouble visualizing a sine wave if he/she is told not to use a calculator. "The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes -- especially at the undergraduate level," said King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area." King also suggests that relevant research should be done investigating the correlation between how and why students use calculators to evaluate the types of learning approaches that students adopt toward problem solving in mathematics have developed a classroom design that gives instructors increased flexibility in how to teach their courses and improves accessibility for students, while slashing administrative ... full story May 21, 2012 — Discipline-based education research has generated insights that could help improve undergraduate education in science and engineering, but these findings have not yet prompted widespread changes in
Bowdon Junction CalculusFrequently used Microsoft PowerPoint during MBA coursework. Can help you use templates, create appealing transitions, and link or embed various items. Tutored Prealgebra topics during high school and college
More About This Textbook Overview Algebra is accessible and engaging with this popular text from Charles "Pat" McKeague! ELEMENTARY AND in every chapter of this user-friendly book highlight the relevance of what you are learning. And studying is easier than ever with the book's multimedia learning resources, including CengageNOW for ELEMENTARY AND INTERMEDIATE ALGEBRA, a personalized online learning companion. Related Subjects Meet the Author includ
Introduction to the Mathematics of Money: Saving and Investing - 07 edition Summary: This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given. Proofs using induction recurrence relations and proofs by contradiction are covered. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are use...show mored. Basic topics in probability and statistics are presented. The student is introduced to elements of saving and investing that are of life-long practical use. These include savings and checking accounts certificates of deposit student loans credit cards mortgages buying and selling bonds and buying and selling stocks. The book is self contained and accessible. The authors follow a systematic pattern for each chapter including a variety of examples and exercises ensuring that the student deals with realities rather than theoretical idealizations. It is suitable for courses in mathematics investing banking financial engineering and related topics. ...show less105.82 +$3.99 s/h VeryGood Herb Tandree Philosophy Books Stroud, Glos, 2006
Add song and dance to your mathematics lessons with this exercise, in which students discover the various covert mathematical relations hidden in the lyrics to the song ?The Twelve Days of Christmas.? In addition to ... A brief introduction to the field of continued fractions, including some basic theory about the subject; the history of continued fractions, tracing some of the major developments in the field in the past 2500 years;... An undergraduate tutorial on wallpaper groups, with illustrations of the 17 wallpapers groups and a wallpaper gallery. Technically, the derivation of all discontinuous groups of rotation and translation in the plane.... Lessons with an Internet component, developed for GirlTECH'95, a program funded by CRPC Rice University and the RUSMP. A Functional Housing Market asks students to search the Internet for housing prices and compare them...
I think it depends largely on what your friend already knows, and how he studies, but if your just looking for a list of books, there are many similar questions out there with responses in this form. – EthanApr 27 '13 at 21:58 3 Answers The Art of Problem Solving website has some great articles and resources with respect to number theory, e.g., the nuts and bolts aimed to very bright young folks. That would be a terrific start. (Author: Naoki Sato). See also this post for some great references, all varying with respect to complexity. One recommendation on that post that seems particularly relevant is the book by Rosen: Elementary Number Theory. Also, this post offers recommendations for the self study of elementary number theory geared to high-achieving high-school students, and undergraduates. There are plenty of answers on past SE threads, but I'll recommend my favourite anyways: The Higher Arithmetic by H. Davenport is a quite difficult and very rigorous introduction, but is a wonderful text that can also serve as a reference. It assumes no prior knowledge at all and has exercises ranging from introductory to very difficult. John Stillwell, Elements of Number Theory (Springer 2002). This is by a masterly expositor, and is particularly approachable. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (OUP 1938, and still going strong with a 6th edition in 2008). Also aimed at beginning undergraduate mathematicians and pleasingly accessible. Alan Baker, A Comprehensive Course in Number Theory (CUP 2012) is a nice recent textbook (shorter than its title would suggest, too). An Introduction to the Theory of Numbers is a wonderful, fascinating text, but I wouldn't say it's a good introduction, though it can be interesting to read even when it becomes hard to follow. Unfortunately it has no exercises and moves very very quickly at some stages, requiring some alternative reference for all but the most Ramanujanish beginners, in my opinion. – user54147Apr 27 '13 at 22:07 @LevLivnev You could be right! The students round my ivory tower are an unrepresentative lot – Peter SmithApr 27 '13 at 22:30
Questions About This Book? The Used copy of this book is not guaranteed to inclue any supplemental materials. Typically, only the book itself is included. Summary Today's algebra students want to know thewhybehind what they are learning and it is this that motivates them to succeed in the course. By focusing on algebra in a real-world context, Gary Rockswold gracefully and succinctly answers this need. As many topics taught in today's college algebra course aren't as crucial to students as they once were, Gary has developed this streamlined text, covering linear, quadratic, nonlinear, exponential, and logarithmic functions and systems of equations and inequalities, to get to the heart of what students need from this course. By answering thewhyand streamlining thehow, Rockswold has created a text to serve today's students and help them to truly succeed.
And not even because this is essentially the ultimate in cheating on your homework, but because being able to see the work that leads to the correct answer would have helped me understand everything so much better. I… I never have to take a math class ever again in my life but oh my god?? this would have saved me so many tears??? holy shit I'm so unhappy that this is only a thing right now internet you've actually failed me Sophie, in case you have to take any other math classes. oh man this would have made the first 3 years of high school so much easier Awesome! Another great one is if you're interested ^^ It gives you some of the answers along with work from your textbook if they have it!
Transit of Venus Mathematics -- This problem book covers 17 specific mathematics problems that are common to studying the transit of venus more critically. These include: geometry of chords and triangles, working with angular measure, unit conversion exercises, parallax and angular size, angular speed, time zone conversions, working with proportions, a few problems involving trig! The majority of the problems may be undertaken by grade 7-9 students. This resource is also helpful for the amateur astronomer wishing to gain a better understanding of parallax, transit geometry and how the Astronomical Unit can be deduced.
Rent Textbook Buy New Textbook Currently Available, Usually Ships in 24-48 Hours $235Engineers looking for an accessible approach to calculus will appreciate Precalculus, 2nd Edition. The book offers a clear writing style that helps reduce any math anxiety they may have while developing their problem-solving skills. It incorporates Parallel Words and Math boxes that provide detailed annotations which follow a multi-modal approach. Your Turn exercises reinforce concepts by allowing them to see the connection between the exercises and examples. A five-step problem solving method is also used to help engineers gain a stronger understanding of word problems
A 15,000 sq. ft. state of the art model factory is the centerpiece of the Advanced Integrated Manufacturing (AIM) Center, which offers workforce development, production support, and process improvement services for... A unit designed to improve students' understanding and appreciation of basic geometric shapes used in architecture. It describes various plane geometric figures and discusses in detail the properties of several of these... Part of MIT\'s innovative OpenCourseWare Project, that provides materials from MIT classes to the public on the web, the site includes materials from an introductory expository writing course that focuses on theAn unusual and attractive edition of Euclid was published in 1847 in England, edited by an otherwise unknown mathematician named Oliver Byrne. It covers the first 6 books of Euclid, which range through most of...
Sufficiently rigorous for introductory or intermediate graduate courses, this text helps students acquire valuable investigative tools, presenting a comprehensive scope of the techniques and limitations of statistical mechanics. Clear and readable, the treatment follows a logical development from ele... read more Introduction to Knot Theory by Richard H. Crowell, Ralph H. Fox Appropriate for advanced undergraduates and graduate students, this text by two renowned mathematicians was hailed by the Bulletin of the American Mathematical Society as "a very welcome addition to the mathematical literature." 1963
Reinforces student understanding with detailed explanations, worked-out examples, and practice problems. Lists key ideas to master and builds problem-solving skills. There is a section in the Study Guide corresponding to each section in the text.
Calculus is one of the greatest achievements of the human intellect. Inspired by problems in astronomy, Newton and Leibniz developed the ideas of calculus 300 years ago. Since then, each century has demonstrated the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the social and biological sciences. Calculus has been so successful both because its central theme-change-is pivotal to an analysis of the natural world and because of its extraordinary power to reduce complicated problems to simple procedures. The Calculus 3 Tutor Volume 1 is a 10 hour DVD course that picks up right where the very popular Advanced Calculus 2 Tutor ends and teaches students how to do well in Calculus 3 by fully worked example problems. This DVD course is essential for any student taking Calculus 3 at the university level. The Calculus 3 Tutor Volume 2 is a 11 hour DVD course that picks up right where the very popular Calculus 3 Tutor: Volume 1 ends and continues to teach students how to do well in Calculus 3 by fully worked example problems. This DVD course is essential for any student taking Calculus 3 at the university levelCalculus can be an intimidating subject if you don't have a good teacher. It is important to have a teacher that takes things step-by-step so the students don't get lost. That is exactly what this DVD set provides. I have tutored many many people in Math through Calculus and I have found that if you start off with the basics and take things one step at a time - anyone can learn complex Math topics. This 2-DVD set contains 8 hours of fully worked example problems in Calculus 1 and 2. After viewing this DVD course in Calculus 1 & 2 you'll discover that the material isn't hard at all if it is presented in a clear manner. No knowledge is assumed on the part of the student. Each example builds in complexity so before you know it you'll be working the 'tough' problems with ease!Understanding Calculus: Problems, Solutions, and Tips immerses you in the unrivaled learning adventure of this mathematical field in 36 half-hour lectures that cover all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college. With crystal-clear explanations of the beautiful ideas of calculus, frequent study tips, pitfalls to avoid, and-best of all-hundreds of examples and practice problems that are specifically designed to explain and reinforce major concepts, this course will be your sure and steady guide to conquering calculus. Calculus II is the payoff for mastering Calculus I. This second course in the calculus sequence introduces you to exciting new techniques and applications of one of the most powerful mathematical tools ever invented.
Module Description The module gives an introduction to practical techniques for carrying out numerical computations on a range of mathematical problems. Students will be expected to have an elementary acquaintance with Matlab. (An introductory session on Matlab will be provided for those who need it.) 1. An Introduction to Practical Computation - Algorithms - Simple examples - Pitfalls 2. Solving single nonlinear equations - Bisection Method - Regula Falsi - Newton's Method 3. Numerical linear algebra - Gaussian Elimination - Partial pivoting - Iterative methods 4. Numerical solution of ordinary differential equations - Euler's Method - Runge-Kutta Methods - Linear Multi-step Methods 5. Simple approximation - Polynomial interpolation - Optimal interpolation points - Fourier and Trigonometric Series On completion of the module students should be able to: - appreciate the processes and pitfalls of mathematical approximation - demonstrate knowledge and understanding of mathematical computing - motivate and describe the derivation of the numerical algorithms covered in the module - carry out simple numerical processes "by hand" - implement and execute algorithms in Matlab - evaluate, contrast and reflect upon the numerical results arising from different algorithms. Learning & Teaching Methods This module runs at 3 hours per week in the Spring term. There are two lectures and one class / laboratory each week. In the Summer term 3 revision lectures are given. Assessment 30 per cent Coursework Mark, 70 per cent Exam Mark Coursework: The coursework will involve five practical assignments, one associated with each of the five major headings in the module description above. Other details: Information about coursework deadlines can be found in the "Coursework Information" section of the Current Students, Useful Information Maths web pages: Coursework and Test Information
's Basic Math and Pre-Algebra for the Clueless STUDENT TESTED AND APPROVED! If you suffer from math anxiety, then sign up for private tutoring with Bob Miller! Do mathematics and algebraic ...Show synopsisSTUDENT TESTED AND APPROVED! If you suffer from math anxiety, then sign up for private tutoring with Bob Miller! Do mathematics and algebraic formulas leave your head spinning? If so, you are like hundreds of thousands of other students who face math-especially, algebra-with fear. Luckily, there is a cure: Bob Miller's Clueless series! Like the teacher you always wished you had (but never thought existed), Bob Miller brings knowledge, empathy, and fun to math and pre-algebra. He breaks down the learning process in an easy, non-technical way and builds it up again using his own unique methods. Meant to bridge the gulf between the student, the textbook, and the teacher, Basic Math and Pre-Algebra for the Clueless is packed with all the latest information you need to conquer basic math and pre-algebra, including: Anxiety-reducing features on every page Quick tips for solving difficult problems Full explanations of basic principles to make hard problems easy Bite-sized math portions that short study sessions (and attention spans) "I am always delighted when a student tells me that he or she hated math ! but taking a class with me has made math understandable ! even enjoyable." Now it's your turn. Sharpen your #2 pencils, and let Bob Miller show you how to never be clueless again
Curriculum report on mathematics places a renewed emphasis on the practical uses of mathematical skills and ways of thinking. This work reflects this emphasis in its range of entries. Whilst retaining the information required to understand topics such as directed numbers, mean, root and vector, the book now includes entries on, for example, estimate, half life, hypothsis, linear programming and tree diagram. In addition, the layout of the series has been redesigned for the new edition, and the text completely reset. Illustrations are now expanded upon in linked caption text, increased inter-entry spacing is easier on the eye, and running heads at the top of each page improve quick reference. Richard Browne has revised the text of Gem "Basic Facts: Mathematics" for this edition. He has taught mathematics at secondary level for 11 years, and is presently Professional Officer with responsibility for Mathematics at the School Examinations and Assessment Council.
Larson's PRECALCULUS WITH LIMITS is known for delivering the same sound, consistently structured explanations and exercises of mathematical concepts as the market-leading PRECALCULUS, Ninth Edition, w
More About This Textbook Overview The Second Edition of this popular text provides an insightful introduction to the use of finite difference and finite element methods for the computational solution of ordinary and partial differential equations. Readers gain a thorough understanding of the theory underlying the methods presented in the text. The author emphasizes the practical steps involved in implementing the methods, culminating in readers learning how to write programs using FORTRAN90 and MATLAB to solve ordinary and partial differential equations. Related Subjects Meet the Author GRANVILLE SEWELL, PHD, is Visiting Professor of Mathematics at Texas A&M University and Professor of Mathematics at the University of Texas at El Paso. He is the principal developer of PDE2D, a general-purpose partial differential equation solver. Dr. Sewell has written three books and published more than fifty articles on numerical methods
Elementary Linear Algebra with Applications - 3rd edition This book is intended for the first course in linear algebra, taken by mathematics, science, engineering and economics majors. The new edition presents a stronger geometric intuition for the ensuing concepts of span and linear independence. Applications are integrated throughout to illustrate the mathematics and to motivate the student.Edition/Copyright: 3RD 96 Cover: Hardback Publisher: Saunders College Division Published: 09/08/1995 International: No View Table of Contents Preface. List of Applications. 1. Introduction to Linear Equations and Matrices. Introduction to Linear Systems and Matrices. Gaussian Elimination. The Algebra of Matrices: Four Descriptions of the Product. Inverses and Elementary Matrices. Gaussian Elimination as a Matrix Factorization. Transposes, Symmetry, and Band Matrices: An Application. Numerical and Programming Considerations: Partial Pivoting, Overwriting Matrices, and Ill-Conditioned Systems. Review Exercises133.58 +$3.99 s/h New Premier Books BN Romulus, MI 0030103479 Brand New. Exact book as advertised. Delivery in 4-14 business days (not calendar days). We are not able to expedite delivery. $146.25 +$3.99 s/h New bargainforce Naperville, IL 0030103479 New. Looks like an interesting title!
The Guide was designed to support educators by providing an overview of the new test design. Information about how the assessment shifts informed test development and how the CCLS will be measured on the new Regents Exam in Algebra I (CC) is specified. The three webcasts provide: (1) background on the Common Core State Standards along with teaching principles and tools that are crucial for implementing changes in pedagogy and a mathematics curriculum (2) information for teachers on the test design of the Regents Exam in Algebra I (Common Core) and how it measures the Common Core Learning Standards (CCLS) and (3) background information on the sample questions for Algebra I (Common Core). As NY guidance documents have indicated we must simplify radicals as we solve quadratics using the quadratic formula. Anyone using the modules will have to add this supplement to module 4 as they care covering the quadratic formula. You do not need to do extensive work around this, but enough so that students can utilize it as they solve quadratics. New modules are posted for most grade levels. The full Module 1 is up for 6-10 and most module 2′s at the k-5 level. I have a busy week, but will work on getting everything up next week. Until then here is the link to engageny. In the past 20 months of attending training events in Albany this was the best one that I have attended so far. The biggest reason for this is that they had information to share with us and it looks like they will meet their deadline of having Algebra I Modules (units) ready by July 31. For grades 6 – pre-calculus, we were provided a quick look at the first module of every grade level. This quick look included the table of contents for the unit, the standards addressed in that module, a mid-module assessment, and an end of unit assessment. During the presentation we did get some informal looks at actual lessons, but there were none included in the actual module overviews at this time. Direct links to each grade level's module 1 overview and the overall view for 6-8 and 9-12 can be found onthis page on EngageNY. The Algebra 1 modules looked like they are in line with our current thoughts on Algebra and the Common Core. We will need to look at them in more detail when they come out this summer. The Geometry modules place a large emphasis on transformational geometry and make it the foundation for the course. The first two modules of the year are transformational geometry and they use transformations to define and prove congruence, similarity, and triangle congruence theorems. This is a significant shift for us in geometry and is one that we don't have to make. We will have to have a conversation as a PLC and a department about this shift as we look at the Geometry modules next year. The Algebra 2 and pre-calculus modules were not a focus as they did not have the same importance for us, or them, at this point as they are a couple of years away. I did ask if they had talked with the community colleges about the pre-calculus module due to the number of schools that utilize dual credit. They did not have an answer of me. Below is the schedule that they have published for when the modules will be ready and published. Algebra I Common Core Regents Exam The state shared quite a bit of information regarding the Common Core Algebra I exam scheduled to be given next June. Kate Gerson and Ken Slentz both stated that they are unsure of whether the Regents will continue to be public tests or will become secure assessments with limited items released similar to the AP exams and the NYS 3-8 assessments. They did state that either way the tests will continue to be written by NYS teachers as they have been in the past. Kate and two of the psychometricians working on the Algebra I assessment shared the test construction and blueprint information in the assessment packet that I emailed out last Wednesday. These can be found at the beginning of the document and lay out how the test will be constructed. It appears to be in line with their philosophy on the 3-8 exams of assessing more of the major standards and significantly less on the supporting and additional standards. They provide additional details on how they will assess some of the standards in the Standards Clarification section and said that this section will be a living document that will get added to as they continue to construct items and the assessment. They also laid out the number and types of items for the Algebra I, Geometry, and Algebra II exams. This same assessment packet also includes the guiding documents that they use when constructing and reviewing items. These documents can be found in the middle of the document and provide a great insight into how the state is constructing their items to assess standards in a variety of ways. I also asked what they did with their questions that they discard throughout the process and was told that they just sit in their offices. I then suggested that they share those discarded items with a brief rationale as to why it was discarded – too difficult, too easy, bad wording, not appropriate for a regents, etc. We could then take these questions and use them in our own assessment and instruction. They thought this was a great idea and said that they are going to see how they can share them in this manner. At the end of the same assessment packet, you will find some sample items for the Common Core Algebra I regents exam. These sample items are somewhat hidden after the previous regents exam questions that are provided for comparison. It is important to note, that not only do they provide sample items, but also standards, item review information, and scoring guides for each item. We did not get any information on when we can expect to see a similar document for Geometry or Algebra II.
Prentice Hall Algebra 1, Geometry, Algebra 2 makes it easy to teach by providing the teacher with a wealth of resources to meet the needs of a diverse classroom. From extra practice, to performance tasks, to activities, games, and puzzles, Pearson is your one-stop shop for all teaching resources. Teaching resources are available in print, online, and on a DVD. Our digital resources open in a Word document and are completely editable. Example of an Editable Chapter Test Sampler of all Teaching Resources Technology Overview Our Online Lesson Planner is powerful tool and a time saver for all teachers! Just drag and drop the lesson into the calendar and each lesson plan is already created. All the lesson content is completely editable. The teaching resources are also linked to the plan, and teachers have the option to include their own. Easy access to all textbook answers and solutions. Choose formats for printing or for whiteboard display. Personalize and save answer files for the students in all of your classes. Program Facts Support Professional Development
Rent Textbook Buy Used Textbook Buy New Textbook This is a hard-to-find title. We are making every effort to obtain this item, but do not guarantee stock. $185.36 eTextbook We're Sorry Not Available14Enhanced WebAssign - Start Smart Guide for Students Essentials of Trigonometry Essentials of Trigonometry: Student Ilrn Tutorial Student-Essentials of Trigonometry Summary Intended for the freshman market, this book is known for its student-friendly approach. It starts with the right angle definition, and applicationsinvolving the solution of right triangles, to help students investigate and understand the trigonometric functions, their graphs, theirrelationships to one another, and ways in which they can be used in a variety of real-world applications. The book is not dependent upon agraphing calculator.
Description of Saxon Algebra 2: Home School Kit by Saxon Publishers Geared specifically toward the homeschool classroom, Saxon Algebra 2 is a college-prep course designed to build the mathematical foundation necessary for students to transition successfully into higher-level math courses. Students completing Algebra 2 will have studied the equivalent of one semester of informal geometry. Kit includes Student Textbook with 129 lessons, glossary and index; Homeschool Packet with 32 test forms; and Answer Key with answers to all student textbook problem sets and all tests. (Solutions Manual not included). Product: Saxon Algebra 2: Home School Kit Manufactured by: Saxon Publishers Edition Number: 3 Series: Saxon Algebra Binding Type: Hardcover Media Type: Book Minimum Grade: 9th Grade Maximum Grade: 12th Grade Weight: 3.79 pounds Length: 11.22 inches Width: 8.39 inches Height: 1.35 inches Publisher: Saxon Publishers Publication Date: September 2006 Subject: Algebra, Calculus & Trig, Math Curriculum Name: Saxon Learning Style: Auditory, Kinesthetic, Visual Teaching Method: Charlotte Mason, Classical, Traditional, Unit Study, Unschooling There are currently no reviews for Saxon Algebra 2: Home School Kit. Average Rating Parent Rating Comments My son has used Saxon for most of his school years and scored high in math on his ACT test after 8th and 9th grade. This helped my first two children to be ready for calculus in college--they were able to test out of college algebra. My youngest child needed to go at a slower rate and had to take college Math in college. So it is great for those who have a bent for math and science. If the parent is rusty in algebra and cannot be ready to fully explain the concepts when the book explanation is insufficient, then I recommend the DIVE lecture CDs which go along with the Saxon books. It adds a visual and auditory component, and the explanation is usually just enough different to give the student a new perspective and get them through the lesson. Our daughter had difficulty with math when we started using Saxon math curriculum. The spiral organization of the curriculum never gave her the opportunity to 'forget' previously learned concepts, but challenged her to continually progress in each concept. Math is now her top subject.
Holt McDougal Larson Algebra 1 Student Edition - 12 edition Summary: Fully Aligned to the Common Core The Student Editions provide clear, comprehensive coverage of the new Common Core State Standards, with a special focus on the Standards for Mathematical Practice. Focused on Depth of Understanding Lessons and activities are designed to emphasize conceptual understanding, focus on critical thinking and reasoning, and to model mathematics for solving problems that arise in everyday life. Organized to Suppo...show morert the Common Core Teacher resources include correlations to the Common Core State Standards, differentiated pacing guides, and course planners with teaching support to ensure complete coverage of the Common Core State Standards. ...show less 0547647131 WE HAVE NUMEROUS COPIES. HARDCOVER. Mild-Moderate wear to cover,edges, and corners. Minor cardboard showing on top corner of back cover. 1/2 inch tear on front cover along spine has been r...show moreepaired with tape. Minor tears along ends of spine and barcode on spine. Student markings on outside edge of pages. Generally clean inside, minimal writing/highlighting with no detracting writing inside book cover separating from binding, binding still strong ...show less $79.08 +$3.99 s/h Good SellBackYourBook Aurora, IL 0547647131 Item in good condition. Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!! $82.95 +$3.99 s/h New PoplarBooks Carlyle, IL 2012 Hardcover New Unused/unmarked student text. $96
If you have any question about the materials in class, please feel free to ask. The class time is for learning and questions are usually a better way to learn than just taking notes. Class Attendance:You are expected to be at every class. If you are not in class, you are responsible for attaining any notes that were given in class from another student. Frequently missing class will make it hard to be successful in any academic learning. I will hand out an attendance sheet every class and give some extra credits at the end of the semester for good attendance (number of missing class <=5). Grading Policy: Homework 30% Quizzes 20% In-class Exams 30% Final Exam 20% No make-up exams and quizzes will be given. Grading Scale: A:90-100%B:80-89%C:70-79%D:60-69%F:0-59% Homework: Homework will be assigned and done weekly on MyLabsPlus, an online assessment tool to practice the material in textbook. You will need to register on MLP as soon as possible. No late homework will be accepted. Every homework is due Tuesday at 1am. When you are entering answers, you need to make sure your answers are in the form requested. You have 3 tries for each answer. After you enter your answer and click CHECK, if the answer is right, move on to the next problem. If not, try to correct your mistake, it might be a syntax or mathematics error. 11 homework will be assigned in the semester, and the lowest score will be dropped in the final grade. Extra Credit: The Mathematics Department sponsors a Problem of the Week. This problem may be completed for extra credit. Solutions to these problems should come with complete work, not just answers.Each complete and correct Problem of the Week will be worth 0.2% extra credit in the final grade. Quizzes And Exams:There will be a Departmental Final, four in-class hourly exam, and 8 weekly quizzes this semester. There will be a short 15-20 minute quiz given at the end of every Friday class. These quizzes will cover the material over the week. A lowest quiz score and a lowest in-class exam score will not be counted in the final grade.The departmental final exam will cover all material throughout the semester. No make-up quizzes and exams will be given expect under extreme circumstances such as serious illness or family emergency—you should contact me as soon as such an emergency arises and depending on the circumstances, documentation may be requested. In addition no student is to be excused or exempted from taking the Final Exam, the Final Exam must be taken. Calculators:A (graphing) calculator is allowed on all tests unless otherwise specified but cannot also be a communication device (i.e., your iPhone is not a calculator). You are always required to show your work in order to get full credit for a problem. Academic Dishonesty:CheIf you need extra help in any aspect of this course, you can stop by my office to ask questions in my office hours or by appointment. Math help room (Carver 385) is open M-F 9am-4pm, begins 3 Sep. Furthermore, you can also get help from supplemental Instruction (SI): . Students With Disability:If the instructor early in the semester so that your learning needs may be appropriately met. Note, this syllabus is subject to change at any time. If any changes are made, they will be announced in class, and an updated syllabus will be posted on MyLabsPlus.
Full Content is available to subscribers Solutions and Dosage ABSTRACT An understanding of simple arithmetic, through "percentage" and "ratio and proportion," is essential to the accurate preparation and administration of drugs. This textbook provides a clear presentation of simple mathematical relations, explained in terms of both the apothecary and metric systems of units. It contains chapters on arithmetic review, on the preparation of solutions and on dosage, with many exercises and experiments designed to develop facility in accurate calculation. The problems deal with drugs used in modern therapy. This book should prove useful to student nurses, pharmacy students and physicians who wish to review simple arithmetic
Essentials of College Algebra with Modeling and Visualization - 4th edition Gary Rockswold teaches algebra in context, answering the question, &''Why am I learning this?&'' By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswold&'s focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses. This streamlined text covers linear, qu...show moreadratic, nonlinear, exponential, and logarithmic functions and systems of equations and inequalities, which gets to the heart of what students need from this course. A more comprehensive college algebra text isalso26.10 +$3.99 s/h VeryGood Bookbyte-OR Salem, OR Has minor wear and/or markings. SKU:9780321715289-3-0 $26.11 +$3.99 s/h Good SellBackYourBook Aurora, IL 0321715284 Item in good condition. Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!! $27.30 +$3.99 s/h LikeNew Bookbyte-OR Salem, OR Almost new condition. SKU:9780321715289-2-0
Math Trek 04/01/05 The NECTAR Foundation's new Math Trek suite of products ( uses curriculum-based programs that cover the foundations of math for grades 1-12. These engaging programs feature sound, graphics, animation and music clips through interactive tutorials, problem-solving activities, assessment components and student tracking. The Macintosh- and Windows-compatible learning aids include a comprehensive teacher resource document with print support materials, as well as individual, group and culminating performance tasks that incorporate many skills into a meaningful context. The NECTAR Foundation also offers specialized programs for algebra I, calculus and trigonometry
Arithmetic the Easy Way - 4th edition Summary: Everybody uses arithmetic on virtually a daily basis, and this book serves as a handy brush-up for general readers while it also helps students master basic skills that they need before moving up to high-school-level math and beyond. It reviews addition, subtraction, multiplication, and division, then moves on to calculating with fractions, decimals, and percentages. A concluding chapter reviews units of measurement and word problems. Chapters are filled with short p...show moreractice exercises, all of which are answered at the back of the book. The book features many tables, charts, and line illustrations. Barron's Easy Way books focus on both practical and academic topics, presenting fundamental subject matter in clear, understandable language. Equally popular as self-teaching manuals and supplementary texts for classroom use, they are written to help students improve their grades and review subject matter before tests. They are also useful for introducing general readers to a new career-related skill. Easy Way titles cover virtually all subjects that are taught on advanced high school and college-101 levels. New subjects are periodically added, and existing titles are frequently updated to keep them timely and relevant to students' needs. Subject heads and key phrases are set in a second color. ...show less 4Books Squared Dallas, TX Book selection as BIG as Texas. $9
Print Book Key Features @bul:* In-depth coverage of classical number theory * Thorough discussion of the theory of groups and rings * Includes application of Taylor polynomials * Contains more advanced material than other texts * Illustrates the results of a theorem with an example * Excellent presentation of the standard computational exercises * Nearly 1000 problems--many are proof-oriented, several others require the writing of computer programs to complete the computations * Clear and well-motivated presentation * Provides historical references noting distinguished number theory luminaries such as Euclid, de Fermat, Hilbert, Brun, and Lehmer, to name a few * Annotated bibliographies appear at the end of all of the chapters Description Elements of the Theory of Numbers teaches students how to develop, implement, and test numerical methods for standard mathematical problems. The authors have created a two-pronged pedagogical approach that integrates analysis and algebra with classical number theory. Making greater use of the language and concepts in algebra and analysis than is traditionally encountered in introductory courses, this pedagogical approach helps to instill in the minds of the students the idea of the unity of mathematics. Elements of the Theory of Numbers is a superb summary of classical material as well as allowing the reader to take a look at the exciting role of analysis and algebra in number theory. Part I The Fundamentals Introduction: The Primes The Fundamental Theorem of Arithmetic and Its Consequences An Introduction to Congruences Polynomial Congruences Primitive Roots Residues Multiplicative Functions Part II Special Topics Representation Problems An Introduction to Number Fields Partitions Recurrence Relations Quotes and reviews @qu:"I definitely appreciate the unified approach. I think it is important that the students realize that mathematics does not consist of separate entities." @source:--Maureen Fenrick, Mankato State University @qu:"The authors communicate successfully the joy they find in number theory. Students will be excited by learning from this (text)." @source:--Frank DeMeyer, Colorado State University @qu:"The book's biggest advantage is its thorough integration of the relevant algebra into the development. It's about time!" @source:--Thomas McLaughlin, Texas Tech University
Linear Algebra Course Description Prerequisite: MATH 106 or equivalent. Introduces linear algebra with emphasis on interpretation and the development of computational techniques. Topics include systems of equations; matrices are utilized for the interpretation of vector spaces, subspaces, independence bases, dimension, inner product, outerproduct, orthogonal and orthonormal sets. Also the transformation of matrices, matrix operations, inverses, conditions for invertibility, determinants and their properties. The characteristics equation and its eigenvalue are used for problem solving and the development of linear transformations. Three class hours weekly.
Math 360 Assignment, math grammatically Math 360 is "Foundations of Higher Mathematics" and required for Math majors with a pre-req of calc I. It's mainly proofs and logic, set theory and what not. Being the first day, I assume we start off basic.
CSET Single Subject Math Test Preparation There are two possibilities for Math CSET: Foundational level subtests I Algebra and II Geometry for teaching those content areas as well as general math, probability and statistics, and consumer math; and Math CSET including subtest III, Calculus and History of Math, that authorizes teaching of all math courses. Print and study carefully the Mathematics Subject Matter Requirements and use them as your guide preparing for CSET. The theory on CSET is particularly tricky so spend time mastering all theories mentioned in the test descriptions. It is essential to know how to solve the CSET sample questions that you can download from the CSET website. The books mentioned above will help you to understand and solve these sample questions. For Geometry, you should look for ways to prove theorems taught in high school (see frameworks above).
CAS – Computer Algebra System – you can perform both symbolic and numeric calculations, factor and expand expressions and solve for common denominators, learn mathematically-rich lessons in less time. The Texas Instruments TINSPIRECAS Graphic Calculator allows you to save and review work in documents, similar to a computer. This is your perfect companion for college entrance and advanced placement exams. See it Multiple representations let you manipulate and view up to four representations of a math problem at once. Grab it Grab and Move features give you the ability to navigate seamlessly between lines, parabolas and more. Download it The student software (sold separately) enables your home computer to function just like the handheld graphing calculator. So you can use it on the larger screen to easily create, edit and transfer TI-Nspire documents between computer and handheld. Save it Just like a computer, you can save your documents with the Texas Instruments TINSPIRECAS Graphic Calculator. Spec Display Notation Graphic Global Product Type Calculators-Graphing Power Source(s) 4 AAA Batteries Size 1.13" H x 3.94" W x 7.81" L love my nSpire CAS Pros: Ti, nSpire, CAS (Computer Algebra System), great calculator does what you want it to do the first time Cons: CAS = not for standerdized testing, some test will accept it, but it is ;iffy. Does not do 3D graphing. Games and app support is minimal due to restricted programing capability. Other Thoughts: Upgrade to OS 2.0 before initial use. This calculator is great for teaching, learning, and standered practice. It's a calculator, not a gaming device or mp3 player; use it for what it was intended and you'll be happy
Beginning Algebra (Gr 6-8) Shop SchooDoodle.com to buy and access free digital ebook reader pdf downloads for children and schools from our online store and library. We offer over 10,000 electronic book titles to download for elementary school, middle school, and high school students. Preview the books for free on the sample links, download immediately, and own forever! Each month we will be featuring an educational eBook from our catalog for you to use to enhance your classroom curriculum and offer a free download of the book. We now offer over 10,000 electronic book titles to download for elementary school, middle school, and high school students. Preview the books for free on the sample links, download immediately, and own forever! Since we have added this feature, the demand for this technology has grown substantially. Why are eBooks the smart choice? 1. eBooks are less costly than traditional books 2. eBooks are available instantly 3. eBooks take up no physical storage space and minimal electronic storage space Our featured eBook this month is Beginning Algebra (Grades 6-8). This 48 page eBook will help you give your students all the essential tools for a solid introduction to algebra! The skills required to master basic algebra are introduced in Algebra I and developed further in the more advanced Algebra II. A variety of rules, theorems, and processes are presented along with easy-to-follow examples. Games and puzzles use answers to practice problems to reinforce learning and make algebra fun. We think you will love this eBook so much that we would like to offer it to you for free. We have a wide selection of other Pre - Algebra, Beginning Algebra, Algebra II etc. so please see all of our Algebra ebooks that are available for download so teachers and parents can access them any time. For other subjects and grade levels (K-12) see our entire eBook Catalog. We have 10,000 titles for educators to choose from that range in all subject areas. You can preview the books for free on the sample links, download immediately, and own forever!

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