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Based on the author's junior-level undergraduate course, this introductory textbook is designed for a course in mathematical physics. Focusing on the physics of oscillations and waves, A Course in Mathematical Methods for Physicists helps students understand the mathematical techniques needed for … Within a unifying framework, Diffusion: Formalism and Applications covers both classical and quantum domains, along with numerous applications. The author explores the more than two centuries-old history of diffusion, expertly weaving together a variety of topics from physics, mathematics, … Designed for a one-semester course, Introduction to Numerical Analysis and Scientific Computing presents fundamental concepts of numerical mathematics and explains how to implement and program numerical methods. The classroom-tested text helps students understand floating point number … Introduction to Mathematical Modeling and Chaotic Dynamics focuses on mathematical models in natural systems, particularly ecological systems. Most of the models presented are solved using MATLAB®. The book first covers the necessary mathematical preliminaries, including testing of stability. It … Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. Suitable for those who have completed the standard … A Readable yet Rigorous Approach to an Essential Part of Mathematical ThinkingBack by popular demand, Real Analysis and Foundations, Third Edition bridges the gap between classic theoretical texts and less rigorous ones, providing a smooth transition from logic and proofs to real analysis. Along … While there are many available textbooks on quantum information theory, most are either too technical for beginners or not complete enough. Filling this gap, Elements of Quantum Computation and Quantum Communication gives a clear, self-contained introduction to quantum computation and communication … New Textbook! Taking an applied mathematics approach, An Introduction to Exotic Option Pricing illustrates how to use straightforward techniques to price a wide range of exotic options within the Black–Scholes framework. These methods can even be used as control variates in a Monte Carlo simulation of a stochastic volatility model. Celebrate the life of Alan Turing with us by checking out this new book: The Universal Computer: The Road from Leibniz to Turing. Alan Turing Year2012 marks the scientific contributions of this pioneer on the 100th anniversary of his birth on June 23, 1912. Turing's influence on computer science, artificial intelligence, and developmental biology, along with his important work on code-breaking during the Second World War paved the way for a host of future discoveries and innovations.
and self-contained, this penetrating study of the theory of canonical matrices presents a detailed consideration of all the theory's principal features. Topics include elementary transformations and bilinear and quadratic forms; canonical reduction of equivalent matrices; subgroups of the group of equivalent transformations; and rational and classical canonical forms. The final chapters explore several methods of canonical reduction, including those of unitary and orthogonal transformations. 1952 edition. Index. Appendix. Historical notes. Bibliographies. 275 problems.
This algebra lesson from Illuminations involves using linear equations and graphs in a real world context. Students will graph a line based on data points, find the equation of the line, identify y-intercept and slope,... This algebra lesson from Illuminations has students collect data about the national debt, and then plot that data by decade. They will then determine whether an exponential curve is a good fit for the data. The... This interdisciplinary math lesson from Illuminations may be used in conjunction with geography education. Students will consider strategies for figuring out the distance to the horizon from various heights. A formula... This algebra lesson from Illuminations allows students to define and practice metacognition. Sample word problems and equations are included to practice the model. Polya's four steps of problem solving are integrated... This algebra lesson from Illuminations helps students develop their understanding of mathematical functions and modeling using spreadsheets, graphing calculators, and computer graphing utilities. The differences between...
About: Basic Operations with Real Numbers: Scientific Notation Metadata Name: Basic Operations with Real Numbers: Scientific Notation ID: m21879 Language: English (en) Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of \|x\|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples. Objectives of this module: be able to convert a number from standard form to scientific form and from scientific form to standard form, be able to work with numbers in scientific notation.
Algebraic geometry is a central field within mathematics which is often viewed as difficult for outsiders to enter. Recent years have seen many books which present various topics of algebraic geometry from a computational viewpoint, thereby making the subject more accessible. The book under review, RAC for short, is part of this positive trend. Its main focus is parametrizing rational plane algebraic curves. It is a graduate level text aimed at fairly wide readership, including for example readers interested primarily in computer-aided geometric design. To give an idea of the content and level of RAC, I will slowly work out its Exercise 4.20: The problem has been posed in purely algebraic terms and, as we'll see, can be solved in purely algebraic terms. But ignoring geometry would be missing half of algebraic geometry! So we type the given polynomial into a computer algebra system to draw part of C in the x-y plane. We thereby obtain a monochromatic version of Figure 1. We learn immediately that at three points in this window, the curve C looks locally like the letter X. The three crossing points are called singularities, and they will play an important role in our algebraic solution. Figure 1 draws these points as small black disks. Continuing with geometry, a problem with Figure 1 is that it shows only part of C. To remedy this problem, we can use the coordinates (u,v) = (x,y)/(r2 + x2 + y2)1/2 for any positive real r we find convenient. Via these coordinates, we can draw the entire x-y plane as the open unit disk U in the u-v plane. Exactly half the area of U comes from the disk of radius r about the origin in the x-y plane. Accordingly, one can take r small or large according to whether one wants to dedicate visual space to the part of the x-y plane near or far from the origin. In Figure 2, we view all of C in this way, taking r=3. So we learn that Figure 1 missed a whole piece of our C, the part drawn in orange at the lower right of Figure 2. In fact, Figure 1 missed this piece by a lot, since the closest the orange part of C comes to the origin is at (x,y) ≈ (20,–45). Figure 2 has actually given us more insight than we asked of it, because the circle ∂U bounding U also plays a role. In fact, if we identify opposite points of this boundary circle then we have just abstractly sewn the closed unit diskU into the "projective plane." Our affine curve C has gained four new points to become its projective completion C. Now let's mix algebraic and geometric thinking, as even beginning algebraic geometers should! In elementary algebraic terms, Exercise 4.20 is asking for non-constant rational functions x(t) and y(t) of minimal degree such that f(x(t),y(t)) is identically zero. Geometrically, one can think of t as time, and then (x(t),y(t)) should be thought of as a moving point. The numbers near the curve on Figure 2 capture the solution we will be algebraically producing. At t= –∞, the point starts at the point (1,1) which is labeled ±∞ on Figure 2. Then, as t increases, the point (x(t),y(t)) moves first upwards on the red arc. It goes straight through points at infinity, and also straight through singular points, moving mostly from left to right. Finally at t=∞ the point returns to (1,1), having visited each singular point twice and all other points exactly once. Thus — despite our color scheme! — the curve C forms a single loop. If it shocks your mathematical intuition that a plane curve C given by a random f(x,y) should be so parametrizable, then you are right! Only very special curves are parametrizable. If you are intimidated also about passing from abstract existence to actually finding (x(t),y(t)), then you are again reacting properly. In practice, it would be impossible to find (x(t),y(t)) by naive algebraic fiddling with variables. A systematic geometry-inspired approach is required, and that is the subject of RAC! To do at least some justice to the systematic approach of RAC, let's consider a general polynomial f(x,y) with real coefficients. Let d be its degree, i.e. the largest i+j appearing among its terms axiyj. Then a sufficient condition for parametrizability is that the corresponding complete curve C consists of a single loop which crosses itself at (d–1)(d–2)/2 singular points in the x-y plane. This sufficient condition is satisfied in our case, since d=4 and (4–1)(4–2)/2=3. A weaker but similar condition, involving complex numbers among other things, is necessary and sufficient for parametrizability. One of RAC's central algorithms is parametrization-by-adjoints in Section 4.7. Actually using this algorithm to parametrize a curve meeting our sufficient condition is a very attractive mix of algebra and geometry. The algorithm comes in several variants, as the adjoint curves involved can have degree d–2, d–1 or d. We will use the d–2 variant for our C, simultaneously indicating how it works for general d ≥ 3. First, one locates the singularities of C by finding the common roots of f(x,y) and its partial derivatives fx(x,y) and fy(x,y). This is a standard computer algebra task. In our case, the three singularities are (0,0), (–√2,1–√2), and (√2,1+√2). Second, one chooses d–3 non-singular points of C. In our case, we need just one point and we choose (1,1). Now let V be the vector space of polynomials of degree ≤ d–2. In our case, the general element of V has the form g(x,y) = a x2 + b x y + c y 2 + d x + e y + f. Let A be the subspace of V consisting of those g(x,y) which vanish on all (d–1)(d–2)/2 singular points and also on the d–3 chosen non-singular points. A key feature of this construction, expected by a naïve dimension count, is that A always has dimension two. Let g0(x,y) and g1(x,y) be a basis for A, and form the one parameter family of polynomials gt(x,y) = (1–t) g0(x,y) + t g1(x,y). In our case, suitable choices yield gt(x,y) = 2 x2 + (–5 – t) x y + 2 y 2 + (1 – t) x + 2 t y. Let Dt be the solution curve of gt(x,y)=0. The Dt are the adjoint curves in question. Another key feature of this construction, a consequence of Bezout's theorem this time, is that for all but finitely many t, the curves C and Dt meet at exactly one point beyond the imposed intersection points. The remaining intersection point (x(t),y(t)), unlike the imposed ones, varies with t. To find x(t), one eliminates y from the system f(x,y) = gt(x,y)=0 and solves for x in terms of t. This is standard computer algebra, essentially a single call to a resultant command. Likewise, to find y(t) one eliminates x and solves for y. In our case, the final answer is As RAC rightly emphasizes, a parametrization for a curve gives one much better control over the curve than one has from a defining equation alone. For example, finding points (x,y) on our C with rational coordinates is difficult from the original description f(x,y)=0. It is trivial from the parametrization, as one can simply plug in rational t. Likewise, the colors in Figures 1 and 2 were easy to draw only after we had the parametrization. The roots of c(t) are at approximately t=–14.9, –9.2, –1.0, and 7.0; these roots serve as start and end times for our color intervals. In some ways, RAC has a classical feel. For example, as the authors indicate, the main idea of the parametrization-by-adjoints algorithm is already present in book form in Walker's classic 1950 text. In fact, Theorem III.5.1 there gives the d–1 version. However RAC is very modern in its emphasis on computational issues. For example, the issue of solving problems like Exercise 4.20 without ever writing down computer-unfriendly irrationalities like √2 is thoroughly treated. I am looking forward to a future when algebraic geometry has thoroughly lost its aura of inaccessibility. Books like RAC are hastening the day. If you understood this review, you are ready to read RAC. If you were annoyed at how I suppressed complex numbers, you are more than ready! References: Algebraic Curves , by Robert J. Walker. Dover Publications, 1950. David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.
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8/12When the answer at the back of the book is simply not enough, then you need the Student Solutions Manual. With fully worked-out solutions to all odd-numbered text problems, the Student Solutions Manual lets you "learn by example" and see the mathematical steps required to reach a solution. Worked-out problems included in the Solutions Manual are carefully selected from the textbook as representative of each section's exercise sets so you can follow-along and study more effectively. The Student Solutions Manual is simply the fastest way to see your mistakes, improve learning, and get better grades.
Understanding Real Analysis - 10 edition Summary: In this introduction to real analysis, a branch of mathematical analysis dealing with the set of real numbers and their operations, Paul Zorn (St. Olaf College) aims to develop ideas already familiar from elementary calculus in a rigorous manner. The text will help students deeply understand some basic but crucial mathematical ideas, and to see how definitions, proofs, examples, and other forms of the mathematical 'apparatus' work together to create a unified theory. A key feature of...show more the book is that it includes substantial treatment of some foundational material, including general theory of functions, sets, cardinality, and basic proof techniques. ...show less 1568814151
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CALC CALC is a number theory calculator program which uses arbitrary precision integer arithmetic. It would be useful in a first course in number theory. It is written in ANSI C and Yacc, along the lines of the calculator programs hoc1,2,3, Kernighan and Pike, The Unix Programming Environment, 233-254, 1984. The source comes with two makefiles, one for Unix and one for Windows XP. The Windows XP version and C source can be downloaded here. I am grateful to Jean-Jacques Delgove for making changes to my code, so as to allow compilation under Visual C++ 2005. He also pointed out that CALC now can utilise AWK scripts such as the following, invoked by typing awk -f calc.awk in a Unix window: Note: Any previous values for the array will be erased and a new array will be created with the values in the curly braces. When one wishes to output the array a[ ], all that must be typed is: > a[ ] alone on a line. If one uses the former method, the size of the array will be set to the largest subscript entered. Any subscripts that have not been defined and which are less than this subscript, are initialised to zero. eg. > a[1]=2 > a[2]=3 > a[4]=10 > a[ ] [0]:0 [1]:2 [2]:3 [3]:0 [4]:10 Some functions require an mxn integer matrix to be entered, either at the keyboard, or from a file. The format of such a file is as follows: m n a11 ... a1n ... am1 ... amn In other words, the first line contains the row and column size, separated by a space, while the entries on each row are separated by spaces and each row is terminated by a newline. Calc is now capable of parsing polynomials. A polynomial can be entered using the reserved symbol X (capital x) in the following manner: euclid(a,b,&q[],&r[],&s[],&t[],&m) Here m=n+1 where we return four arrays arising from Euclid's algorithm are returned: q[0]=NULL,...,q[n],q[n+1]=NULL (quotients) r[0]=a,r[1]=b,...,r[n+1] (remainders) s[0]=a,s[1]=0,...,s[n+1] t[0]=a,t[1]=1,...,t[n+1] r[k]=r[k+1]q[k+1]+r[k+2], 0 < r[k+2] < r[k+1], s[k]=-q[k-1]s[k-1]+s[k-2], t[k]=-q[k-1]t[k-1]+t[k-2], r[n]=gcd(a,b)=s[n]a+t[n]b. These are printed in euclid.out. z=gcd(x,y) This returns the gcd of x and y. z=gcdv(x,y,&u,&v) As well as returning z=gcd(x,y), it gives numbers u and v arising from Euclid's algorithm and satisfying the equation z = ux+vy. z=gcda(a[]) z = gcd(a[0],...,a[n-1]), where values for a[0],...,a[n-1] having been previously entered. z=gcdav(a[],&b[]) z = gcd(a[0],...,a[n-1]). Also gives integers b[0],...,b[n-1] satisfying z = b[0]a[0]+...+b[n-1]a[n-1]. egcd() This is an implementation of Algorithm 1 of a recent paper by Havas, Majewski and Matthews. Like lllgcd( ), it finds short multipliers for the gcd of m numbers, using LLL ideas. It also finds all shortest vectors, unlike lllgcd( ), which lists only one shortest multiplier. The file of m integers should have as its first line m, then the integers should be listed on separate lines. An m x m matrix whose rows are X[1],...,X[m-1],P is sent to an output file eg multipliers are sent to an output file called egcdmult.out. Sometimes the multipliers delivered by egcd() are shorter that those of lllgcd( ). There is also an option to find all the shortest multipliers. sgcd(N) This performs the LLL algorithm on [In\|NA], where A is a column vector of positive integers. This is Algorithm 2 of paper. If N is sufficiently large, the last column will be reduced to ±NdEn, where d = gcd(a[1],...,a[n]). Output is sent to sgcdbas.out. lllgcd() This performs a modification of LLL which is essentially a limiting form of sgcd(N) for large N. It is superior to egcd() in that it avoids inputting a large initial unimodular matrix and instead builds one from the identity matrix at the outset. This is Algorithm 3 of a recent paper of Havas, Majewski and Matthews. The file of m integers should have as its first line m, then the integers should be listed on separate lines. An m x m matrix whose rows are X[1],...,X[m] is sent to an output file lllg inhomogeneous version of the Fincke-Pohst algorithm (see [Po2][191]) can then be used as an option to find a shortest multiplier vector by solving the inequality \|\|X[m] - x1X[1]- ··· -xm-1X[m-1]\|\|2 ≤ \|\|X[m]\|\|2 in integers x1,...,xm-1. Each time a shorter multiplier vector Q = X[m]-x1X[1]- ··· -xm-1X[m-1] is found, X[m] is replaced by Q, until the shortest Q is found. The multipliers are sent to an output file called lllgcdmult.out. The unimodular matrix P of a recent paper of Havas, Majewski and Matthews, is sent to lllgcdmat.out. In verbose mode, the intermediate steps are printed. Note: if the shortest vector option is chosen, the last row of P has been replaced by this vector. lllgcd0() This in general gives a better multiplier than lllgcd and is based on an algorithm in a recent manuscript of the author. jacobigcd() This performs Jacobi's extended gcd algorithm of 1869. In verbose mode the intermediate steps are printed out. z=lcm(x,y) z=lcma(x) z = lcm(x[0],...,x[n-1]). (n is the size of the array) z=length(n) z is the number of decimal digits of n. z=pollard(x) This attempts to return a factor of a composite x using Pollard's p-1 method. z=nprime(x) This finds the first integer after x which passes the strong base 2 pseudoprime test and the Lucas pseudoprime test. (See [Pom].) This integer is likely to be prime. z=nprimeap(a,b,m) This finds the first p, p=b(mod a), m ≤ p, which passes the strong base 2 pseudoprime test and the Lucas pseudoprime test. Here a must be even, b odd, 1 ≤ b < a, gcd(a,b)=1, b ≤ m. z=jacobi(x,y) z is the value of the Jacobi symbol (x/y). z=peralta(a,p) Peralta's algorithm is used to return a square root z of a (mod p). Here a is a quadratic residue mod p. (See [Per].) x=congr(a,b,m,&n) Returns the solution x of the congruence ax=b(mod m). Also n=m/gcd(a,m) is returned. x=chinese(a,b,m,n,&l) Returns the solution x(mod l) of the system of congruences x=a(mod m) and x=b(mod n). Also l=lcm(m,n) is returned. x=chinesea(a[],m[],&l) Returns the solution x(mod l) of the system of congruences x=a[i](mod m[i]), 0 ≤ i < n. Also l=lcm(m[0],...,m[n-1]) is returned. (n is the size of the array) z=mthroot(x,m) The integer part of the m-th root of x is returned. (See [Mat].) mthrootr(x,y,m,r) The m-th root of x/y is computed to r decimal places. z=fund(d,&x,&y) x and y are returned, where x+y is the fundamental unit of Q(√d). z=Norm(x+y ) is also returned. z=pell(d,e,&x,&y) The continued fraction expansion of √d is periodic after the first term: a[0],a[1],...,a[n-1],2a[0],a[1],...,a[n-1],2a[0],.... Also the section a[1],...,a[n-1] is palindromic. We print a[0] and half the palindrome, iff e is nonzero, sending the output to a file called pell.out. The least solution x and y of Pell's equation x2-dy2=epsilon is returned, where epsilon=1 or -1. z=epsilon is also returned. (See [Ros][382], [Ven][62], [Knu][359], [Po1][359], [Sie][296], [Dav][109]. Also see manuscript.) z=surd(d,t,u,v,&a[],&u[],&v[],&p[],&q[]) The continued fraction of the quadratic irrationality (u+t√d)/v, d > 1, nonsquare, v > 0, is determined as far as the period. The period length z is returned. a[] is the array of partial quotients, u[] and v[] give the complete convergents (u[i]+√d)/v[i], p[] and q[] give the convergents p[i]/q[i]. Output is sent to a file surd.out. (See [Ros][379-381] and [Knu][359].) z=mpower(a,b,c) z=ab(mod c) is returned. z=inv(a,m) z=a-1(mod m) is returned. (See [Knu][325].) z=elliptic(n,m,p) The elliptic curve method is used to try to find a factor z of a composite number n. Here m, p < 232 and 1279 ≥ m > 10, p ≥ 1. z=factor(n) The factorization of n is performed using the multiple polynomial quadratic sieve on numbers with < 55 digits. The elliptic curve algorithm is used on numbers of ≥ 55 digits. If succesful, the primes are certified and z= (n), the number of distinct prime factors of n. There is an option to choose the number p of elliptic curves used. (p=7 suffices to factor 10100+1. (See [Sil][325].) z=tau(n) The divisor function z=d(n) is returned. z=sigma(n) z=σ(n), the sum of the divisors of n, is returned. z=mobius(n) The Möbius function z=μ(n) is returned. z=euler(n) Euler's function z=φ(n) is returned. z=lprimroot(n) The least primitive root mod p, an odd prime, is returned. z=orderm(a,n) The order of a (mod n) is returned. z=lucas(n) n is subjected to a strong pseudoprime test to base 2, together with a Lucas pseudoprime test. If z=0 is returned, n is composite, while if z=1 is returned, then n is a Lucas probable prime, as well as a base 2 strong pseudoprime. (See [Pom].) serret(p,&x,&y) Here p is a prime of the form 4n+1. Serret's algorithm is used to return integers x and y such that p=x2+y2. (See [Wag].) collatz(x,n) Collatz' 3x+1 conjecture is tested. (See [Lag] and 3x+1 page.) The iterates x,T(x),... are printed iff n is nonzero. cycle() This finds cycles for the generalised Collatz mapping T, arising from starting numbers p, \|p\| ≤ RANGE/2. Trajectories containing an iterate whose magnitude exceeds a prescribed value INFINITY are deemed not to be ultimately cycling. T(x)=int(m[i]x/d)+X[i] if x=i (mod d). Here the d nonzero moduli m[i] and d shifts X[i] are entered from the keyboard. (See 3x+1 page.) If when x ≡ i (mod d), T(x) is given in the form T(x)=(m[i]x+r[i])/d, where d divides m[i]i+r[i], then X[i]=(m[i]i+r[i])/d-int(m[i]i/d)=r[i]/d+{m[i]i/d}, where {t} denotes the fractional part of t. In particular, if k is odd, the "3x+k" mapping is given by m[0]=1, m[1]=3, X[0]=0, X[1]=(k+1)/2. The output is sent to collatz.out. miller(m,b) Here m > 1,b > 1 and m does not divide b. Miller's test to base b is applied to m. hermite( ) The Hermite normal form HNF(A) of an integer matrix A is found using the Kannan-Bachem algorithm. (See [Sim][349-357].) The matrix can be entered either from the keyboard or from a file such as 2 3 1 -4 -7 3 2 4 in the case of a 2 x 3 matrix. The Hermite normal form is sent to a file called hermite.out. A unimodular integer matrix P such that PA = HNF(A) is sent to a file hermitep.out, if desired. The last line of this file contains the value of rank(A). improvep() The output unimodular matrix contained in the file hermitep.out is improved using LLL-Babai method, followed by Gauss lattice reduction. The output is sent to a file improvep.out. lllhermite() This performs a recent LLL based Hermite normal form algorithm due to Havas, Majewski and Matthews. The output unimodular matrix P of a recent paper is sent to lllhermitetrans.out. The HNF(A) is sent to lllhermitebas.out. In verbose mode, the intermediate steps are printed out. shermite(N) This performs the LLL algorithm on [Im\|NnA1\|...\|NAn], where A is an mxn matrix of integers. The columns are scaled down by the corresponding power of N for viewing convenience. The output matrix, in scaled form, is sent to shermitebas.out. smith() The Smith normal form SNF(A) of an integer matrix A is found using a pivoting strategy due to G. Havas. A cutoff value is requested. When coefficients grow above this value in size, the MLLL algorithm is used to reduce coefficient explosion. MLLL is also used at the start of searching for each invariant factor, as it often yields small vectors with potential small invariant factor components. Unimodular matrices P and Q are found such that PAQ = SNF(A). the invariant factors, P and Q are sent to files smith.out, smithp.out and smithq.out, respectively, if desired. mlll() The MLLL algorithm of M. Pohst is applied to an integer matrix whose first row is non-zero. (See [Po2][209-210].) The reduced matrix and transforming matrix are sent to files mlllbas.out and mllltran.out, respectively. axb() This solves the linear system AX=B, where A,X,B have integer coefficients and the first column of A is nonzero. In the case of solubility, file axb.out is either the unique solution or a short solution or the shortest solution, depending on choice; axbbas.out is a short basis for the nullspace of A. (See preprint.) fp() This is the simplest form of the Fincke-Pohst algorithm (see [Po2][190]). This takes an integer matrix with LI rows and a positive integer C as input and finds all lattice vectors X with \|\|X\|\|2 ≤ C. The vectors found are sent to a file called fp.out. It is a good idea to start with a matrix whose rows are LLL reduced. slv() This finds the shortest vectors in the lattice spanned by the LI family b[1],...,b[m]. It applies Fincke-Pohst to examine all nonzero X in L satisfying \|\|X\|\|2 < C = \|\|b[1]\|\|2. If it finds an X shorter than b[1], the new bound C=\|\|X\|\|2 is chosen. At the end, Finke-Pohst has only the shortest vectors to enumerate. Only the vectors with highest positive coefficient of b[i] are given. The output is also sent to slv.out. inhomfp() The inhomogeneous version of the Fincke-Pohst algorithm. (See [Po2][190].) Input: An m x M integer matrix A whose first m-1 rows are LI and which are preferably in LLL reduced form. A positive integer C is also entered. Let L be the lattice spanned by the first m-1 rows of A and let P be the last row of A. Output: All lattice vectors X of L such that \|\|X-P\|\|2 ≤ C. The vectors found are sent to a file called inhomfp.out. e=rsae(p,q) Here p and q are distinct odd primes, each > 355142 and e is the least integer such that gcd(e, (p-1)(q-1))=1 and 32e> pq. e is for use as an RSA encryption modulus. encode(e,n) Here n=pq, a product of primes each greater than 355142. (pq > 126126126126.) e is the RSA encryption modulus, found using rsae( ) above. A string of non-control characters when entered from the keyboard or from a file consisting of lines each containing less than 500 characters. These characters have ascii values in the range 32-126. The message string is encoded using the RSA algorithm: every 4 characters are converted to ascii, joined as strings and the resulting large number m is encoded as n = me(mod pq). The encoded numbers are sent to a file "encoded.out", terminated by an entry -1. decode(e,p,q) This calculates the decryption modulus d, then decodes each number n in the file "encode": m = nd(mod pq). The ascii characters are split off and the original string of message characters is reformed. The encoded message is sent to a file "decoded.out", as well as to the screen. addcubicr() The sum of two points on the elliptic curve y2+a1xy+a3y=x3+a2x2+a4x+a6 is calculated. The discriminant is also calculated. powercubicr() The point nP, where P is on the elliptic curve y2+a1xy+a3y=x3+a2x2+a4x+a6 is calculated. The discriminant is also calculated. ordercubicr() The order of point P on the elliptic curve: y2+a1xy+a3y=x3+a2x2+a4x+a6 is calculated. The discriminant is also calculated. addcubicm() The sum of two points on the elliptic curve mod p: y2+a1xy+a3y=x3+a2x2+a4x+a6 is calculated. The discriminant is also calculated. powercubicm() The point nP, where P is on the elliptic curve mod p: y2+a1xy+a3y=x3+a2x2+a4x+a6 is calculated. The discriminant is also calculated. ordercubicm() The order of point P on the elliptic curve mod p: y2+a1xy+a3y=x3+a2x2+a4x+a6 is calculated. The discriminant is also calculated. convergents(a[],&p[],&q[]) The convergents p[0]/q[0],...,p[n]/q[n] of the continued fraction [a[0];a[1],...,a[n]] are returned as arrays p[] and q[]. Here a[0] is an integer, a[1],...,a[n] are positive integers. lagrange("poly",&a[],m) "poly" is a polynomial with integer coefficients, having no rational roots and having exactly one real positive root x, this being > 1. The method of Lagrange (1769) is used to find the the first m+1 partial quotients aa[0],...,aa[m] of x. (See Knuth, Art of computer programming, volume 2, problem 13, 4.5.3. Also S. Lang and H. Trotter,Continued fractions for some algebraic numbers J. für Math. 255 (1972) 112-134; Addendum 267 (1974) ibid. 219-220 and page 261, Number Theory with Applications, by R. Kumanduri and C. Romero. P. Shiu, Computation of continued fractions without input values, Math. Comp. 64 (1995), no. 211, 1307-1317, and D.G. Cantor, P.H. Galyean, H.G. Zimmer [Can]. (If the root is rational, Lagrange's algorithm still finds the partial quotients and we exit at the appropriate point. We check that poly is squarefree, has leading coefficient positive, does not vanish at x=0 or x=1, is not linear and has exactly one positive real root t and that t > 1. z=perfectpower(n) Here n > 1. z= x if n = xk, for some x, k > 1, NULL otherwise. See E. Bach and J. Sorenson, "Sieve algorithms for perfect power testing", Algorithmica 9 (1993) 313-328. z=leastqnr(p) Returns np, the least quadratic non-residue (mod p), if np < 65536, otherwise returns 0. sturm(f(X)) Prints rational open intervals that are guaranteed to each contain only one real root of the polynomial f(X). It is assumed that f(X) has no rational roots. The output is sent to the file sturm.out and to the screen. See ([Akr],page 341.) rootexp(f(X),m) Finds the first m partial quotients of the continued fraction expansion of all real roots of a squarefree polynomial f(X) with no rational roots, using Lagrange's method and a variation presented in D.G. Cantor, P.H. Galyean, H.G. Zimmer [Can], especially pp. 785-787. The output is sent to a file rootexp.out and to the screen. Also see D. Rosen, J. Shallit [Rosen-Shallit.] Programmed by Sean Seefried in January 2000. content(f(X)) Returns the content of a polynomial f(X) (the gcd of the coefficients of f(X) x sign of the leading coefficient of f(X)). eg. > content(-3X^2+6) -3 primitive(f(X)) Returns the primitive part pp(f(X)) of a polynomial f(X), where f(X)=content(f(X))pp(f(X)). eg. >primitive(-3X^2+6) X^2-2 >content(-3X^2+6) -3 z=sqroot(a,n,&s[],&m) This solves x2 ≡ a (mod n). The number z of solutions mod n is returned. The solutions have the form ±s[0],...,±s[r] (mod m). If there is no solution, z=0 and s[0]=NULL are returned. cornacchia(a,b,m) See L'algorithme de Cornacchia, A. Nitaj, Expositiones Mathematicae, 358-365. This returns the positive primitive solutions (x,y) of ax2+by2=m, x ≥ y, where a,b,m are positive integers, with m ≥ a+b, gcd(a,m)=1=gcd(a,b). (Actually Nitaj also requires gcd(b,m)=1, but this does not seem to be necessary.) patz(d,n) This returns the positive primitive fundamental solutions of the diophantine equation x2-dy2=±N, where d > 1 is not a perfect square. The algorithm is from a recent paper of the author and is in essence due to Lagrange 1770. z=congq(a,b,c,n,&s[]) This solves the quadratic congruence ax2+bx+c ≡ 0 (mod n), where gcd(a,n)=1, n > 0. The solutions are s[0],...,s[z-1]. z=binform(a,b,c,n,e) This solves the diophantine equation ax2+bxy+cy2=N, N non-zero, where D=b2-4ac > 0 and is not a perfect square. One solution from each equivalence class is printed, along with the corresponding solution n of the congruence n2 ≡ D (mod 4\|N\|), -\|N\| < n ≤ \|N\|. e=1 is verbose, e=0 is terse. Output is sent to binform.out. The algorithm goes back to Lagrange (1770) and described in a paper to be submitted by the author. z=absmod(a,b) z=r=a(mod b) if r ≤ b/2, otherwise z=r-b. z=ceil(a,b) This finds the least integer z not less than a/b, where a and b are integers, b nonzero. log(a,b,d,r,&m[],&l) This delivers with a high degree of certainty, partial quotients a[s] of logba, where a > b > 1, d> 1, r > 0. l is the number of partial quotients returned. Output is sent to log.out. (See paper.) testlog(a,b,d,m,n) This runs the testlog1(a,b,d,r) program for r=m,...,n and is recommended in order to get a good idea of the correct partial quotients of logba. Output is sent to testlog.out. (See paper.) r=resultant(p,q) This returns the resultant of non-constant polynomials p and q. r=discriminant(p) This returns the discriminant of a non-linear polynomial p. q=deriv(p) This returns the derivative of polynomial p. c=primes(m,n) This prints the c primes, (c ≥ 0), in the interval [m,n], where 1 < m,n < 1010. Output is sent to primes.out. c=sturmsequence(f,b,e) This produces the Sturm sequence for the polynomial f, evaluates the sequence at x=b and calculates the number c of sign-changes. No output if e=0, output if e=1. f is checked to see if it is squarefree and nonlinear and that f(b) is nonzero. p=cyclotomic(n) This returns the nth cyclotomic polynomial, n < 65536. See H. Lüneburg Galoisfelder, Kreisteilungskörper und Schieberegisterfolgen, BI Mannheim 1979 or MP473 notes. h=classnop(d) This returns the class number h=h(d) of a real quadratic field Q(√d). Here 1 < d < 106 is squarefree and D is the field discriminant. We locate all reduced irrationals of the form (b+√D)/(2\|c\|), where c is negative and 4 D, where a=(b2-D)/(4c). We are able to also determine if the Pell equation x2-Dy2=-4 has a solution, thereby finding the norm of the fundamental unit. (See Henri Cohen's A course in computational number theory, page 260, First Edition.) h=classnon(d,e) Here d < 0 and 1 < \|d\| < 106 is squarefree and d=0 or 1(mod 4). This is Henri Cohen's Algorithm 5.3.5, p. 228, for finding the class number h(d) of binary quadratic forms of discriminant d, when d < 0. h(d) is returned in each case. If e=1, we print only the primitive forms. If e=0, we print primitive and imprimitive forms. If d is the discriminant of an imaginary quadratic field K, then the primitive forms class-number h(d) is also the class number of K. Davenport's Higher Arithmetic has a table of forms, which lists the imprimitive ones with an asterisk. z=nearint(a,b) Here b > 0. Then z is the nearest integer to a/b and where z=t, if a/b=1/2+t, t an integer. In fact nearint(a,b)=(a-absmod(a,b))/b. h=reduceneg(a,b,c) This is Gauss's algorithm for reducing a positive definite binary quadratic form. See L.E. Dickson, Introduction to the theory of numbers, page 69. The reduced form (A,B,C) satisfies -A < B ≤ A, C ≥ A, with B ≥ 0 if C = A. The number h of steps taken in the reduction is returned. h=reducepos(a,b,c) Given an indefinite binary quadratic form ax2+bxy+cy2, we use the PQa continued fraction algorithm to determine a reduced form and thence a cycle of reduced forms. A unimodular matrix is constructed which converts (a,b,c) to reduced form. The output is sent to reducepos.out. Note: d=b2-4ac > 0, d is not a perfect square and we assume d < 106. (See explanatory note and Henri Cohen's A course in computational number theory, Edition 1, pp. 257-261.) h=classnop0(d) Here 1 < d < 106 is not a perfect square and d is 0 or 1 (mod 4). Returns the number h of classes of binary quadratic forms of discriminant d A complete set of reduced binary forms is given. We locate all reduced irrationals of the form (b+√d)/(2\|c\|), where c is negative and 4* d, where a=(b2-d)/(4c). We are able to also determine if the Pell equation x2-dy2=-4 has a solution, by using the fact that the equation is soluble iff at least one of the above cycles is odd. If there is no solution, the reduced forms (-a,b,-c) have to be counted as well. (See G.B. Mathews, Theory of Numbers, 80-81.) (Also see Henri Cohen's A course in computational number theory, page 260, First Edition.) h=tableneg(m,n) Here 1 ≤ m ≤ n < 106. Calculates h(-d) for all squarefree d with m ≤ d ≤ n. The number h of squarefree d in the range is returned. The output is also sent to tableneg.out. h=tablepos(m,n) Here 2 ≤ m ≤ n ≤ 106. Calculates h(d) for all squarefree d with m ≤ d ≤ n. The number h of squarefree d in the range is returned. The output is also sent to tablepos.out. h=davison(l,m,n) Here l,m > 0, 105 ≥ n ≥ 0. We find h partial quotients a[i] of el/m. We cannot predict the value of h. The a[i] are also sent to davison.out. The program stops if 106 partial quotients a[i] are found. h=raney(p,q,r,s) Input: a non-singular matrix A=[p,q;r,s], p,q,r,s>=0, A!=I2, A!=[0,1;1,0]. With L=[1,0;1,1] and R=[1,1;0,1], we express A uniquely as a product of non-negative powers of L and R, (at least one is positive) followed by a row-balanced B. B=[a,b;c,d] is row-balanced if (a < c & b > d) or (c < a & d > b) and a,b,c ≥ 0. The number h of powers of L and R is returned. The maximum number of partial quotients returned is 106. Also BCMATH version. (See G.N. Raney, On continued fractions and finite automata, Math, Annalen 206 (1973) 265-283.) h=unimodular(p,q,r,s) This program expresses a unimodular matrix A = [p,q;r,s] ≠ I2 or U = [0,1;1,0] with non-negative coefficients, as a product of the type P, UP, PU, or UPU, where P is a product of matrices of the form Ua=[a,1;1,0], a>0. The representation is unique. (See Kjell Kolden, Continued fractions and linear substitutions, Arch. Math. Naturvid. 50 (1949), 141-196. Also see the corresponding BCMATH program.) twoadicsqrt(b,n,&a[]) Here b > 0, b=8k+1. Finds n terms a[0]...a[n-1]of the 2-adic square root x of b, x ≡ 1 (mod 4). Output also sent to 2-adic.out. padicsqrt(b,n,p,&a[]) Here b > 0, b a quadratic residue (mod p), p an odd prime. First finds a square root u of b (mod p), 0 < u < p, and finds n terms a[0]...a[n-1]of the p-adic square root x of b, x ≡ u (mod p). Output also sent to p-adic.out. See lectures and solutions. u=ramanujan(n) Returns the value of Ramanujan's tau function, 0 < n < 216. z=repdefinite(a,b,c,m,print_flag) This solves the diophantine equation m=ax2+bxy+cy2, where d=b2-4ac < 0, a > 0, c > 0. See BCMATH version for the algorithm due to Gauss. print_flag = 1 lists solutions and unimodular transformations, while print_flag = 0 lists only the solutions. The output is sent to repdefinite.out. powerd(a,b,d,n,&aa,&bb). Here (a + b√d)n = aa + bb√d. z=euclid1(a,b). z is the length of Euclid's algorithm. Here a and b are positive integers. z=rcfperiod(d). z is the length of the period of the regular continued fraction expansion of √d. z=nscfperiod(d). z is the length of the period of the nearest square continued fraction expansion of √d. z=nicfperiod(d). z is the length of the period of the nearest integer continued fraction expansion of √d. algcycle(). This looks for cycles for the d-branched generalised Collatz mapping T: ℤ[√d]→ℤ[√d]. The output is sent to alg_cycle.out. See examples in Section 10 of survey. spiral(n,&x,&y). This produces the spiral point (x,y) corresponding to a non-negative n. See page 99 of Concrete Mathematics, Graham, Knuth, Pataschnik. n=spiralinverse(x,y). This produces the n corresponding to spiral point (x,y). See page 99 of Concrete Mathematics, Graham, Knuth, Pataschnik. z=nicfperiod0(d,e,&x,&y). z is the length of the period of the nearest integer continued fraction expansion of √d. (x,y) is the fundamental solution of x2-dy2=±1. e=1 prints the type of midpoint criteria. z=nscfperiod0(d,e,&x,&y). z is the length of the period of the nearest square continued fraction expansion of √d. (x,y) is the fundamental solution of x2-dy2=±1. e=1 prints the type of midpoint criteria. z=rcfperiod0(d,e,&x,&y). z is the length of the period of the regular continued fraction expansion of √d. (x,y) is the fundamental solution of x2-dy2=±1. e=1 prints the type of midpoint criteria. carmichael(n). This solves φ(x) = n. See BCMath description. carnielli(). This finds cycles for Walter Carnielli's generalization of the 3x+1 mapping. See BCMath description. The output is sent to carnielli_cycle.out. lupei(). This finds cycles for Lu Pei's generalization of the 3x-1 mapping. See BCMath description. The output is sent to lupei_cycle.out. tangent(n). This computes the n-th tangent number t(n) for 1 ≤ n ≤ 2000. Also see BCMATH program. bernoulli(n,&x,&y). This computes the n-th Bernoulli number B(n) for 0 ≤ n ≤ 4000. Also see BCMATH program. partition(n). This computes the n-th partition number p(n) for 1 ≤ n ≤ 65535. Also see BCMATH program. twocycle( ). This tests for cycles of the 2-branched mapping T(x)=int(aix/mi)+xi, where i = mod(x,2), a0,a1 are nonzero multipliers, m0,m1 are nonzero positive divisors and x0,x1 are arbitrary shifts. mcycle( ). This tests for cycles of a 3x+1 type mapping due to Benoit Cloitre. Here f(x)=int((m+1)/m)x) and T(x)=f(x)/2 or (3f(x)+1)/2, according as f(x) is even or odd. pqcycle( ). This tests for cycles of a 3x+1 type mapping due to Benoit Cloitre. Here f(x)=int(p*x/q), p > q > 1, q not dividing p, and T(x)=f(x)/2 or (3f(x)+1)/2, according as f(x) is even or odd. h=fg(p,q). This calculates h=p(q(X)). nagell(d,n). This finds the fundamental solutions u+v√d of x2 - dy2 = n, where d > 0 is not a perfect square and n is non-zero; it also gets a basis for the non-negative solutions. The algorithm works only for small d and is based on upper estimates for v due to Nagell, Tchebicheff and Frattini. See note. frattini(d,n). This finds a set of non-negative solutions u+v√d of x2 - dy2 = n, where d > 0 is not a perfect square and n is non-zero. These solutions generate all non-negative solutions by multiplication by εn, n≥ 0, where ε=x1+y1√d is the fundamental solution of Pell's equation x2-dy2=n. The algorithm works only for small d and n and is based on an upper estimate v < y1√n due to Frattini 1891-92. The author is grateful to Peter Adams for help in understanding the mysteries of Yacc and C programming. Sean Vickery helped in getting the PC version up and running. 1999/2000 summer vacation scholar Sean Seefried vastly improved the error handling, added polynomials to the parser and coded sturm(), rootexp() and a much improved lagrange(). CALC is maintained by Keith Matthews. The software is intended as a service to the number theory community, but the author cannot be held responsible for any consequences, either direct or indirect, which the use of this package may have. It can be freely copied for non-commercial purposes.
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Post navigation Starting Calculus with Area Functions So I decided to try a new beginning to (non-AP) calculus this year. Instead of doing an algebra bootcamp and diving into limits, I decided to teach kids a new kind of function transformation. I'd say this is something that makes my classroom uniquely mine (this is my contribution to Mission 1 of Explore the MTBoS). I don't think anyone else I know does something like this. You see, I was talking with a fellow calculus teacher, and we had a big realization. Yes, calculus is hard for kids because of all the algebra. But also, calculus involves something that students have never seen before. It involves transformations that morph one graph into another graph. And not just standard up, down, left, right, stretch, shrink, reflect transformations. Although they do transform functions, they don't make them look too different from the original. Given a function and a basic up, left, reflect, shrink transformation of it, you'd be able to pair them up and say they were related… But." So I wanted to start out with a unit on abstract and weird function transformations. Turns out, even though the other teacher and I had brainstormed 5 different abstract function transformations, I got so much mileage out of one of them that I didn't have to do anything else. You see: I introduced my kids to integrals, without ever saying the word integrals. Well, to be fair, I introduced them to something called the area transformation and the only difference between this and integrals is that we can't have negative area. [1] That packer was just the bare backbones of what we did. There was a lot of groupwork in class, a lot of conceptual questions posed to them, and more supplemental documents that were created as I started to realize this was going to morph into a much larger unit because I was getting so much out of it. (I personally was finding so much richness in it! A perfect blend of the concrete and the abstract!) It's conceptual, so those kids who aren't strong with the algebraic stuff gain confidence at the start of the year Kids start to understand the idea of integration as accumulation (though they don't know that's what they are doing!) Kids understand that something can be increasing at a decreasing rate, increasing at a constant rate, or increasing at an increasing rate. They discovered those terms, and realized what that looks like graphically. Kids already know why the integral of a constant function is a linear function, and why the integral of a linear function is a quadratic function. Kids are talking about steepness and flatness of a function, and giving the steepness and flatness meaning… They are making statements like "because the original graph is close to the x-axis near x=2, not much area is being added as we inch forward on the original graph, so the area function will remain pretty flat, slightly increasing… but over near x=4, since the original function is far from the x-axis, a lot of area is being added as we inch forward on the original graph, so the area function shoots up, thus it is pretty steep" Once we finish investigating the concept of "instantaneous rate of change" (which is soon), kids will have encountered and explored the conceptual side of both major ideas of calculus: derivatives and integrals. All without me having used the terms. I'm being a sneaky teacher… having kids do secretlearning. I mean… I worked these kids hard. Here is a copy of my assessment so you can see what was expected of them. First of all, thanks for being a part of the MTBoS leadership that planned this blogging event. I like to blog, but this is the exact time of the year when it gets hard to do so admist other responsibilities. The nudge is appreciated. I teach 8th grade, so I won't be able to incorporate this awesome idea into my classroom. I do have a ton of former students come back to visit who are mightily struggling in calculus and I wish they could be in your class. You're clearly meeting your students where they are and making calc "totes" fun. I love how this gets students thinking about the concepts of integrals without getting them tied up in all the formal algebra. I definitely never would have thought about doing it at the beginning of the year, but I can see how it will help students later as they tie the ideas together with the calculus more formally. It sounds like it got them thinking about some awesome topics! Sorry, I guess my question wasn't clear. Most calculus courses begin with slope. I'm curious about what made you decide to start with area instead. I am intrigued, and want to think about this. (To me, it seems so much easier to start with slopes and rate of change. I am allegedly supposed to use the same textbook the rest of my department uses. I provide suggested homework from the book, but I skip around wildly, because I don't want to do limits at the beginning of the course – and many other choices I've made that don't go with the book. If I tried to do area first, I can't imagine how I'd use the textbook. But, if it seems like a good enough idea, maybe I can find a way.) 1) I really want kids to realize that calculus — the big ideas anyway — aren't "hard." So I figure by starting the year with something like "area functions" and getting them to think about some of the ideas without all the gross mathematical language or symbols which can be scary was a pretty neat idea. And then when we get to integrals, I'm hoping that the initial foray into area functions will stick with them. And the language they used to describe things "you're adding area at an increasing rate" or "you're adding area at a decreasing rate" (and thus, the area graph looks like BLAH or BLAH). 2) I really wanted to start my kids off this year thinking about "strange functions" that are based off of original functions. So eventually they will learn the derivative fxn. But the original fxn and the derivative fxn look very different — almost like they are not related. But there are deep connections between the two that make them intimately related. I was thinking if I could use a different "strange function" at the beginning of the year to get them thinking like that… and I realized area functions were perfect. (I had a few others in mind, but once I saw the good thinking the kids were doing, I just focused on that). Yes! You got it! I was going to make mine an activity, and it turned into a week and a half exploration because they were saying good things. But that's about it. Then we went straight into "normal" calculus (slopes). "But."" Well, it depends on what you mean by "related." If related means "similar" in the usual geometric sense, and if that's all you've got, then no. Obviously, rigid transformations don't change the shape of the graph, so all rigid transformations, even in combination, produce "similar" (even congruent, if lifting into 3-space is permissible) graphs from whatever you start with. Adding dilation to the mix makes it a little harder, but you still have similarity to go with, and while you might mistakenly see to graphs as similar, you probably aren't going to go WAY wrong. But the relationship between the graphs of f(x) and f'(x) have a relationship, of course. It's simply not one that kids are used to seeing in math class (and while they may have seen it in science classrooms, they probably weren't thinking too heavily about the relationships among the position, velocity, and acceleration graphs for some object in motion, nor dreaming of how the functions can be derived from each other through differentiation or integration. How could they? What they certainly can learn to do is see that the graph of f'(x) shows how the slope of the graph of f{x} is changing. And there is software, for one thing, to show that relationship. There are hands-on exercises and other physical ways to model and even experience the changes and the relationship. I say none of this to take away from what you describe doing, which is fine. But I'm not sure there is anything weird about the transformation of the function graph into the graph of its derivative. It does take some visualization skills that might not be easy for everyone, but I suspect that traditional calculus instruction has made use of far fewer "tools" and models than I would expect from a contemporary, creative, progressive teacher, particularly a teacher who recognizes that left to his/her own devices, it is not very likely that a beginning calculus student who doesn't know that there is a relationship between the derivative of a function and the slope of that function at a given value of x is going to leap to the conclusion that the graph of f'(x) shows the change in slope of the graph of f(x) as x changes. I think we're saying the exact same thing! Kids aren't exposed to transformations that do anything but preserve the same general shape. That's all that I meant by "weird" — it's weird and strange to them on first learning. I don't (obviously) deny that there is a relationship between f(x) and f'(x). I just started this year off trying to highlight that there can be relationships between graphs that aren't intuitive on first glance, but there are deep connections between the two graphs! Sam, I'm sure we're basically on the same page here, and what you're doing is interesting. What really worries me is how weak kids are (haven't been asked to think about, haven't be led to understand) the relationship between graphs and what they might be describing in "the real world." So asking kids to put stories to graphs or match graphs with appropriate stories catches many of them completely off-guard. It's worth noting, however, that there was a study done at the University of Michigan about a dozen years ago in which they interviewed successful calculus students (I think that means kids who finished at least one calculus course with an A, but I heard Hy Bass talk about this long enough ago that some of the details have slipped into the ether). They were each asked to explain in their own words what the derivative of a function meant. The striking thing was that virtually none of them could say that the derivative of a function gives the slope of the function f(x) at a given value of x. They could explain how to FIND the derivative of a polynomial (that was the most common response, but the meaning was not accessible, if in fact they even knew it at all. That, to me, is a frightening indictment of K-14 math education. We. So it's not terribly surprising that they aren't inclined to extend their thinking past whatever transformation relationships they've been taught explicitly. Their assumption, I suspect, if they think about it at all, is that "This is all" + "This is all I need know if this isn't all." I love when someone comes up with an idea s/he hasn't been taught or led to. My son came into the living room a few years ago, when he was in Algebra II, and asked me if there could be quadratic or higher degree asymptotes. I'm embarrassed to admit that I dismissed the question as silly without giving it any real thought. He came back about 20 minutes later and showed me a quadratic asymptote on his graphing calculator. I've never been more pleased to be wrong. This first unit is EXACTLY getting kids to articulate what you are talking about…"We." If you look at the assessment, it requires kids to explain the reasons undergirding everything. Nothing is taken for granted. So as long as your indictments aren't directed towards this unit, I feel good. Because this unit is actually going against everything you're saying is wrong with traditional calc education. The request to "explain in their own words what the derivative of a function meant" is terribly vague. I wouldn't know whether the questioner was looking for the graphical interpretation that Michael wants, the limit definition, the formula for the specifc derivative, a physical interpretation like velocity, or something else entirely. I've always hated that sort of vague question in math classes. Note: I don't see that sot of vagueness in Sam's handout. Where he asks for an explanation, it is for a specific phenomenon, and the explanation can be made in concrete terms without much ambiguity. "You see, I was talking with a fellow calculus teacher, and we had a big realization. Yes, calculus is hard for kids because of all the algebra. But also, calculus involves something that students have never seen before." This is such a money statement. Calculus is hard for students for a lot of reasons. Specifically, it's hard for students who may not be used to having a math class be hard. This is something that I'm seeing so far this year. Some students are struggling for the first time. They've never looked at a progress report and seen a "C+" so they are having to explore new emotions and develop new habits. @gasstation: I attended that talk more than a decade ago. I'm sure the question was better-phrased than my attempt to recall it. I know that it was pretty damned clear what the researchers were NOT asking: how to find a derivative, but that was pretty much what people offered, or vague ideas that danced around the subject from a procedural perspective, not at all in the ballpark of what was asked. I don't find that surprising. And these were not low-level students at a third tier university barely scraping by, but kids who had excelled in the standard undergraduate calculus at U of Michigan. I think if you search on-line, you may find the actual question that was posed, but in any event, I don't want you to get the impression that it was ill-posed just because I haven't remembered it accurately. I love this approach, too. In fact, I used it this past spring when I taught calculus to nurses and biologists. It was a great success, conceptually (algebraically, they were still devastatingly awful). Instead of referring to it as an area function, I called it an accumulation function. This made it more natural to talk about negative quantities (I used a weight function as my primary example, and the students all could relate to gaining and losing weight). It took us about three (50-minute) sessions to get to a point where we could relate the accumulation function to the rate of change function via the fundamental theorem of calculus (stated without any of the usual symbology, of course). I love that we think alike!!! Did you post anything about this — or do you have any materials you could share? When you did the weight fxn, was your graph's y-axis the "change in weight" or was it just "weight"? I REALLY like the idea of using weight because it's so concrete. I hope hope HOPE I remember this when I get to integrals. I have a terrible memory… For weight (which we talked about in class constantly, but which does not appear on the handouts), the y-axis was the change in weight. The beauty was that by day three, when I asked what the accumulation of the change in weight function was, the class almost unanimously agreed that it was the net change in weight. When I gave them the FTOC handout, they asked why something so obvious was a fundamental theorem. Giggle giggle.
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Related Resources OK. Let's get started, please. First, I wanted to remind you that tomorrow is the first exam in this class. It is taking place here if your last name starts with S or beyond, and it is taking place in Walker top floor if your last name starts from A-R. If you are left-handed and you prefer to have a real table, feel free to go to Walker. Otherwise, try to go to the correct room. It is better. And, well, let's see. We will be basically grading the exam in the afternoon evening and then posting the scores on the online system, so you can check there. And there will be makeup exams on Tuesday, Wednesday and Thursday next week. For times and places, I will announce on the Web page. That is in case things don't go well or in case somehow you cannot make it tomorrow. Remember, if you have an absence for like a sports tournament or something like this, please let us know now. What else? Hopefully you have seen the two practice exams. And one of them you probably talked about in recitation yesterday. The other one we are going to see a bit now. I have had some of you comment that the practice exams are much easier than the problem sets. And, well, let's say the problem sets are harder than the exams because they are designed to make you think and to truly help you put things together, see how you would use the concepts in a real world situation. The exams are a bit easier. The actual exam is comparable to the practice exams in difficulty. It is a little bit, maybe a tiny, tiny bit harder, especially when practice exam 1B was a little bit on the easier side of thing, but not much harder. So, if you do fine on the practice exams, you should do fine tomorrow. As you have seen, the format is basically five or six problems that cover all the material we have seen. And, well, there is a bit of everything. Some of the problems are very easy and shouldn't take more than a minute or two to finish. Some of those will take a bit more thinking. Remember, you only have 50 minutes so you have to think about how to use your time. The exam is designed so that most of you can finish it in 50 minutes, but don't stay stuck on the same problem for a half an hour. Switch to the next page. Let me just tell you first about the list of topics. Basically, the list of topics is simple. It is everything. I mean, everything we have seen so far is on the exam. But let me just remind you of the main topics that we have seen. First of all, we learned about vectors, how to use them, and dot-product. At this point, you probably should know that the dot-product of two vectors is obtained by summing products of components. And geometrically it is the length of A times the length of B times the cosine of the angle between them. And, in particular, we can use dot-product to measure angles by solving for cosine theta in this equality. And most importantly to detect whether two vectors are perpendicular to each other. Two vectors are perpendicular when their dot-product is zero. Any questions about that? No. Is everyone reasonably happy with dot-product by now? I see a stunned silence. Nobody happy with dot-product so far? OK. If you want to look at Practice 1A, a good example of a typical problem with dot-product would be problem 1. Let's see. We are going to go over the practice exam when I am done writing this list of topics. I think probably we actually will skip this problem because I think most of you know how to do it. And if not then you should run for help from me or from your recitation instructor to figure out how to do it. The second topic that we saw was cross-product. When you have two vectors in space, you can just form that cross-product by computing its determinant. So, implicitly, we should also know about determinants. By that I mean two by two and three by three. Don't worry about larger ones, even if you are interested, they won't be on the test. And applications of cross-product, for example, finding the area of a triangle or a parallelogram in space. If you have a triangle in space with sides A and B then its area is one-half of the length of A cross B. Because the length of A cross B is length A, length B sine theta, which is the same as the area of the parallelogram formed by these two vectors. And the other application of cross-product is to find a vector that's perpendicular to two given vectors. In particular, to find the vector that is normal to a plane and then find the equation of a plane. Another application is finding the normal vector to a plane and using that finding the equation of a plane. Basically, remember, to find the equation of a plane, ax by cz = d, what you need is the normal vector to the plane. And the components of the normal vector are exactly the coefficients that go into this. And we have seen an argument for why that happens to be the case. To find a normal vector to a plane typically what we will do is take two vectors that lie in the plane and will take their cross-product. And the cross-product will automatically be perpendicular to both of them. We are going to see an example of that when we look at problem 5 in practice 1A. I think we will try to do that one. Another application, well, we will just mention it as a topic that goes along with this one. We have seen also about equations of lines and how to find where a line intersects a plane. Just to refresh your memories, the equation of a line, well, we will be looking at parametric equations. To know the parametric equation of a line, we need to know a point on the line and we need to know a vector that is parallel to the line. And, if we know a point on the line and a vector along the line, then we can express the parametric equations for the motion of a point that is moving on the line. Actually, starting at point, at time zero, and moving with velocity v. To put things in symbolic form, you will get a position of that point by starting with a position of time zero and adding t times the vector v. It gives you x, y and z in terms of t. And that is how we represent lines. We will look at problem 5 in a bit, but any general questions about these topics? No. Do you have a question? Do we have to know Taylor series? That is a good question. No, not on the exam. [APPLAUSE] Taylor series are something you should be aware of, generally speaking. It will be useful for you in real life, probably not when you go to the supermarket, but if you solve engineering problems you will need Taylor series. It would be good not to forget them entirely, but on the 18.02 exams they probably won't be there. Let me continue with more topics. And then we can see if you can think of other topics that should or should not be on the exam. Third topics would be matrices, linear systems, inverting matrices. I know that most of you that have calculators that can invert matrices, but still you are expected at this point to know how to do it by hand. If you have looked at the practice tests, both of them have a problem that asks you to invert a matrix or at least do part of it. And so it is very likely that tomorrow there will be a problem like that as well. In general, when a kind of problem is on both practice tests it's a good indication that it might be there also on the actual exam. Unfortunately, not with the same matrix so you cannot learn the answer by heart. Another thing that we have learned about, well, I should say this is going to be problem 3 on the test and will on the practice test. On the actual test, too, I think, actually. Anyway, we will come back to it later. A couple of things that you should remember. If you have a system of the form AX equals B then there are two cases. If a determinant of A is not zero then that means you can compute the inverse matrix and you can just solve by taking A inverse times B. And the other case is when the determinant of A is zero, and there is either no solution or there is infinitely many solutions. In particular, if you know that there is a solution, for example, if B is zero there always is an obvious solution, X equals zero, then you will actually have infinitely many. In general, we don't really know how to tell whether it is no solution or infinitely many. Questions about that? Yes? Will we have to know how to rotate vectors and so on? Not in general, but you might still want to remember how to rotate a vector in a plane by 90 degrees because that has been useful when we have done problems about parametric equations, which is what I am coming to next. What we have seen about rotation matrices, that was the homework part B problem, you are not supposed to remember by heart everything that was in part B of your homework. It is a good idea to have some vague knowledge because it is useful culture, I would say, useful background for later in your lives, but I won't ask you by heart to know what is the formation for a rotation matrix. And then we come to, last by not least, the problem of finding parametric equations. And, in particular, possibly by decomposing the position vector into a sum of simpler vectors. You have seen quite an evil exam of that on the last problem set with this picture that maybe by now you have had some nightmares about. Anyway, the one on the exam will certainly be easier than that. But, as you have seen -- I mean, you should know, basically, how to analyze a motion that is being described to you and express it in terms of vectors and then figure out what the parametric equation will be. Now, again, it won't be as complicated on the exam as the one in the problem set. But there are a couple of those on the practice exam, so that gives you an idea of what is realistically expected of you. And now once we have parametric equations for motion, so that means when we know how to find the position vector as a function of a parameter maybe of time, then we have seen also about velocity and acceleration, which the vector is obtained by taking the first and second derivatives of a position vector. And so one topic that I will add in there as well is somehow how to prove things about motions by differentiating vector identities. One example of that, for example, is when we try to look at Kepler's law in class last time. We look at Kepler's second law of planetary motion, and we reduced it to a calculation about a derivative of the cross-product R cross v. Now, on the exam you don't need to know the details of Kepler's law, but you need to be able to manipulate vector quantities a bit in the way that we did. And so on practice exam 1A, you actually have a variety of problems on this topics because you have problems two, four and six, all about parametric motions. Probably tomorrow there will not be three distinct problems about parametric motions, but maybe a couple of them. I think that is basically the list of topics. Anybody spot something that I have forgotten to put on the exam or questions about something that should or should not be there? You go first. Yeah? How about parametrizing weird trigonometric functions? I am not sure what you mean by that. Well, parametric curves, you need to know how to parameterize motions, and that involves a little bit of trigonometrics. When we have seen these problems about rotating wheels, say the cycloid, for example, and so on there is a bit of cosine and sine and so on. I think not much more on that. You won't need obscure trigonometric identities. You're next. Any proofs on the exam or just like problems? Well, a problem can ask you to show things. It is not going to be a complicated proof. The proofs are going to be fairly easy. If you look at practice 1A, the last problem does have a little bit of proof. 6B says that show that blah, blah, blah. But, as you will see, it is not a difficult kind of proof. So, about the same. Yes? Are there equations of 3D shapes that we should know at this point? We should know definitely a lot about the equations of planes on lines. And you should probably know that a sphere centered at the origin is the set of points where distance to the center is equal to the radius of the sphere. We don't need more at this point. As the semester goes on, we will start seeing cones and things like that. But at this point planes, lines. And maybe you need to know about circles and spheres, but nothing beyond that. More questions? Yes? If there is a formula that you have proved on the homework then, yes, you can assume it on the test. Maybe you want to write on your test that this is a formula you have seen in homework just so that we know that you remember it from homework and not from looking over your neighbor's shoulder or whatever. Yes, it is OK to use things that you know general-speaking. That being said, for example, probably there will be a linear system to solve. It will say on the exam you are supposed to solve that using matrices, not by elimination. There are things like that. If a problem says solve by using vector methods, things like that, then try to use at least a vector somewhere. But, in general, you are allowed to use things that you know. Yes? Will we need to go from parametric equations to xy equations? Well, let's say only if it is very easy. If I give you a parametric curve, sin t, sin t, then you should be able to observe that it is on the line y equals x, not beyond that. Yes? Do we have to use -- Yes. I don't know if you will have to use it, but certainly you should know a little bit about the unit tangent vector. Just remember the main thing to know that the unit tangent vector is velocity divided by the speed. I mean there is not much more to it when you think about it. Yes? Kepler's law, well, you are allowed to use it if it helps you, if you find a way to squeeze it in. You don't have to know Kepler's law in detail. You just have to know how to reproduce the general steps. If I tell you R cross v is constant, you might be expected to know what to do with that. I would say -- Basically, you don't need to know Kepler's law. You need to know the kind of stuff that we saw when we derived it such as how to take the derivative of a dot-product or a cross-product. That is basically the answer. I don't see any questions anymore. Oh, you are raising your hand. Yes. How to calculate the distance between two lines and the distance between two planes? Well, you have seen, probably recently, that it is quite painful to do in general. And, no, I don't think that will be on the exam by itself. You need to know how to compute the distance between two points. That certainly you need to know. And also maybe how to find the compliment of a vector in a certain direction. And that is about it, I would say. I mean the more you know about things the better. Things that come up on part Bs of the problem sets are interesting things, but they are usually not needed on the exams. If you have more questions then you are not raising your hand high enough for me to see it. OK. Let's try to do a bit of this practice exam 1A. Hopefully, everybody has it. If you don't have it, hopefully your neighbor has it. If you don't have it and your neighbor doesn't have it then please raise your hand. I have a couple. If you neighbor has it then just follow with them for now. I think there are a few people behind you over there. I will stop handing them out now. If you really need one, it is on the website, it will be here at the end of class. Let's see. Well, I think we are going to just skip problems 1 and 2 because they are pretty straightforward and I hope that you know how to do them. I mean I don't know. Let's see. How many of you have no problem with problem 1? How many of you have trouble with problem 1? OK. How many of you haven't raised your hands? OK. How many of you have trouble with problem 2? OK. Well, if you have questions about those, maybe you should just come see me at the end because that is probably more efficient that way. I am going to start right away with problem 3, actually. Problem 3 says we have a matrix given to us \|1 3 2; 2 0 - 1; 1 1 0\|. And it tells us determinant of A is 2 and inverse equals something, but we are missing two values A and B and we are supposed to find them. That means we need to do the steps of the algorithm to find the inverse of A. We are told that A inverse is one-half of \|1 ... ...; - 1 - 2 5; 2 2 - 6\|. And here there are two unknown values. Remember, to invert a matrix, first we compute the minors. Then we flip some signs to get the cofactors. Then we transpose. And, finally, we divide by the determinant. Let's try to be smart about this. Do we need to compute all nine minors? No. We only need to compute two of them, right? Which minors do we need to compute? Here and here or here and here? Yeah, that looks better. Because, remember, we need to transpose things so these two guys will end up here. I claim we should compute these two minors. And we will see if that is good enough. If you start doing others and you find that they don't end up in the right place then just do more, but you don't need to spend your time computing all nine of them. If you are worried about not doing it right then, of course, you can maybe compute one or two more to just double-check your answers. But let us just do those that we think are needed. The matrix of minors. The one that goes in the middle position is obtained by deleting this row and that column, and we are left with a determinant \|3 2;1 0\|, 3 times 0 minus 1 times 2 should be - 2 should be - 2. Then the one in the lower left corner, we delete the last row and the first column, we are left with \|3 2; 0 - 1\|. 3 times (- 1) is negative 3 minus 0. We are still left with negative three. Is that step clear for everyone? Then we need to go to cofactors. That means we need to change signs. The rule is -- We change signs in basically these four places. That means we will be left with positive 2 and negative 3. Then we take the transpose. That means the first column will copy into the first row, so this guy we still don't know, but here we will have two and here we will have minus three. Finally, we have to divide by the determinant of A. And here we are actually told that the determinant of A is two. So we will divide by two. But there is only one-half here so actually it is done for us. The values that we will put up there are going to be 2 and negative 3. Now let's see how we use that to solve a linear system. If we have to solve a linear system, Ax equals B, well, if the matrix is invertible, its determinant is not zero, so we can certainly write x equals A inverse B. So we have to multiply, that is one-half \| 1 2 - 3; - 1 - 2 5; 2 2 - 6\|. Times B [ 1, - 2, 1]. Remember, to do a matrix multiplication you take the rows in here, the columns in here and you do dot-products. The first entry will be one times one plus two times minus two plus minus three times one, one minus four minus three should be negative six, except I still have, of course, a one-half in front. Then minus one plus four plus five should be 8. Two minus four minus six should be -8. That will simplify to [- 3, 4, - 5]. Any questions about that? OK. Now we come to part C which is the harder part of this problem. It says let's take this matrix A and let's replace the two in the upper right corner by some other number C. That means we will look at 1 3 C; 2 0 - 1; 1 1 0\|. And let's call that M. And it first asks you to find the value of C for which this matrix is not invertible. M is not invertible exactly when the determinant of M is zero. Let's compute the determinant. Well, we should do one times that smaller determinant, which is zero minus negative one, which is 1 times 1 minus three times that determinant, which is zero plus one is 1. And then we have plus C times the lower left determinant which is two times one minus zero is 2. That gives us one minus three is - 2 2C. That is zero when C equals 1. For C equals 1, this matrix is not invertible. For other values it is invertible. It goes on to say let's look at this value of C and let's look at the system Mx equals zero. I am going to put value one in there. Now, if we look at Mx equals zero, well, this has either no solution or infinitely many solutions. But here there is an obvious solution. Namely x equals zero is a solution. Maybe let me rewrite it more geometrically. X 3 y z = 0. 2x - z = 0. And x y = 0. You see we have an obvious solution, (0, 0, 0). But we have more solutions. How do we find more solutions? Well, (x, y, z) is a solution if it is in all three of these planes. That is a way to think about it. Probably we are actually in this situation where, in fact, we have three planes that are all passing through the origin and all parallel to the same line. And so that would be the line of solutions. To find it actually we can think of this as follows. The first observation is that actually in this situation we don't need all three equations. The fact that the system has infinitely many solutions means that actually one of the equations is redundant. If you look at it long enough you will see, for example, if you multiply three times this equation and you subtract that one then you will get the first equation. Three times (x y) - (2x - z) will be x 3y z. Now, we don't actually need to see that to solve a problem. I am just showing you that is what happens when you have a matrix with determinant zero. One of the equations is somehow a duplicate of the others. We don't actually need to figure out how exactly. What that means is really we want to solve, let's say start with two of the equations. To find the solution we can observe that the first equation says actually that <x, y, z> dot-product with <1, 3, 1> =0. And the second equation says <x, y, z> dot-product with <2, 0,- 1> is zero. And the third equation, if we really want to keep it, says we should be also having this. Now, these equations now written like this, they are just saying we want an x, y, z that is perpendicular to these vectors. Let's forget this one and let's just look at these two. They are saying we want a vector that is perpendicular to these two given vectors. How do we find that? We do the cross-product. To find x, y, z perpendicular to <1, 3, 1> and <2, 0, - 1>, we take the cross-product. And that will give us something. Well, let me just give you the answer. I am sure you know how to do cross-products by now. I don't have the answer here, so I guess I have to do it. That should be <- 3, probably positive 3, and then - 6>. That is the solution. And any multiple of that is a solution. If you like to neatly simplify them you could say negative one, one, negative two. If you like larger numbers you can multiply that by a million. That is also a solution. Any questions about that? Yes? That is correct. If you pick these two guys instead, you will get the same solution. Well, up to a multiple. It could be if you do the cross-product of these two guys you actually get something that is a multiple -- Actually, I think if you do the cross-product of the first and third one you will get actually minus one, one, minus two, the smaller one. But it doesn't matter. I mean it is really in the same direction. This is all because a plane has actually normal vectors of all sizes. Yes? I don't think so because -- An important thing to remember about cross-product is we compute for minors, but then we put a minus sign on the second component. The coefficient of j in here, the second component, you do one times minus two times one. That is negative three indeed. But then you actually change that to a positive three. Yes? Well, we don't have parametric equations here. Oh, solving by elimination. Well, if it says that you have to use vector methods then you should use vector methods. If it says you should use vectors and matrices then you are expected to do it that way. Yes? It depends what the problem is asking. The question is, is it enough to find the components of a vector or do we have to find the equation of a line? Here it says find one solution using vector operations. We have found one solution. If you wanted to find the line then it would all the things that are proportional to this. It would be maybe minus 3t, 3t minus 6t, all the multiples of that vector. We do because (0, 0, 0) is an obvious solution. Maybe I should write that on the board. You had another question? Not quite. Let me re-explain first how we get all the solutions and why I did that cross-product. First of all, why did I take that cross-product again? I took that cross-product because I looked at my three equations and I observed that my three equations can be reformulated in terms of these dot-products saying that x, y, z is actually perpendicular these guys and these guys have normal vectors to the planes. Remember, to be in all three planes it has to be perpendicular to the normal vectors. That is how we got here. And now, if we want something that is perpendicular to a bunch of given vectors, well, to be perpendicular to two vectors, an easy way to find one is to take that cross-product. And, if you take any two of them, you will get something that is the same up to scaling. Now, what it means geometrically is that when we have our three planes and they all actually contain the same line -- And we know that is actually the smae case because they all pass through the origin. They pass through the origin because the constant terms are just zero. What happens is that the normal vectors to these planes are, in fact, all perpendicular to that line. The normal vectors -- Say this line is vertical. The normal vectors are all going to be horizontal. Well, it is kind of hard to draw. By taking the cross-product between two normal vectors we found this direction. Now, to find actually all the solutions. What we know so far is that we have this direction <-3 3 - 6>. That is going to be parallel to the line of intersections. Let me do it here, for example, <3 3 - 6>. Now we have one particular solution. 0, 0, 0. Actually, we have found another one, too, which is <- 3, 3, - 6>. Anyway, if a line of solutions -- -- has parametric equation x = - 3t, y = 3t, z = - 6t, anything proportional to that. That is how we would find all the solutions if we wanted them. It is almost time. I think I need to jump ahead to other problems. Let's see. I think problem 4 you can probably find for yourselves. It is a reasonably straightforward parametric equation problem. You just have to find the coordinates of point P. And for that it is a very simple trick. Problem 5. Find the area of a spaced triangle. It sounds like a cross-product. Find the equation of a plane also sounds like a cross-product. And find the intersection of this plane with a line means we find first the parametric equation of the line and then we plug that into the equation of the plane to get where they intersect. Does that sound reasonable? Who is disparate about problem 5? OK. Let me repeat problem 5. First part we need to find the area of a triangle. And the way to do that is to just do one-half the length of a cross-product. If we have three points, P0, P1, P2 then maybe we can form vectors P0P1 and P0P2. And, if we take that cross-product and take the length of that and divide by two, that will give us the area of a triangle. Here it turns out that this guy is <1, 1, 2>, if I look at the solutions, so you will end up with square root of 6 over 2. The second is asking you for the equation of a plane containing these three points. Well, first of all, we know that a normal vector to the plane is going to be given by this cross-product again. That means that the equation of plane will be of a form x plus y plus 2z equals something. If a coefficient is here it comes from the normal vector. And to find what goes in the right-hand side, we just plug in any of the points. If you plug in P0, which is (2, 1, 0) then two plus one seems like it is 3. And, if you want to double-check your answer, you can take P1 and P2 and check that you also get three. It is a good way to check your answer. Then the third part. We have a line parallel to the vector v equals one, one, one through the point S, which is (- 1, 0, 0). That means you can find its parametric equation. X will start at - 1, increases at rate 1. Y starts at zero, increases at rate one. Z starts at zero, increases at rate one. You plug these into the plane equation, and that will tell you where they intersect. Is that clear? And now, in the last one minute, on that side I have one minute, let me just say very quickly -- Well, do you want to hear about problem 6 anyway very quickly? Yeah. OK. Problem 6 is one of these like vector calculations. It says we have a position vector R. And it asks you how do we find the derivative of R dot R? Well, remember we have a product rule for taking the derivative. UV prime is U prime V plus UV prime. It also applies for dot-product. That is dR by dt dot R plus R dot dR by dt. And these are both the same thing. You get two R dot dR/dt, but dR/dt is v for velocity vector. Hopefully you have seen things like that. Now, it says show that if R has constant length then they are perpendicular. All you need to write basically is we assume length R is constant. That is what it says, R has constant length. Well, how do we get to, say, something we probably want to reduce to that? Well, if R is constant in length then R dot R is also constant. And so that means d by dt of R dot R is zero. That is what it means to be constant. And so that means R dot v is zero. That means R is perpendicular to v. That is a proof. It is not a scary proof. And then the last question of the exam says let's continue to assume that R has constant length, and let's try to find R dot v. If there is acceleration then probably we should bring it in somewhere, maybe by taking a derivative of something. If we know that R dot v equals zero, let's take the derivative of that. That is still zero. But now, using the product rule, dR/dt is v dot v plus R dot dv/dt is going to be zero. That means that you are asked about R dot A. Well, that is equal to minus V dot V. And that is it. Simple manipulations. Good luck on the exam tomorrow. I will see you tomorrow
Summary: This text is for a one-term course in intermediate algebra, for students who have had a previous elementary algebra course. A five- step problem-solving process is introduced, and interesting applications are used to motivate students. Coverage progresses from graphs, functions, and linear equations to sequences, series, and the binomial theorem. New to this edition are sections on connecting concepts, study tips, and exercises designed to foster intuitive problem so...show morelving. Bittinger teaches at Indiana University; Ellenbogen at Community College of Vermont. ...show less 2 CD'S INCLUDED GOOD CONDITION ...show less8.86 +$3.99 s/h Good Bookmans AZ Tucson, AZ 2009 Hardcover Good Satisfaction 100% guaranteed. $8.8618.22 +$3.99 s/h VeryGood Follett School Solutions, Inc. Woodridge, IL 0321557182
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Algebra I Workbook This workbook is a collection of practice problems crucial for reinforcing Algebra I concepts necessary to exceed State expectations for required coursework and standardized... More > testing. There are a total of 117 Algebra I practice worksheets, which includes two worksheets for all 50 Bench Mark topics, designed to reinforce concepts within the Algebra I Curriculum. This workbook can also be used to supplement any textbook approved for use in the State. This is a valuable instrument for teaching and a convenient bank of problems to check for understanding and to assist in mastering Algebra I.< Less Hands-On Algebra: Patterns, Equations, and Functions is an easy, step-by-step activity book designed for Kindergarten - Third grade teachers. The book is loaded with games, investigations, and... More > blackline masters that are quick and easy. The focus is to provide all children with hands on approaches to understanding algebraic concepts. Included in this activity book is also a diagnostic assessment on algebraic concepts according to the NCTM standards. If you are as busy, stressed and pressed for time as most teachers, this book will quickly become another best friend.< Less
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And finally, let the students remember that success is 1% inspiration and 99% perspiration. Hard work can never fail and will certainly help them reap rich rewards. Success will then become a habit for them. "Seeing much, suffering much, and studying much, are the three pillars of learning" SELF PRACTICE 16.If 2, ½ are the zeros of px2+5x+r, prove that p= r. 17.If m, n are zeroes of ax2-5x+c, find the value of a and c if m + n = m.n=10 (Ans: a=1/2 ,c=5) 18. What must be subtracted from 8x4 + 14x3 – 2x2 + 7x –8 so that the resulting polynomial is exactly divisible by 4x2+3x-2. (Ans: 14x – 10) 19. What must be added to the polynomial p(x)= x4 + 2x3 – 2x2 + x –1 so that the resulting polynomial is exactly divisible by x2+2x-3. (Ans: x-2) UNIT-3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Like the crest of a peacock so is mathematics at the head of all knowledge. 1.At a certain time in a deer park, the number of heads and the number of legs of deer and human visitors were counted and it was found there were 39 heads & 132 legs. Find the number of deer and human visitors in the park. (Ans:27,12) 3. Find the value of p and q for which the system of equations represent coincident lines 2x +3y = 7, (p+q+1)x +(p+2q+2)y = 4(p+q)+1 Ans: a1 = 2, b1 = 3, c1 = 7 a2 = p + q + 1 , b2 = p + 2q + 2 , c2 = (p + q )+ 1 For the following system of equation the condition must be a1 = b1 = c1 a2b2c2 => 2 =3= p + q +1p + q + 24(p + q) +1 => 2 =7 p + q +14(p + q) +1 = 5p - 2q - 11 = 0 ----------------(2) p + q + - 5 = 0 5p - 2q - 11 = 0 From (1) and (2) 5p + 5q - 25 = 0 5p - 2q - 11 = 0 Solve it, to get q = 2 Substitute value of q in equation (1) p + q - 5 = 0 On solving we get, p = 3 and q = 2 4.Students are made to stand in rows. If one student is extra in a row there would be 2 rows less. If one student is less in a row there would be 3 rows more. Find the number of students in the class. Ans: No. of rows be y Let the number of students be x Number of students in the class will be = xy 6.A train covered a certain distance at a uniform speed. If the train would have been 6km/hr faster, it would have taken 4hours less than the scheduled time. And if the train were slower by 6km/hr, it would have taken 6 hours more than the scheduled time. Find the distance of the journey. Ans: Let the speed of the train by x km/hr And the time taken by it by y Now distance traveled by it is x x y = xy 8. In an election contested between A and B, A obtained votes equal to twice the no. of persons on the electoral roll who did not cast their votes & this later number was equal to twice his majority over B. If there were 18000 persons on the electoral roll. How many voted for B. 11.A cyclist, after riding a certain distance, stopped for half an hour to repair his bicycle, after which he completes the whole journey of 30km at half speed in 5 hours. If the breakdown had occurred 10km farther off, he would have done the whole journey in 4 hours. Find where the breakdown occurred and his original speed. (Ans: 10km, 10km/hr) Ans: Let x be the place where breakdown occurred y be the original speed x + 30 − x = 5 2 x +1030 − (x +10) += 4 2 x60 − 2 x += 5 yy On solving, we get, x = 10 km and y = 10 km/h 12.The population of the village is 5000. If in a year, the number of males were to increase by 5% and that of a female by 3% annually, the population would grow to 5202 at the end of the year. Find the number of males and females in the village. 4.A dealer sells a toy for Rs.24 and gains as much percent as the cost price of the toy. Find the cost price of the toy. Ans: Let the C.P be x ∴Gain = x% ⇒ Gain = x. x 100 S.P = C.P +Gain SP = 24 ⇒ x + x2 =24 100 On solving x=20 or -120 (rej) ∴C.P of toy = Rs.20 5.A fox and an eagle lived at the top of a cliff of height 6m, whose base was at a distance of 10m from a point A on the ground. The fox descends the cliff and went straight to the point A. The eagle flew vertically up to a height x metres and then flew in a straight line to a point A, the distance traveled by each being the same. Find the value of x. Q 6.A lotus is 2m above the water in a pond. Due to wind the lotus slides on the side and only the stem completely submerges in the water at a distance of 10m from the original position. Find the depth of water in the pond. 10.A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left over. When he increased the size of the square by one student he found he was short of 25 students. Find the number of students. 11.A pole has to be erected at a point on the boundary of a circular park of diameter 13m in such a way that the differences of its distances from two diametrically opposite fixed gates A & B on the boundary in 7m. Is it possible to do so? If answer is yes at what distances from the two gates should the pole be erected. 13.X and Y are centers of circles of radius 9cm and 2cm and XY = 17cm. Z is the centre of a circle of radius 4 cm, which touches the above circles externally. Given that ∠XZY=90o, write an equation in r and solve it for r. 5.A man arranges to pay a debt of Rs.3600 in 40 monthly instalments which are in a AP. When 30 instalments are paid he dies leaving one third of the debt unpaid. Find the value of the first instalment. 8.Raghav buys a shop for Rs.1,20,000.He pays half the balance of the amount in cash and agrees to pay the balance in 12 annual instalments of Rs.5000 each. If the rate of interest is 12% and he pays with the instalment the interest due for the unpaid amount. Find the total cost of the shop. 11.Balls are arranged in rows to form an equilateral triangle .The first row consists of one ball, the second two balls and so on. If 669 more balls are added, then all the balls can be arranged in the shape of a square and each of its sides then contains 8 balls less than each side of the triangle. find the initial number of balls. 21.The ratio of the sum of first n terms of two AP's is 7n+1:4n+27. Find the ratio of their 11th terms . Ans: Let a1, a2… and d1, d2 be the I terms are Cd's of two AP's. Sn of one AP = Sn of II AP m[] 2a1 + (n −1)d1 =7n +1 m[+ (n −1)d2 ]4n + 27 2a2 2 ⇒ 2a1 + (n − 1)d1 = 7n +1 2a2 + (n −1)d24n + 27 We have sub. n = 21. ⇒ a1 +10d1 = 148 a2 +10d2 111 ∴ ratio of their 11th terms = 4 :3. 22.If there are (2n+1)terms in an AP ,prove that the ratio of the sum of odd terms and the sum of even terms is (n+1):n HEIGHTS AND DISTANCES 1. If the angle of elevation of cloud from a point 'h' meters above a lake is and the angle of depression of its reflection in the lake is , prove that the height of the cloud is . Ans : If the angle of elevation of cloud from a point 'n' meters above a lake is ∝ and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is h tanβ+ tanα    Let AB be the surface of the lake and Let p be an point of observation such that AP = h meters. Let c be the position of the cloud and c' be its reflection in the lake. Then ∠CPM = ∝ and ∠ MPC1 = β. Let CM = x. 2. From an aero plane vertically above a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aero plane are observed to be α and β. Show that the height of the aero plane above the road is . 3. Two stations due south of a tower, which leans towards north are at distances 'a' and 'b' from its foot. If α and β be the elevations of the top of the tower from the situation, prove that its inclination 'θ' to the horizontal given by 4.The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is α. On advancing 'p' meters towards the foot of the tower, the angle of elevation becomes β. show that the height 'h' of the tower is given by h= 5.A boy standing on a horizontal plane finds a bird flying at a distance of 100m from him at an elevation of 300. A girl standing on the roof of 20 meter high building finds the angle of elevation of the same bird to be 450. Both the boy and the girl are on opposite sides of the bird. Find the distance of the bird from the girl. (Ans: 42.42m) 6.From a window x meters high above the ground in a street, the angles of elevation and depression of the top and the foot of the other house on the opposite side of the street are α and β respectively. Show that the height of the opposite house is Meters. 7.Two ships are sailing in the sea on either side of a lighthouse; the angles of depression of two ships as observed from the top of the lighthouse are 600 and 450 respectively. If the distance between the ships is meters, find the height of the lighthouse. (Ans:200m) Ans: In right ∆ ABC h Tan 60 = BC h 3 = BC ⇒ H = 3 BC In right ∆ ABD h Tan 45 = BD h = BD = 200m ∴ height of light house = 200m 8.A round balloon of radius 'a' subtends an angle θ at the eye of the observer while the angle of elevation of its centre is Φ. Prove that the height of the center of the balloon is a sin θ cosec Φ /2. Ans: Let θ be the centre of the ballon of radius 'r' and 'p' the eye of the observer. Let PA, PB be tangents from P to ballong. Then ∠APB = θ . ∴∠APO = ∠BPO = Let OL be perpendicular from O on the horizontal PX. We are given that the angle of the elevation of the centre of the ballon is φ i.e., ∠OPL = φ In ∆OAP, we have sin θ = OA 9.The angle of elevation of a jet fighter from a point A on the ground is 600. After a flight of 15 seconds, the angle of elevation changes to 300. If the jet is flying at a speed of 720 km/hr, find the constant height at which the jet is flying.(Use 3 =1.732 (Ans: 2598m) 10.A vertical post stands on a horizontal plane. The angle of elevation of the top is 60o and that of a point x metre be the height of the post, then prove that x = 2h . 3 Self Practice 11. A fire in a building B is reported on telephone to two fire stations P and Q, 10km apart from each other on a straight road. P observes that the fire is at an angle of 60o to the road and Q observes that it is an angle of 45o to the road. Which station should send its team and how much will this team have to travel? (Ans:7.32km) Self Practice 12. A ladder sets against a wall at an angle αto the horizontal. If the foot is pulled away from the wall through a distance of 'a', so that is slides a distance 'b' down the wall making an angle βwith the horizontal. Show that cosα− cosβ= a . 13.Two stations due south of a leaning tower which leans towards the north are at distances a and b from its foot. If α, β be the elevations of the top of the tower from these stations, prove that its inclination ϕis given by cotϕ= bcotα− acotβ. b − a 14.In Figure, what are the angles of depression from the observing positions O1 and O2 of the object at A? (Ans: 30o ,45o) Self Practice 15. The angle of elevation of the top of a tower standing on a horizontal plane from a point A is α. After walking a distance d towards the foot of the tower the angle of elevation is found to be β. Find the height of the tower. (Ans: d ) cotα− cotβ Ans: Let BC = x tan β = AB CB tan β = h x h x = tan β x = h cot β ---------(1) tan α = AB DC + CB tan α = h d + x h d + x = = hcotα tanα x = h cot α - d ----------(2) from 1 and 2 h cot β = h cot α - d h (cot α - cot β ) = d d h = cotα− cot β 16. A man on a top of a tower observes a truck at an angle of depression α where tanα = and sees that it is moving towards the base of the tower. Ten minutes later, the angle of depression of the truck is found to be β where tan β=√5, if the truck is moving at a uniform speed, determine how much more time it will take to reach the base of the tower... 10 minutes=600sec A Ans: Let the speed of the truck be x m/sec CD=BC-BD In right triangle ABC 2.Determine the ratio in which the line 2x + y -4 = 0 divide the line segment joining the points A (2,-2) and B (3, 7).Also find the coordinates of the point of division. [Ans:2 : 9 , ( )] Ans : Let the ratio be k:1 Let the co-ordinates of point of division be (x, y) k(3) +1.23k + 2 ∴ x = = k +1k +1 20.Plot the points A(2,0) and B (6,0) on a graph paper. Complete an equilateral triangle ABC such that the ordinate of C be a positive real number .Find the coordinates of C (Ans: (4,2√3) Ans : Proceed by taking C(x, y) AC = BC = AB 21.Find the ratio in which the line segment joining A(6,5) and B(4,-3) is divided by the line y=2 (Ans:3:5) Ans : Let the ratio be k:1 x = 4k + 6 k +1 y = − 3k + 5 k +1 − 3k + 5 = 2 k +1 On solving we get k = 3 : 5 22.The base BC of an equilateral triangle ABC lies on the y-axis. The coordinates of C are (0,-3). If the origin is the midpoint of BC find the coordinates of points A and B. Ans : Hint : The point A will lie on the x axis. Find A using AB = BC = AC. Coordinates of B (0, 3) UNIT-8 SIMILAR TRIANGLES Geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art. 1.ABC is a right-angled triangle, right-angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 6cm and 8 cm. Find the radius of the in circle. (Ans: r=2) 2.ABC is a triangle. PQ is the line segment intersecting AB in P and AC in Q such that PQ parallel to BC and divides triangle ABC into two parts equal in area. Find BP: AB. Ans: Refer example problem of text book. 3.In a right triangle ABC, right angled at C, P and Q are points of the sides CA and CB respectively, which divide these sides in the ratio 2: 1. Prove that 9AQ2= 9AC2 +4BC2 9BP2= 9BC2 + 4AC2 9 (AQ2+BP2) = 13AB2 4.P and Q are the mid points on the sides CA and CB respectively of triangle ABC right angled at C. Prove that 4(AQ2 +BP2) = 5AB2 Self Practice 5. In an equilateral triangle ABC, the side BC is trisected at D. Prove that 9AD2 = 7AB2 Self Practice 6.There is a staircase as shown in figure connecting points A and B. Measurements of steps are marked in the figure. Find the straight distance between A and B. (Ans:10) Ans: Apply Pythagoras theorem for each right triangle add to get length of AB. 7.Find the length of the second diagonal of a rhombus, whose side is 5cm and one of the diagonals is 6cm. (Ans: 8cm) Ans: Length of the other diagonal = 2(BO) where BO = 4cm ∴ BD = 8cm. 8.Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle. Ans: To prove 3(AB2 + BC2 + CA2) = 4 (AD2+ BE2 + CF2) In any triangle sum of squares of any two sides is equal to twice the square of half of third side, together with twice the square of median bisecting it. If AD is the median  AB2 + AC2 = 2AD 2 + BC2   12.Prove that in any triangle the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median, which bisects the third side. 222 1 BC2 Ans: To prove AB + AC = 2AD + 2 16.In the given figure, ∠AEF=∠AFE and E is the mid-point of CA. Prove that BDBF = CDCE Ans: Draw CG ║DF In ∆ BDF CG ║ DF BDBF ∴= ………….(1) BPT CDGF In ∆AFE ∠AEF=∠AFE ⇒AF=AE ⇒AF=AE=CE…………..(2) In ∆ ACG E is the mid point of AC ⇒ FG = AF ∴ From (1) & (2) BDBF = CDCE Hence proved 17. ABCD is a parallelogram in the given figure, AB is divided at P and CD and Q so that AP:PB=3:2 and CQ:QD=4:1. If PQ meets AC at R, prove that AR= AC. D Q C Ans: ∆APR ∼ ∆CQR (AA) ⇒ AP = PR = AR CQQRCR ⇒ AP = AR & AP= 3 AB CQCR5 ⇒ 3AB = AR & CQ= 4 CD = 4 AB 5CQCR55 ⇒ AR = 3 CR4 ⇒ CR = 4 AR3 CR + AR4 =+1 AR3 ⇒ AC = 7 AR3 ⇒ AR = AC Hence proved 18.Prove that the area of a rhombus on the hypotenuse of a right-angled triangle, with one of the angles as 60o, is equal to the sum of the areas of rhombuses with one of their angles as 60o drawn on the other two sides. Ans: Hint: Area of Rhombus of side a & one angle of 60o 19.An aeroplane leaves an airport and flies due north at a speed of 1000 km/h. At the same time, another plane leaves the same airport and flies due west at a speed of 1200 km/h. How far apart will be the two planes after 1½ hours. (Ans:300√61Km) 4.In figure, the incircle of triangle ABC touches the sides BC, CA, and AB at D, E, and F respectively. Show that AF+BD+CE=AE+BF+CD= (perimeter of triangle ABC), Ans: Since the length of tangents from an external point to are equal ∴AF = AE FB = BD EC = CD 10.If PA and PB are tangents to a circle from an outside point P, such that PA=10cm and ∠APB=60o. Find the length of chord AB. Self Practice 11. The radius of the in circle of a triangle is 4cm and the segments into which one side is divided by the point of contact are 6cm and 8cm. Determine the other two sides of the triangle. (Ans: 15, 13) Ans: a = BC = x + 8 b = AC = 6 + 8 = 14cm c = AB = x + 6 a + b + c Semi – perimeter = 14.Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Ans: Same as question No.5 15.QR is the tangent to the circle whose centre is P. If QA \|\| RP and AB is the Self Practice UNIT-10 CONSTRUCTIONS Questions for self practice 1.Draw a line segment AB of length 4.4cm. Taking A as centre, draw a circle of radius 2cm and taking B as centre, draw another circle of radius 2.2cm. Construct tangents to each circle from the centre of the other circle. 2.Draw a pair of tangents to a circle of radius 2cm that are inclined to each other at an angle of 900. 3.Construct a tangent to a circle of radius 2cm from a point on the concentric circle of radius 2.6cm and measure its length. Also, verify the measurements by actual calculations. (length of tangent =2.1cm) 4.Construct an isosceles triangle whose base is 7cm and altitude 4cm and then construct another similar triangle whose sides are1 times the corresponding sides of the isosceles triangle. 5.Draw a line segment AB of length 8cm. taking A as center, draw a circle of radius 4cm and taking B as centre, draw another circle of radius 3cm. Construct tangents to each circle from the center of the other circle. MENSURATION AREAS RELATED TO CIRCLES The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. 1.In the adjoining figure ∆ABC right angled triangle right angled at A. Semi circles are drawn on the sides of the triangle ∆ABC . Prove that area of the Shaded region is equal to area of ∆ABC BC Ans: Refer CBSE paper 2008 2.The sum of the diameters of two circles is 2.8 m and their difference of circumferences is 0.88m. Find the radii of the two circles (Ans: 77, 63) 4.Find the area enclosed between two concentric circles of radii 3.5cm, 7cm. A third concentric circle is drawn outside the 7cm circle so that the area enclosed between it and the 7cm circle is same as that between two inner circles. Find the radius of the third circle 6.From a sheet of cardboard in the shape of a square of side 14 cm, a piece in the shape of letter B is cut off. The curved side of the letter consists of two equal semicircles & the breadth of the rectangular piece is 1 cm. Find the area of the remaining part of cardboard. (Ans: 143.5 cm 2 ) 10.A circular disc of 6 cm radius is divided into three sectors with central angles 1200, 1500,900. What part of the circle is the sector with central angles 1200. Also give the ratio of the areas of three sectors. (Ans: (Area of the circle) 4 : 5 : 3) 19.The boundary of the shaded portion in the adjoining figure consists of our half-circles and two quarter-circles. Find the length of the boundary and the area of the shaded portion, if OA=OB=OC=OD=14cm. (Ans:132 cm, 308 sq cm) Ans: Proceed as in sum no 18. 20.The adjoining figure shows the cross-section of a railway tunnel. The radius of the tunnel is 3.5m (i.e., OA=3.5m) and ∠AOB=90o. Calculate : i.the height of the tunnel. ii.the perimeter of its cross section, including base. iii.the area of the cross-section iv. the internal surface area of the tunnel, excluding base, if its length is 50m. 1.A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 4 cm. The solid is placed in a cylindrical tub, full of water, in such a way that the whole solid is submerged in water. If the radius of the cylindrical tub is 5 cm and its height is 10.5 cm, find the volume of water left in the cylindrical tub (use π = ] (Ans: 683.83 cm³) Ans: No. of solid = vol of cone + vol of hemisphere 1 2 π r3 = π r h + 2.A bucket of height 8 cm and made up of copper sheet is in the form of frustum of right circular cone with radii of its lower and upper ends as 3 cm and 9 cm respectively. Calculate i) the height of the cone of which the bucket is a part ii) the volume of water which can be filled in the bucket iii) the area of copper sheet required to make the bucket (Leave the answer in terms of π(Ans: 129 π cm2) Ans: Let total height be h h3 => = (similar ∆'s ) h+8 9 => h = 4 cm ∴ ht. of cone which bucket is a part = 4 cm Substitute to get Ans.: for ii) iii) 3.A sphere and a cube have equal surface areas. Show that the ratio of the volume of the sphere to that of the cube is 6 : π. 5.Water in a canal 30 dm wide and 12 dm deep is flowing with a velocity of 10 km/h. How much area will it irrigate in 30 minutes if 8 cm of standing water is required for irrigation? (Ans: 225000 cu. m) 6.A cylindrical vessel of diameter 14 cm and height 42 cm is fixed symmetrically inside a similar vessel of diameter 16 cm and height 42 cm. The total space between two vessels is filled with cork dust for heat insulation purposes. How many cubic centimetres of cork dust will be required? (Ans:1980 cu.cm) 7.An ice-cream cone has a hemispherical top. If the height of the cone is 9 cm and base radius is 2.5 cm, find the volume of ice cream cone. (Ans: 91 cu.cm) Ans: Do yourself 8.A building is in the form of a cylinder surrounded by a hemispherical vaulted dome and contains 41 cu m of air. If the internal diameter of the building is equal to its total height above the floor, find the height of the building. ( Ans : 4m) 9.The height of the Cone is 30 cm A small cone is cut of f at the top by a plane parallel to its base if its volume be of the volume of the given cone at what height above the base is the section cut (Ans:20 cm) 10.A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is th of the curved surface of the whole cone, find the ratio of the line segments into which the cone's altitude is divided by the plane. 13.A shuttlecock used for playing badminton has the shape of a frustum of a Cone mounted on a hemisphere. The external diameters of the frustum are 5 cm and 2 cm, and the height of the entire shuttlecock is 7cm. Find the external surface area. (Ans: 74.26cm2) Ans: r1 = radius of lower end of frustum = 1 cm r2 = radius of upper end = 2.5 cm h = ht of frustum = 6cm l = h2 +(r2 −r1)2 = 6.18 cm External surface area of shuttlecock = π (r1 + r2) l + 2π r2 1 On substituting we get, = 74.26 cm2 14.A Solid toy in the form of a hemisphere surmounted by the right circular cone of height 2cm and diameter of the base 4 cm .If a right circular cylinder circumscribes the toy, find how much more space than the toy it will cover. (Ans: 8π) Ans : Self practice 15.A conical vessel of radius 6cm and height 8cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed as shown in the figure. What fraction of water flows out. [Ans: ] Ans: This problem can be done in many ways Let "r" be the radius of sphere In right triangle Tan θ = 6 = 3 16.A golf ball has a diameter equal to 4.1cm. Its surface has 150 dimples each of radius 2mm. Calculate the total surface area which is exposed to the surroundings assuming that the dimples are hemispherical. (Ans: 71.68) 17.A solid metallic circular cone 20cm height with vertical angle 60 is cut into two parts at the middle point of its height by a plane parallel to the base. If the frustum, so obtained be drawn into a wire of diameter cm Find the length of the wire. (Ans:7964.4m) 18.If the areas of the circular bases of a frustum of a cone are 4cm2 and 9cm2 respectively and the height of the frustum is 12cm. What is the volume of the frustum. (Ans:44cm2). Ans: Self practice 19.The lower portion of a hay stack is an inverted cone frustum and the upper part is a cone find the total volume of the hay stack. cm Ans: Self practice 20.A vessel in shape of a inverted cone is surmounted by a cylinder has a common radius of 7cm this was filled with liquid till it covered one third the height of the cylinder. If the height of each part is 9cm and the vessel is turned upside down. Find the volume of the liquid and to what height will it reach in the cylindrical part. (Ans:924π cu cm, 6cm) Ans: Volume of liquid in the vessel = π (7)2 (9) + π (7)2 (3) = 924 cu cm 924 height of cylindrical part = = 6 cm × 49 UNIT 13 STATISTICS AND PROBABILITY Statistics are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man. 5.The mean of 'n' observation is x , if the first term is increased by 1, second by 2 n +1 and so on. What will be the new mean. (Ans: x +) 2 Ans: I term + 1 II term +2 III term + 3 . . n term + n n(n +1) 7.The mode of a distribution is 55 & the modal class is 45-60 and the frequency preceding the modal class is 5 and the frequency after the modal class is 10.Find the frequency of the modal class. (Ans:15) Number of Multiples of 6 or 8 = 50 P(Multiples of 6 or 8) = 50 / 200 = 1/4 2.A box contains 12 balls out of which x are black .if one ball is drawn at random from the box what is the probability that it will be a black ball ? If 6 more black balls are put in the box ,the probability of drawing a black ball is now double of what it was before. Find x. (Ans: x = 3) 12 Total number of possible outcomes (2)if 6 more black balls are put in the bag, then The total number of black balls = x + 6 Total number of balls in the bag = 12 + 6 = 18 According to the question Probability of drawing black ball is second case = 2 X probability drawing of black ball in first case x + 6x = 2 () 1812 x + 6x = 186 6 x + 36 = 18x x = 3 hence number of black balls = 3 3.A bag contains 8 red balls and x blue balls, the odd against drawing a blue ball are 2: 5. What is the value of x? (Ans:20) 2x = 40 ∴ x = 20 4.A card is drawn from a well shuffled deck of cards (i) What are the odds in favour of getting spade? (Ans: 1:3, 3:1, 3:10, 1:25) (ii) What are the odds against getting a spade? (iii)What are the odds in favour of getting a face card? (iv)What are the odds in favour of getting a red king Ans: Total cards 52 Spade = 13 Remaining cards 39 i)The odds in favour of getting spade 13 The odds is not in favour of getting spade 39 = = 1 : 3 ii)The odds against getting a spade 39 The odds not against getting a spade 13 = = 3 : 1 iii)The odds in favour of getting a face card 12 The odds not in favour of getting a face card 40 = = 3 :10 iv)The odds in favour of getting a red king 2 The odds not in favour of getting a red king 50 = = 1 : 25 5 A die is thrown repeatedly until a six comes up. What is the sample space for this experiment? HINT ;A= {6} B={1,2,3,4,5,} Ans: The sample space is = {A, BA, BBA, BBBA, BBBBA……..}) 6.Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a foot ball match? Ans: equally likely because they are mutually exclusive events . 7.A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball , determine the number of blue balls in the bag. (Ans:10) Ans: Let the number of blue balls is the bag be x Then total number of balls is the bag = 5 + x ∴ Number of all possible outcomes = 5 + x Number of outcomes favourable to the event of drawing a blue ball = x (Q there are x blue balls) x ∴ Probability of drawing a blue ball 5 + x 5 Similarly, probability of drawing a red ball = 5 + x According to the answer x5 = 2 () 5 + x5 + x x= 10 8.A box contains 12 balls out of which x are black. If one ball is drawn at random from the box, what is the probability that it will be a black ball? If 6 more black balls are put in the box the probability of drawing a black ball is now double of what it was before. Find x? (Ans: 3) Ans: Number of all possible outcomes = 12 Number of outcomes favourable to the event of drawing black ball = x x Required probability = 12 Now when 6 more black balls are put in the box, Number of all possible outcomes = 12 + 6 = 18 Number of outcomes favourable to the event of drawing a black ball = x + 6 x + 6 ∴ Probability of drawing a black ball = 18 According to the question, x+ 6x = 2 () 1812 ∴ x = 3 9.If 65% of the populations have black eyes, 25% have brown eyes and the remaining have blue eyes. What is the probability that a person selected at random has (i) Blue eyes (ii) Brown or black eyes (iii) Blue or black eyes (iv) neither blue nor brown eyes (Ans: ) 10.Find the probability of having 53 Sundays in (i) a leap year (ii) a non leap year (Ans: ) Ans: An ordinary year has 365 days i.e. 52 weeks and 1 day This day can be any one of the 7 days of the week. ∴ P(that this day is Sunday) = Hence, P(an ordinary year has 53 Sunday) = A leap year 366 days i.e. 52 weeks and 2 days This day can be any one of the 7 days of the week ∴ P (that this day is Sunday) = Hence, P(a leap year has 53 Sunday) = 11.Find the probability that the month June may have 5 Mondays in (i) a leap year (ii) a non leap year (Ans: ) Self Practice 12. Find the probability that the month February may have 5 Wednesdays in (i) a leap year (ii) a non leap year (Ans: ,0) Self Practice 13.Five cards – the ten, jack, queen, king and ace, are well shuffled with their face downwards. One card is then picked up at random. (i)What is the probability that the card is a queen? (ii)If the queen is drawn and put aside, what is the probability that the second card picked up is a (a) an ace (b) a queen (Ans: ,0) Ans : Here, the total number of elementary events = 5 (i) Since, there is only one queen ∴ Favourable number of elementary events = 1 ∴Probability of getting the card of queen = (ii) Now, the total number of elementary events = 4 (a)Since, there is only one ace ∴ Favourable number of elementary events = 1 ∴Probability of getting an ace card = (b)Since, there is no queen (as queen is put aside) ∴ Favourable number of elementary events = 0 ∴Probability of getting a queen = = 0 14.A number x is chosen at random from the numbers -3, -2, -1, 0 1, 2, 3. What is the probability that x < 2 (Ans: ) 15.A number x is selected from the numbers 1,2,3 and then a second number y is randomly selected from the numbers 1,4,9. What is the probability that the product xy of the two numbers will be less than 9? (Ans: ) Ans : Number X can be selected in three ways and corresponding to each such way there are three ways of selecting number y . Therefore , two numbers can be selected in 9 ways as listed below: (1,1), (1,4), (2,1), (2,4), (3,1) ∴ Favourable number of elementary events = 5 Hence, required probability = 16.In the adjoining figure a dart is thrown at the dart board and lands in the interior of the circle. What is the probability that the dart will land in the shaded region. 17. In the fig points A ,B ,C and D are the centres of four circles ,each having a radius of 1 unit . If a point is chosen at random from the interior of a square ABCD ,what is the probability that the point will be chosen from the shaded region . 18. In the adjoining figure ABCD is a square with sides of length 6 units points P & Q are the mid points of the sides BC & CD respectively. If a point is selected at random from the interior of the square what is the probability that the point will be chosen from the interior of the triangle APQ. D Q C 19.In a musical chair game the person playing the music has been advised to stop playing the music at any time within 2 minutes after she starts playing. What is the probability that the music will stop within the half minute after starting. (Ans: ) Ans: Here the possible outcomes are all the numbers between 0 and 2. This is the portion of the number line from 0 to 2 as shown in figure. Let A be the event that 'the music is stopped within the first half minute.' Then, outcomes favorable to event A are all points on the number line from O to Q i.e., from 0 to 1/2 . Q P 0 ½ 1 2 The total number of outcomes are the points on the number line from O to P i.e., 0 to 2. ∴ P (A) = Length of OQ = 1 / 2 = 1 Length of OP 2 4 20.A jar contains 54 marbles each of which is blue , green or white. The probability of selecting a blue marble at random from the jar is and the probability of selecting a green marble at random is . How many white marbles does the jar contain? (Ans:12)
Introduction to Analysis 9780387943695 ISBN: 0387943692 Publisher: Springer Summary: This book is intended to serve as a textbook for an introductory course in mathematical analysis. In preliminary form it has been used in this way at the University of Michigan, Indiana University, and Texas A&M University. The book addresses the needs of a beginning graduate student, that is a student who has completed an undergraduate program with a mathematics major. Brown, Alren is the author of Introduc...tion to Analysis, published under ISBN 9780387943695 and 0387943692. Five hundred fourteen Introduction to Analysis textbooks are available for sale on ValoreBooks.com, ninety used from the cheapest price of $15.64, or buy new starting at $72.22 provides an introduction to mathematical analysis, intended to serve the needs of the beginning graduate student in mathematics or p [more] This item is printed on demand. This book provides an introduction to mathematical analysis, intended to serve the needs of the beginning graduate student in mathematics or physics. It is carefully organized, starting from an overview of preliminary mate da
How to order your own hardcover copy Wouldn't you rather have a bound book instead of 640 loose pages? Your laser printer will thank you! Order from Amazon.com. Chapter 8: The Discrete Fourier Transform Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids. The discrete Fourier transform (DFT) is the family member used with digitized signals. This is the first of four chapters on the real DFT, a version of the discrete Fourier transform that uses real numbers to represent the input and output signals. The complex DFT, a more advanced technique that uses complex numbers, will be discussed in Chapter 29. In this chapter we look at the mathematics and algorithms of the Fourier decomposition, the heart of the DFT.
Mathematics - Calculus (464 results)One of the purposes of the elementary working courses in mathematics of the freshman and sophomore years is to exhibit the bond that unites the experimental sciences. "The bond of union among the physical sciences is the mathematical spirit and the mathematical method which pervade them." For this reason, the applications of mathematics, not to artificial problems, but to the more elementary of the classical problems of natural science, find a place in every working course in mathematics. This presents probably the most difficult task of the text-book writer,- namely, to make clear to the student that mathematics has to do with the laws of actual phenomena, without at the same time undertaking to teach technology, or attempting to build upon ideas which the student does not possess. It is easy enough to give examples of the application of the processes of mathematics to scientific problems; it is more difficult to exhibit by these problems, how, in mathematics, the very language and methods of thought fit naturally into the expression and derivation of scientific laws and of natural concepts.<br><br>It is in this spirit that the authors have endeavored to develop the fundamental processes of the calculus which play so important a part in the physical sciences; namely, to place the emphasis upon the mode of thought in the hope that, even though the student may forget the details of the subject, he will continue to apply these fundamental modes of thinking in his later scientific or technical career. It is with this purpose in mind that problems in geometry, physics, and mechanics have been freely used. The problems chosen will be readily comprehended by students ordinarily taking the first course in the calculus.<br><br>A second purpose in an elementary working course in mathematics is to secure facility in using the rules of operation which must be applied in calculations. The present volume is intended to form a sound introduction to a study of the Integral Calculus, suitable for a student beginning the subject. Like its companion, the Differential Calculus for Beginners, it does not therefore aim at completeness, but rather at the omission of all portions of the subject which are usually regarded as best left for a later reading.<br><br>It will be found, however, that the ordinary processes of integration are fully treated, as also the principal methods of Rectification and Quadrature, and the calculation of the volumes and surfaces of solids of revolution. Some indication is also afforded to the student of other useful applications of the Integral Calculus, such as the general method to be employed in obtaining the position of a Centroid, or the value of a Moment of Inertia. Florian Cajori's A History of Mathematics is a seminal work in American mathematics. The book is a summary of the study of mathematics from antiquity through World War I, exploring the evolution of advanced mathematics. As the first history of mathematics published in the United States, it has an important place in the libraries of scholars and universities. A History of Mathematics is a history of mathematics, mathematicians, equations and theories; it is not a textbook, and the early chapters do not demand a thorough understanding of mathematical concepts. The book starts with the use of mathematics in antiquity, including contributions by the Babylonians, Egyptians, Greeks and Romans. The sections on the Greek schools of thought are very readable for anyone who wants to know more about Greek arithmetic and geometry. Cajori explains the advances by Indians and Arabs during the Middle Ages, explaining how those regions were the custodians of mathematics while Europe was in the intellectual dark ages. Many interesting mathematicians and their discoveries and theories are discussed, with the text becoming more technical as it moves through Modern Europe, which encompasses discussion of the Renaissance, Descartes, Newton, Euler, LaGrange and Laplace. The final section of the book covers developments in the late 19th and early 20th Centuries. Cajori describes the state of synthetic geometry, analytic geometry, algebra, analytics and applied mathematics. Readers who are not mathematicians can learn much from this book, but the advanced chapters may be easier to understand if one has background in the subject matter. Readers will want to have A History of Mathematics on their bookshelves. The aim of this work is to give a brief exposition of some of the devices employed in solving differential equations. The book presupposes only a knowledge of the fundamental formulæ of integration, and may be described as a chapter supplementary to the elementary works on the integral calculus.<br><br>The needs of two classes of students, with whom the author has been brought into contact in the course of his experience as a teacher, have determined the character of the work. For the sake of students of physics and engineering who wish to use the subject as a tool, and have little time to devote to general theory, the theoretical explanations have been made as brief as is consistent with clearness and sound reasoning, and examples have been worked in full detail in almost every case. Practical applications have also been constantly kept in mind, and two special chapters dealing with geometrical and physical problems have been introduced.<br><br>The other class for which the book is intended is that of students in the general courses in Arts and Science, who have more time to gratify any interest they may feel in this subject, and some of whom may be intending to proceed to the study of the higher mathematics. The present small volume is intended to form a sound introduction to a study of the Differential Calculus suitable for the beginner. It does not therefore aim at completeness, but rather at the omission of all portions which are usually considered best left for a later reading. At the same time it has been constructed to include those parts of the subject prescribed in Schedule I. of the Regulations for the Mathematical Tripos Examination for the reading of students for Mathematical Honours in the University of Cambridge.<br><br>Particular attention has been given to the examples which are freely interspersed throughout the text. For the most part they are of the simplest kind, requiring but little analytical skill. Yet it is hoped they will prove sufficient to give practice in the processes they are intended to illustrate. This little book has been written for two classes of persons: those who wish, for purposes of culture, to know, in as simple and direct a way as possible, what the calculus is and what it is for; and students primarily engaged in work in chemistry, astronomy, economics, etc., who have not time or inclination to take long courses in mathematics, yet who would like to know how to use a tool as fine as the calculus. The pure mathematician will note the omission of various subjects that are important from his point of view; but for him there are admirable and lengthy treatises on pure calculus. Also the student whose experience has led him to conceive of mathematical study as the doing of interminable lists of exercises, will be surprised and, possibly, disappointed. This book is a reading lesson in applied mathematics. Fancy exercises have been avoided. The examples are, for the most part, real problems from mechanics and astronomy. This plan has been pursued in the conviction that such problems are just as good as make-believe ones for purposes of discipline, and a good deal better for purposes of knowledge. In writing the present treatise on the Integral Calculus the object has been to produce a work at once elementary and complete adapted for the use of beginners, and suflScient for the wants of advanced students. In the selection of the propositions, and in the mode of establishing them, I have endeavoured to exhibit fiiUy and clearly the principles of the subject, aud to illustrate all their most important results. The process of summation has been repeatedly brought forward, with the view of securing the attention of the student to the notions which form the true foundation of the Integral Calculus itself, as well as of its most valuable applications. Considerable space has been devoted to the investigations of the lengths and areas of curves and of the volumes of solids, and an attempt has been made to explain those difficulties which usually perplex beginners especially with reference to the limits of integrations. The transformation of multiple integrals is one of the most interesting parts of the Integral Calculus, and the experience of teachers shews that the usual modes of treating it are not free from obscurity. I have therefore adopted a method different from those of previous elementary writers. In revising this book for a second edition, I have endeavoured to give references to, and in some cases accounts of, the numerous original researches in Dynamics which have been published by various investigators since the first edition appeared. I have moreover added some historical matter, and rewritten many sections. It is not necessary to specify these in detail, but perhaps I may mention that the new explanation of the transformation-theory of Dynamics in 125 sprang from a desire to do justice to the earliest great work of Hamilton's genius: that the changes in 69 (the motion of a body about a fixed point under no forces) arose from my opinion that the Jacobian functions are preferable to the Weierstrassian in the numerical computations: and that I should have liked to give a fuller proof of Sundman's theorem (181), but thought it better to give only such an account as might impel the reader to consult Mr Sundman's own accessible and readable memoir.<br><br>I wish again to record my obligations to the staff of the Cambridge University Press. Analytical science, after having been long neglected in these countries as an elementary department of education, has, within a few years, been cultivated by the young aspirants for mathematical celebrity with an ardour; and prosecuted with a rapidity and success, which its warmest admirers could scarcely have hoped for. This change would probably have taken place at an earlier period, but for the obstacle opposed to it by the want of treatises on the subject, in our language, of a sufficiently elementary nature. The restless activity of the human mind in the pursuit of knowledge was not long to be checked by so trifling an impediment, and our students soon found in foreign works that which our own professors had failed to supply; and though the medium of these treatises, analytical science began, and has continued, to be cultivated at the universities with singular success. Ox account of the present disturbed state of public afairs, the publication of the Mathematical Monthly will be discontinued until further notice. Electrotyped md Printed by Welch, Bigelow, md Company. These lectures on the analytical formulæ relating to the motion of the top were delivered on Monday, Tuesday, Wednesday, and Thursday, October 12-15, 1896. They were reported and prepared in manuscript form by Professor H. B. Fine of Princeton University, and the manuscript was revised by Professor Klein. The topics in this book are arranged for primary courses in calculus in which the formal division into differential calculus and integral calculus is deemed necessary. The book is mainly made up of matter from my Infinitesimal Calculus. Changes, however, have been made in the treatment of several topics, and some additional matter has been introduced, in particular that relating to indeterminate forms, solid geometry, and motion. The articles on motion have been written in the belief that familiarity with the notions of velocity and acceleration, as treated by the calculus, is a great advantage to students who have to take mechanics.<br><br>Part of the preface of my Infinitesimal Calculus applies equally well to this book. Its purpose is to provide an introductory course for those who are entering upon the study of calculus either to prepare themselves for elementary work in applied science or to gratify and develop their interest in mathematics. Little more has been discussed than what may be regarded as the essentials of a primary course. An attempt is made to describe and emphasise the fundamental principles of the subject in such a way that, as much as may reasonably be expected, they may be clearly understood, firmly grasped, and intelligently applied by young students. There has also been kept in view the development in them of the ability to read mathematics and to prosecute its study by themselves.<br><br>With regard to simplicity and clearness in the exposition of the subject, it may be said that the aim has been to write a book that will be found helpful by those who begin the study of calculus without the guidance and aid of a teacher. The Directly-Useful Technical Series requires a few words by of introduction. Technical books of the past have arranged themselves largely under two sections: the Theoretical and the Practical. Theoretical books have been written more for the training of college students than for the supply of information to men in practice, and have been greatly filled with problems of an academic character. Practical books have often sought the other extreme, omitting the scientific basis upon which all good practice is built, whether discernible or not. The present series is intended to occupy a midway position. The information, the problems and the exercises are to be of a directly-useful character, but must at the same time be wedded to that proper amount of scientific explanation which alone will satisfy the inquiring mind. We shall thus appeal to all technical people throughout the land, either students or those in actual practice. In annouacing the commencement of a new series, the Editors desire to explain the modifications which will distinguish it from the former series. The Messenger of Mathematics was projected about ten years ago, chiefly with the view of encouraging original research in the three Universities, among junior graduates and others. It was thought that through the Messenger many valuable papers might be made public which their authors would not have deemed of sufficient interest to communicate to Scientific Societies. An examination of the Five Volumes already published will make it evident that the Editors have throughout endeavoured to keep their original purpose steadily in view. While feeling, however, that they have every reason to be satisfied with the success achieved by the Messenger regarded as a stimulus to original research in junior students, they have also great satisfaction in acknowledging that no inconsiderable proportion of its contents have been supplied by writers of established reputation, who rank amongst the foremost mathematicians of the age; and it is this fact in particular which now induces them to appeal directly to the mathematical world at large, and to remove from their title-page any words which might be supposed to limit the sphere of usefulness of the Messenger. A procedure for the direct numerical integration of the steady-state, elastic scattering neutron transport equation is presented. Conditions for the existence, uniqueness and non-negativeness of the numerical solution are obtained. Under these conditions it is shown that the scheme is stable and the numerical solution converges, with refinement of the mesh, to the solution of the transport equation. To the Society for the Promotion of Engineering Ediication: The committee was appointed at a joint meeting of mathematicians and engineers held in Chicago, December30-31, 1907, under the auspices of the Chicago Section of the American Mathematical Society, and Sections A and Dof the American Association for the Advancement of Science, and on the suggestion of oiBScers of the Society for the Promotion of Engineering Education who were there present, the committee was instructed to report to this Society. The membership of the committee is as follows: Alger, Philip R., t professor of mathematics, U.S. Navy, Annapolis, Md. Campbell Donald F., professor of mathematics, Armour Institute of Technology, Chicago, Hi. Engler, Edmund A., president of the Worcester Polytechnic Institute, Worcester, Mass. Haseins Charles N., assistant professor of mathematics, Dartmouth College, Hanover, N.H. Howe, CharlesS., president. Case School of Applied Science, Cleveland, Ohio. KuiCHLiNG, Bmil, consulting civil engineer. New York City. Magrudbb, William T., professor of mechanical engineering, Ohio State University, Columbus, Ohio. MoDJESKi, Balph, civil engineer, Chicago, Hi. Osgood, William P., professor of mathematics. Harvard University, Cambridge, Mass. SiiiCHTEB, CharlesS., consulting engineer of the U.S. Reclamation Service, professor of applied mathematics. University of Wisconsin, Madison, Wis. For an account of the Chicago meeting, see Scienee for 1908 (July 12, 24, and 31;August 7 and 28;and September4). tDeceased. TlHE theory of functions of a real variable, as developed during the last-- few decades, is a body of doctrine resting, first upon a definite conception of the arithmetic continuum which forms the field of the variable, and which includes a precise arithmetic theory of the nature of a limit, and secondly, upon a definite conception of the nature of the functional relation. The procedure of the theory consists largely in the development, based upon precise definitions, of a classification of function.1-, according as they possess, or do not possess, certain peculiarities, such as continuity, differentiability, c.throughout the domain of the variable, or at points forming a selected set contained in that domain. The detailed consequences of the presence, or of the absence, of such peculiarities are then traced out, and are applied for the purpose of obtaining conditions for the validity of the processes of Mathematical Analysis. These processus, which have boon long employed in the so-called Infinitesimal Calculus, consist essentially in the ascertainment of the existence, and in the evaluation, of limits, and are subject, In every case, to restrictive assumptions which are necessary conditions of their validity, The object to be attained by the theory of functions of a real variable consists then largely in the precise formulation of necessary and sufficient conditions for the validity of the limiting processes of Analysis. A necessary requisite in such formulation is a language descriptive of particular aggregates of values of the variable, in relation to which functions possess definite peculiarities. This language is provided by the Theory of Sets of Points, also known, in its more general aspect, as the Theory of Aggregates, which contains an analysis of the peculiarities of structure and of distribution in the field of the variable which such sets of points may possess. This theory, which had its origin in the exigencies of a critical theory of functions, and has since received wide applications, not only in Pure Analysis, but also in Geometry, must be regarded as an integral part of the subject. A most important part of the theory of functions is the theory of the representation of functions in a prescribed manner, especially by means of secies or sequences of functions of prescribed types. Although I have entitled the present Volume, Dynamics of Material Systems fyet the investigations contained in it are far from comprising all which a complete treatise on that subject requires. They are indeed almost wholly confined to those particular systems in which the internal forces, brought into action, either effectively or potentially, by means of the external forces, enter in equal and opposite pairs; so that they disappear in the equations of motion formed on DAlembert sprinciple. I say almost wholly, because, in the last Chapter but one of the Volume, the motion of the particles of an elastic body is to a certain extent discussed: and herein the elastic forces, which are internal forces, do not disappear, but enter as effective forces, the action of which is determined by Hooke slaw, or by an equivalent assumption of a property of such matter. In all other cases, in a rigid body, in a rigid system which is maintained in a state of relative rest by rigid rods and similar modes of constraint, in systems wherein every mutual action of attraction or repulsion is accompanied by an equal and opposite reaction, the internal forces disappear from the equations of motion. It is now some years since I was requested by the Sjnadics of the University Press to allow my papers on mathematical and physical subjects, which are scattered over various Transactions and scientific Journals, to be reprinted in a collected form. Many of these were written a long time ago, and science has in the mean time progressed, and it seemed to me doubtful whether it was worth while now to reprint a series of papers the interest of which may in good measure be regarded as having passed away. However, several of my scientific friends, and among them those to whose opinions I naturally pay the greatest deference, strongly urged me to have the papers reprinted, and I have accordingly acceded to the request of the Syndics. I regret that in consequence of the pressure of other engagements the preparation of the first volume has been so long in hand. The arrangement of the papers and the mode of treating them in other respects were left entirely to myself, but both the Syndics and my friends advised me to make the reprint full, leaning rather to the inclusion than exclusion of a paper in doubtful cases. I have acted on this advice, and in the first volume, now presented to the public, I have omitted nothing but a few papers which were merely controversial. From the Cambridge and Dublin Mathematical Journal, Vol.in. p.121, March, 1848. Notes On Hydrodynamics. III. On the Dynamical Equations. In reducing to calculation the motion of a system of rigid bodies, or of material points, there are two sorts of equations with which we are concerned; the one expressing the geometrical connexions of the bodies or particles with one another, or with curves or surfaces external to the system, the other expressing the relations between the changes of motion which take place in the system and the forces producing such changes. The equations belonging to these two classes may be called respectively the geometrical, and the dynamical equations. Precisely the same remarks apply to the motion of fluids. The geometrical equations which occur in The series of notes on Hydrodynamics which are printed in Vols. n., in. and rv. of the Cambridge and Dublin Mathematical Journal, were written by agree ment between Sir William Thomson and myself mainly for the use of Students. As far as my own share in the series is concerned, there is little contained hi the notes which may not be found elsewhere. Acting however upon the general advice of my friends, I have included my share of the series in the present reprint. It may be convenient to give here the references to the whole series. I. On the Equation of Continuity (Thomson), Vol. n.p. 282.Ii. On the Equation of the Bounding Surface (Thomson), Vol.in. p.89. IIL (Stokes) as above. IV. Demonstration of a Fundamental Theorem (Stokes), Vol.in. p.209. V.On the Vis Viva of a Liquid in motion (Thomson), Vol. iv.p. 90.Vi. On Waves (Stokes), Vol. rv. p.219. s.n. Introduction. The object of this paper is to study the differential invariants that arise under the continuous transformation group of six parameters, known as the inversion group of the plane, and given by the equations of transformation (Q!iYi+Q!272)(a:+y) + ( i+3m+j32T2)a;-(ata+ffi72-327i)y+9i X= F= (7i +72)(x2+22)+2TiX-272y+l (a27i -aiYz) (x+y)+ (a2+327i -3i72)x+ (ai 3 i7i -3272)y+i8s (7i+72 )(x2+22)+27ia;-272j+l Inversion geometry for the most part has been studied synthetically, in connection with the theory of functions of a complex variable satisfying the general linear fractional transformation group 72 + 0 Thus it is known that the families of minimal lines, circles, and logarithmic equiangular double spirals remain invariant under Ge.t The justification for the study of the group from the standpoint of differential geometry, aside from the classification or codification of the invariants under the group, is the bringing to light of some new properties of the logarithmic double spirals. In part one, the Lie theory is employed in finding the differential invariant (I5) of lowest order, while in part two the interpretation of this invariant is given by making use of a certain family of bicircular quartic curves. Furthermore the differential equation of the logarithmic equiangular double spirals is found See Holzmiiller: Theorie der Isogonal Verwandenschaften; also Cole, Linear Functions of a Complex Variable, Annals of Mathematics, Vol. V:No. 4. tThroughout this paper we shall refer to the inversion group as Gt. Single and Double Position could be solved much more readily by algebra than by the rather absurd rules given for such problems in books on arithmetic. In like manner, I could find no interest in the Differential Calculus till, after wading through two hundred pages of nmtter having no apparent use (and for the most part really useless), I found the calculus available for the ready solution of problems in Maxima and Minima. This little work has been planned with direct reference to my own experience at school and college. The usual method of teaching the Differential and Integral Calculus seems to me almost as absurd (quite as absurd it could scarcely be) as the plan by which children, instead of being taught how to speak whether their own language or another are made to learn by rote rules relating to the philosophy -of language such as not one grammarian in ten thousand ever thinks about in after life.
... More About This Book 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. Editorial Reviews From the Publisher "An antidote to the way math is taught in schools." —Library Journal Library Journal This is not a book for people who are afraid of math, although it is presented as an antidote to the way math is often taught in school. The author expresses a fascination with mathematical principles and a joy in number manipulation that will seem entirely unfamiliar to those who dislike math in the first place. The book offers a concise overview of basic arithmetic and helpful techniques for solving story problems (the bane of every schoolchild's existence). It's not quite as much fun as a book of math puzzles, but a grade-school student who enjoys math might like this book, or it could be used as a workbook for a class. For popular math collections.-- Amy Brunvand, Fort Lewis Coll. Lib., Durango, Col. Booknews Published by Coast Publishing, PO Box 3399, Coos Bay, OR 97420. Basic information on how to simplify computations by reordering, rearranging, and breaking up numbers, using equivalents and identities, and approximating and rounding off 18, 2012 Great! Even though I was already advanced in math, this book provided me some awesome shortcuts. But if I think about it the concepts are pretty simple. In fact, I even knew some of the principles they taught me. This is a good buy. 1 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted April 17, 2013 {<_>} ?.John I am 15 in rp, 13 in rl. What about uNadia to julia Thats Nico your talkin to bcuz i have his email and i can see his comments stuffTy How are u doinJulia to john Real classy nico 0 out of 1 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Math Learning Center Mission Statement Our goal is to bring free, fun, and friendly math help to the students at MSU. We ask that all students bring their MSU ID with them to enter the MLC. Maps of the MLC Flash is required to view the maps. About the MLC Location: The main ann_location of the Math Learning Center is room C126A in Wells Hall. There are also 5 satellite ann_locations in various neighborhood centers (Click here for more information). Get free help for any 1825, 100-level and 200-level math class. Limited assistance is available for 300-level classes. Ask questions in a relaxed, friendly atmosphere. With the help of experienced staff in the MLC, you will come to better understand the concepts covered in your math class. Why You Should Come Soon: Surveys show that students do significantly better if they regularly keep up with their homework and seek help early. MLC tutors do not do homework for you. Instead they help you understand concepts and put you on the right track quickly to help you succeed when you are in class, on your own. Tutors: The MLC is staffed primarily by teaching assistants who are currently teaching the courses in which they are providing assistance. The high quality of this staff is verified by their ratings from student evaluations of the classes they teach. Surveys of those who use the center show they rate our staff to be very friendly and very knowledgeable. The MLC maintains an average ratio of 3 or 4 to 1 (students to tutors) so that students get appropriate assistance. Of course, near exam and assignment times, it will be busier. Effectiveness: Almost all students surveyed indicate that the MLC has helped increase their math grades by anywhere from .5 to 1.5 on the standard 4.0 scale. Special Courses: Please note that upper-level courses, and special courses such as MTH 112, MTH 201, MTH 202, and MTH 299 have restricted tutoring hours available. (Click the Wells Hall link for current information.)
math text! An engineer and a mathematician collaborated for a 'real world' text that is full of examples and exercises. The methods selected are those most frequently used in engineering and the 'hard' sciences. This book is structured to gradually increase in level of difficulty so that even those who are weak in math can master it. Try it, you'll like it! Ideal for an undergrad applied math or intro grad course. Among its features, this edition: -reviews ordinary differential equations, including a number of physical applications -introduces modern matrix theory with a separate chapter on applications -includes series methods, Laplace transforms, vector analysis, Fourier series, partial differential equations, numerical methods using finite differences, and complex variables -presents the material so that step-by-step derivations can be followed easily -contains many examples and problems to reinforce understanding of the mathematical methods presented
0131499793 9780131499799 A First Course in Statistics:For algebra-based Introductory Statistics courses. The Ninth Edition of this best selling introductory text features a stronger emphasis on critical thinking and sound decision making through its extensive coverage of data collection and analysis. In addition, it features a balanced approach to both the theory and application of statistics. Incorporating more pedagogy than ever before, the text helps students develop statistical thinking and teaches students to properly assess the credibility of inferences from the vantage point of both the consumer and the producer. Back to top Rent A First Course in Statistics 9th edition today, or search our site for James T textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
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For examination in June and November 2014 This syllabus is available only to Centers ... Cambridge IGCSEAdditionalMathematics (US) (0459) syllabus content will cover the ... 3.1 Goals Cambridge IGCSEMathematicssyllabus is designed as a two-year course for examination at age 16-plus. The ... 2012-2014. 2 International General Certificate of Secondary Education ... AdditionalMathematics International Mathematics ... TISB has a team of experienced physics teachers, well versed in the Cambridge IGCSEsyllabus For examination in June and November 2014Mathematics (Syllabus D) For Centres in Mauritius 4029 ... • 0581 Cambridge IGCSEMathematics (with Coursework) • 4021 Cambridge O Level Mathematics A ... Additionalsyllabus-specific support is available from our secure Teacher Support website ... together with the basic sciences and mathematics, for more advanced studies in agriculture ... Syllabus content Cambridge IGCSE Agriculture 0600 11 4. Syllabus content 1 General agriculture ... Additionalsyllabus-specific support is available from our secure Teacher Support website IGCSE OPTIONS 2013 – 2014 . 2 ted Page 3: Introduction ... The additionalmathematicssyllabus is intended for high ability candidates who have achieved, or are ... study IGCSEmathematics in Year 10 and Additionalmathematics in Year 11. 0606 Cambridge IGCSEAdditionalMathematics June 2013 0637 Cambridge IGCSE Child Development June 2013 Further revised for ... For further information, please refer to the syllabus for 2014. Latin Cambridge IGCSE Latin, Syllabus 0480 From June 2013 2014IGCSE . CURRICULUM BOOKLET . 2 . Contents Page . ... IGCSEMathematics at SJII is a curriculum that challenges students at all levels. ... Both the International Mathematics and the AdditionalMathematics courses are assessed by external exams at the ... IGCSE Subject Selection for examinations in 2014 page 10. ... New AdditionalMathematics, Pan Pacific Publications (S) Pte Ltd, 2001 ... The course is designed to meet the aims and assessment objectives of the IGCSE Drama syllabus and covers the They then study the AdditionalMathematics Free Standing Maths Qualification (OCR syllabus 6993) and take that exam in June. This is a bridging course ... In April 2014 the History Department ran an iGCSE trip to Munich and there are plans in place for the next History trip to Russia in ... AdditionalMathematics (with coursework) International ... SYLLABUS Cambridge IGCSE ® Mathematics ... 0581 For examination in June and November 2014 *This syllabus is accredited for use in England, Wales and Northern Ireland as a Cambridge International Level 1/Level 2 Certificate ... SEPTEMBER 2013 – JUNE 2014 ... You will then choose an additional four subjects which can be chosen from the range described in this ... Cambridge International Examinations (CIE) IGCSESyllabus: Mathematics 0580 AIMS OF THE COURSE: SEPTEMBER 2013 – JUNE 2014 . ... The main aims of the IGCSEMathematics course are to develop the students' athematical m ... The History IGCSEsyllabus looks at some of the major international issues of the twentieth century, ... 5 MATHEMATICSSyllabus Edexcel's IGCSEsyllabus A (4MA0) Assessment Two terminal two hour examinations, both with calculators A good result at IGCSE is a very valuable commodity, but Mathematics at Godolphin is also seen as In addition to the syllabus for the IGCSE exam, ... Additional course specific Textbook for IGCSE: Edexcel IGCSEMathematics A Student Books 1 and 2 by Turner, Potts, ... IGCSEMathematics regardless of which set they are in. IGCSE Curriculum Booklet 2013 - 2014 ... In addition to the core subjects, learners choose up to four additionalIGCSE subjects from a range of options. The ... International Mathematics Cambridge IGCSE (0607) Aims covers a broad range of skills in that subject. For example, in IGCSEMathematics, students will cover topics from number, algebra, ... 2014 23 First Language English IGCSE / Compulsory ... Studenten volgen de Cambridge IGCSEsyllabus. Cambridge IGCSEMathematicsSyllabus code 0580 Cambridge IGCSEMathematics (with coursework) Syllabus code 0581 For examination in June and November 2012. ... • Tracing paper may be used as an additional material for each of the written papers. • For syllabus 0581, the Coursework components ... They will then follow the AdditionalMathematics Course with OCR and take that exam at the end of ... our pupils we are now studying for the IGCSE as of September 2014 them with a suitable basis for further study and practical ... the syllabus very broad, ...
This lesson helps students understand the concept of inflation in a mathematical context. Students will learn about the Consumer Price Index and will use it to compare the changing worth of a dollar over several years.... 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... With this algebra lesson, students will gather data about different airlines, including flight delays, mishandled baggage and other consumer complaints. The material helps students learn how to effectively analyze data,... This algebra lesson demonstrates exponential growth and decay. The document includes three different ways in which students will retrieve data from the internet, formulate a function, perform calculations and then... This algebra lesson helps students make the connection between functions and their graphs. The model of the level of water in a bathtub is used. Students will watch the graph and a chart of the depth of the water at...
Pre-Statistics Courses A growing number of colleges have begun piloting alternatives to the traditional 3-4 semester developmental Math sequence. The new courses focus not on a review of Algebra concepts and skills, but instead on preparing students for college-level Statistics. Pre-Statistics courses emerged from a widespread recognition that the traditional developmental Algebra sequence is not well-aligned with the study of Statistics. If a student is pursuing a major that includes Statistics rather than Calculus (e.g. fields outside of Science, Technology, Engineering, Math, and Business), the majority of what is covered in elementary and intermediate Algebra courses never comes into play in their college-level Math course. The content is not, in fact, pre-requisite knowledge for the study of Statistics. This misalignment is especially problematic given how many students are lost in long remedial math pipelines. A nationwide study by the Community College Research Center found that among students who begin 3 or more levels below college Math, 90% disappear before ever completing the college-level course. Los Medanos College was the first in the nation to pilot this kind of course with Path2Stats, a one-semester pathway to college Statistics with no minimum placement score. Rather than proceeding step-by-step though all the topics in the traditional math sequence, Path2Stats students engage in statistical analysis from day one. Basic skills remediation occurs in a "just-in-time" fashion, with students reviewing relevant arithmetic and algebraic skills — e.g. calculating percentages, converting ounces to grams — as they are needed for the statistical tasks at hand. Path2Stats students complete college Math at dramatically higher rates that students with comparable starting placements in the traditional sequence. In 2011-12, 7 community colleges are working with the California Acceleration Project to offer redesigned pre-Statistics Courses. They include Diablo Valley College, Cuyamaca College, City College of San Francisco, College of the Canyons, Riverside City College, Moreno Valley College, and Berkeley City College (See video footage of CCSF students talking about their experience.) Some of these courses are open-access; others have arithmetic/pre-Algebra prerequisites. Nationwide, an additional 19 community colleges and 3 state universities are part of the Statway initiative, led by the Carnegie Foundation for the Advancement of Teaching: "a one-year pathway that culminates in college-level statistics…with requisite arithmetic and algebraic concepts taught and applied in the context of statistics." Classroom video: Los Medanos Developmental Math Students Discover an Error in a National Statistics Exam
Summary: These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have a created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan'sAlgebra and Trigonometry: Enhanced with Graphing Utilit...show moreiesgives students a model for success in mathematics. ...show less Hardcover New 013600492X New Condition ~~~ Right off the Shelf-BUY NOW & INCREASE IN KNOWLEDGE... $83.24 +$3.99 s/h VeryGood AlphaBookWorks Alpharetta, GA 013600492109.99 +$3.99 s/h VeryGood Plato1964 Brookline, MA Clean, well cared for textbook with CD! No writing, markings, or highlighting. Cover shows very minor shelf markings but is otherwise in excellent condition! Save $ and Buy Used
UCSMP Function, Statistics, Trigonometry Help students develop into fluent, independent, and highly skilled readers with this Direct Instruction reading intervention program The UCSMP Third Edition curriculum emphasizes problem solving, everyday applications, and the use of technology and reading, while developing and maintaining basic skills. Functions, Statistics, and Trigonometry integrates statistics and algebra concepts, and previews calculus in work with functions and intuitive notions of limits. Enough trigonometry is available to constitute a standard precalculus course in the areas of trigonmetry and circular functions. Technology is assumed available for student use in graphing, algebraic manipulation, modeling and analyzing data, and simulating experiments. Functions, Statistics, and Trigonmetry assumes that students have completed a second-year algebra course. Students who have studied some trigonmetry, such as that found in UCSMP Advanced Algebra, will be at an advantage. The program also provides the background for success in a non-proof oriented calculus course, such as those often taken by business or social studies majors in college, but not for a proof-oriented calculus course. It also provides the background for Advanced Placement AB Calculus and Advanced Placement Statistics. View a sample lesson from the Teacher's Edition of this program by clicking here
Here is a demonstration of the flow of a vector field. You may choose a vector field from the pull down menu. In the graphing... see more the vector field while the rotation of the sides is described by the curl. This applet allows a person to test several numerical integration approximation methods by having the user fill out theleft... see more This applet allows a person to test several numerical integration approximation methods by having the user fill out theleft and right endpoint fields, type in the formula, select the method, and pressing the calculate button. For example trytyping in -1 for the left endpoint, 1 for the right endpoint, and y=x^2 for the formula then press calculate. For each method selected, the estimated integral for that method will appear in the text area at the bottom. Certainfunctions will also appear on the graph on the left. The java graph class is still experimental and does not use equalstretch principles. y = x2 from -1 to 1 will appear the same as y = x2 from -100 to 100. A series of tutorials on basic measurement theory, including such topics as experimental errors, measurement, accuracy and... see more A series of tutorials on basic measurement theory, including such topics as experimental errors, measurement, accuracy and precision, exact numbers, and significant figures (digits). Applet calculators included, along with quizzes. Java applet to demonstrate scentific notation by examining how a view changes upon magnification. From the site: "View the... see more Java applet to demonstrate scentific notation by examining how a view changes upon magnification. From the site: "View the Milky Way at 10 million light years from the Earth. Then move through space towards the Earth in single order of magnitudes until you reach an oak tree just outside the National High Magnetic Field Laboratory in Tallahassee, Florida. Then begin to move from the actual size of a leaf into the microscopic world through leaf cell walls, cell nucleus, DNA and finally, into the subatomic universe of electrons and protons. QuickMath is an automated service for answering common math problems over the internet. ... see more QuickMath is an automated service for answering common math problems over the internet. Think of it as an online calculator that solves equations and does all sorts of algebra and calculus problems - instantly and automatically! When you submit a question to QuickMath, it is processed by Mathematica, the largest and most powerful computer algebra package available today. The answer is then sent back to you and displayed right there on your browser, usually within a couple of seconds. Best of all, QuickMath is 100% free! Here is Zona Land's graphics calculator, EZ Graph. With it you should be able to graph almost any polynomial, rational,... see more Here is Zona Land's graphics calculator, EZ Graph. With it you should be able to graph almost any polynomial, rational, exponential, logarithmic, or trigonometric function. It will allow you to enter variables into your function definition so that you can see the effect of changing coefficients easily.
You are here Math Readiness Course The Math Readiness Course is a non-credit math review, for students who wish to refresh their high school math skills. The course consists of approximately 22 lessons on the following topic groups: algebra functions and graphs geometry trigonometry, and exponential and logarithmic functions Please visit the Math Readiness page at the Department of Mathematics and Statistics or download our 2012 Math Readiness flyer for more information; please note that the fee for 2013 offerings of Math Readiness is $300 plus taxes. To register please call the Centre for Continuing and Distance Education at 306-966-5539 or email ccde.reg@usask.ca. Please note that registration for the summer 2013 Math Readiness course will be open until Friday, August 16. If you need more information, please contact Holly Fraser by email at holly.fraser@usask.ca or by phone at 306-966-2742. Staff from CCDE can help you with administrative questions; please call 306-966-1739 or email math.readiness@usask.ca. Course Offerings In past years, the Math Readiness Course has been offered both as a summer course and an evening course during regular session in Term 1 (September-December) and Term 2 (January-April), provided a sufficient number of students register. Summer course The Department of Mathematics and Statistics has confirmed that Math Readiness will be offered this summer. The exact dates are August 19-30, Monday-Friday. Registration will be available through CCDE after later in May 2013. (Call 966-5539 or email ccde.reg@usask.ca after May 10 to enquire about registration.) If you are looking for a credit course with similar content, please see below ("New! Credit Course"). Evening course Please note that Math Readiness is a cost-recovery course. In general, we usually need to have a minimum number of students registered in order to run the course. The offerings below are tentative. Online Math Readiness is currently available at no charge. For further information or to request an account as a non-university student, please email ulc_onlinemath@usask.ca. New! Credit course option For those who are interested in taking a credit course that reviews concepts from high school math, please check PAWS for "Math 102." Currently, Math 102 is on the slate of offerings for Term 2 of Spring and Summer Session 2013 (i.e. June 28-August 17) and during Quarter 4 (July 23-August 13). As well, Math 102 will be offered during Term 1 of Regular Session 2013-14. Contact the Department of Mathematics and Statistics at 306-966-6081 or by email at math@math.usask.ca for more information. Please note that a credit course means that there will be an official grade issued on your transcript, and that an academic advisor should be consulted to determine how the course will align with your degree program requirements.
Why is probability an under-emphasized subject in most math programs? Why does it seem that the hot topics these days are category theory and algebra? What do you think about the following: A student ... What are some good sources for linear algebra for convex optimization and graph analysis? In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses ...
Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. Along the way, you'll go beyond solving hundreds of repetitive problems, and actually use what you learn to make real-life decisions. Does it make sense to buy two years of insurance on a car that depreciates as soon as you drive it off the lot? Can you really afford an XBox 360 and a new iPhone? Learn how to put algebra to work for you, and nail your class exams along the way. Your time is way too valuable to waste struggling with new concepts. Using the latest research in cognitive science and learning theory to craft a multi-sensory learning experience, Head First Algebra uses a visually rich format specifically designed to take advantage of the way your brain really works.
Step-by-step Tutorials for using GeoGebra ( from beginning stages, to advanced features such as... see more Step-by-step Tutorials for using GeoGebra ( from beginning stages, to advanced features such as interacting with the Web Page, LaTex, and a taste of what's to come with GeoGebra 3.2 (curve fitting). Contains applets, pdf script, and Camtasia demonstrations. Symmetry is used to analyze the patterns in Oriental carpets. Educational activities are provided along with other... see more Symmetry is used to analyze the patterns in Oriental carpets. Educational activities are provided along with other educational resources. This web site is a collaborative project of The Textile Museum and The Math Forum. This is a very nicely presented short course in trigonometry for the student who has already learned algebra and geometry.... see more This is a very nicely presented short course in trigonometry for the student who has already learned algebra and geometry. There are Java applets available throughout the course to illustrate the various concepts. Each section includes text, use of an applet, exercises, hints, and answers. The CCP includes modules that combine the flexibility and connectivity of the Web with the power of computer algebra systems... see more The CCP includes modules that combine the flexibility and connectivity of the Web with the power of computer algebra systems such as Maple, Mathematica, MatLab and MathCad. This particular collection includes single-topic modules on Multivariable Calculus as well as applications in this subject.
Linear Algebra 9780201824797 ISBN: 0201824795 Publisher: Addison-Wesley Longman, Incorporated Summary: Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimi...late. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible. Lay, David C. is the author of Linear Algebra, published under ISBN 9780201824797 and 0201824795. Four Linear Algebra textbooks are available for sale on ValoreBooks.com, and three used from the cheapest price of $4.40
Compressed Zip File Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files. 2.11 MB \| 15 pages PRODUCT DESCRIPTION Students begin the journey into Analytic Trigonometry through exploring basic identities. This lesson contains an eight-page "bound book" style Foldable (C) with an accompanied SmartBoard lesson. There is also a *.pdf file of the completed Foldable and Smart Notes. This teaching method minimizes wasted class time since the "skeleton" of the lesson is pre-printed. Students stay engage and focused
additionrationsSolving a System of Two Linear Equations. Other Systems of Equations. Solving a System of Three Linear Equations. Determinants. Properties of Determinants. Solving a System of Linear Equations Using Determinants. Partial Fractions. The Circle. The Parabola. The Ellipse. The Hyperbola. Translation of Axes. The General Second-Degree Equation. Systems of Quadratic Equations. Polar Coordinates. Graphs in Polar Coordinates. 21. The Derivative. Motion. The Limit. The Slope of a Tangent Line to a Curve. The Derivative. Differentiation of Polynomials. Derivatives of Products and Quotients. The Derivative of a Power. Implicit Differentiation. Proofs of Derivative Formulas. Higher Derivatives. The Indefinite Integral. The Constant of Integration. Area Under a Curve. The Definite Integral. 25. Applications of Integrations. Area Between Curves. Volumes of Revolution: Disk Method. Volumes of Revolution: Shell Method. Center of Mass of a System of Particles. Center of Mass of Continuous Mass Distributions. Moments of Inertia. Work, Fluid Pressure, and Average Value. additionrations
Practical Problems in Mathematics for WeldersTake a practical, clear approach to mathematics as this highly effective text introduces the importance of math to the latest procedures, technologies, and practices in today's welding industry using real-world examples and uncomplicated explanations.
Tool Box Quantitative Reasoning The ability to understand and apply quantitative, mathematical and computational reasoning is an important component in the development of independent and logical thinking. Quantitative literacy is also essential for students to become informed citizens. These courses will introduce students to fundamental mathematical knowledge, including an understanding of the nature of mathematics and quantitative and statistical argumentation. More specifically, students will be able to use quantitative reasoning skills to make informed, analytical decisions on matters that are important to their lives, professions and society. be able to discern when quantitative reasoning and mathematical tools are necessary in reaching fair, equitable and ethical conclusions, and to discern when such tools are misused in arguing such conclusions. be able to comprehend, critique, create and communicate arguments supported by quantitative evidence. For students who arrive at LMU with a higher level of quantitative literacy, as determined by a placement exam, this requirement may be satisfied by other courses emphasizing more abstract mathematical and computational reasoning. Note that there are criteria for two categories of Quantitative Reasoning courses. Most students will satisfy this requirement with a Quantitative Reasoning course. For students who arrive at LMU with a higher level of quantitative literacy, as determined by a placement exam, this requirement may be satisfied by a Mathematical Reasoning course emphasizing more abstract mathematical and computational reasoning, or more advanced methods of quantitative and statistical reasoning.
Free Online Arithmetic Courses Arithmetic Content Navigation Arithmetic is the building block for mathematics, dealing primarily with the use of foundational operations and principles such as addition, subtraction, multiplication, and division. Numerals are used in instead of the variables that are typically seen in more advanced mathematics. One must master arithmetic before pursuing more advanced studies in mathematics. These online arithmetic courses and lectures introduce basic operations in this branch of mathematics, giving you the foundation to manipulate numbers properly at this level.
A+ National Pre-traineeship Maths and Literacy for Retail by Andrew Spencer Book Description Pre-traineeship Maths and Literacy for Retail is a write-in workbook that helps to prepare students seeking to gain a Retail Traineeship. It combines practical, real-world scenarios and terminology specifically relevant to the Retail Industry, and provides students with the mathematical skills they need to confidently pursue a career in the Retail Trade. Mirroring the format of current apprenticeship entry assessments, Pre-traineeship Maths and Literacy for Retail includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Retail Traineeship. Pre-traineeship Maths and Literacy for Retail also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-traineeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level. Buy A+ National Pre-traineeship Maths and Literacy for Retail book by Andrew Spencer from Australia's Online Bookstore, Boomerang Books. You might also like... When Coca-Cola offered the first retail coupon in the 1880s, customers were thrilled. But today, one in four American shoppers will buy something only if it's on sale, and almost half of all merchandise carries a promotional price. This book investigates what happens to markets when everything's negotiable. From his striking window displays during the holidays to his glittering love affairs with the most beautiful women in Europe, this book reveals the secrets behind his success as a tycoon. Filled with revolutionary thoughts about business, leadership, and society, Selfridge will inspire you with powerful aphorisms This resource presents a modern approach to retail and industrial management, suggesting the tools for implementing integrated management based on the open source software, Open ERP. The text describes how to integrate different services and automate tasks including sales and purchases. You already know eBay basics. Now you'd like to go beyond with shortcuts, tricks, and tips that let you work smarter and faster. And because you learn more easily when someone shows you how, this is the book for you
MTHS 662 Introduction to Number Theory Fieldsteel,Adam 09/08/2003 - 12/13/2003 Monday 07:00 PM - 09:30 PM Science Tower 139 The natural numbers, 1, 2, 3, ... are the starting point of mathematics. They form a deceptively simple structure whose investigation has occupied mathematicians for thousands of years. This course will introduce students to some of the basic problems and methods of this subject. We will begin with a study of divisibility and properties of prime numbers. We will go on to study arithmetical functions, the algebra of congruence classes, and Diophantine equations. As time permits, we will discuss special topics such as the distribution of the prime numbers and decimal and continued fraction representations of real numbers
GeoGebra is a dynamic mathematics software that joins geometry, algebra, and calculus. Two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice versa
Mathematics The math teachers at Spring Forest Middle School are committed to helping all students in the acquisition of math skills. Through problem solving, SFMS teachers encourage students to write and talk about the mathematical processes in an effort to help students make connections within and outside mathematics. We all share the belief that "Math is NOT a spectator sport!" Our goal is to create mathematical problem solvers who can analyze any situation, apply their knowledge, and communicate their thinking. Math 6 The primary focal points of Grade 6 math include using ratios to describe proportional relationships involving operations with whole numbers, addition and subtraction of decimals and fractions, geometry, measurement, and probability. 6th grade Math GT Sixth grade GT math combines 6th, 7th, and 8th grade math. All middle school math concepts will be covered with an emphasis on critical thinking and application. Students must master the concepts of each of these grade levels in order to prepare for Algebra I in the 7th grade. (In order to participate in a GT class, the student must be identified as "gifted" by a district screening committee.) The progression of courses for GT math students is: 6th grade 7th grade 8th grade 7th grade GT math Algebra I Geometry Math 7 The primary focal points of Grade 7 math include the modeling and application of operations with whole numbers, decimals, fractions and percent, algebraic reasoning, probability, geometry, measurement and statistics. Students will also be asked to use verbal and written explanations to communicate thinking. Algebra I 7 Pre-AP/GT and Pre-AP Math 8 An emphasis is placed on real world applications to develop concepts including functions, linear equations and inequalities, polynomials, systems of equations and quadratic equations. Critical thinking skills are necessary, so high degrees of self-motivation and a willingness to work outside the classroom are essential for success. Prerequisite: Students enrolled in Algebra I Pre-AP should have mastered 6th, 7th and 8th grade concepts. Students enrolled in GT classes must be identified as gifted through a district screening process. NOTE: Algebra I Prerequisite: Students enrolled in Geometry must have successfully completed Algebra I. Students enrolled in GT classes must be identified as gifted through a district screening process. Note: Geometry
More About This Textbook Overview This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part, Chapters 1-7, is concerned with hyperbolic geometry and discrete groups. The second part, Chapters 8-12, is devoted to the theory of hyperbolic manifolds. The third part, Chapter 13, integrates the first two parts in a development of the theory of hyperbolic orbifolds. There are over 500 exercises in this book and more than 180
Paterson, NJ ACTI Various parts of discrete math were used in all the other math courses I took as well.
Finally figure out physics! Teach Yourself Physics introduces you to physics gradually by building a foundation of appropriate mathematical skills. Easy-to-understand diagrams help you understand key points
This lesson was created by Larry Friesen and Anne Gillis for Butler Community College. It will help physics and calculus students differentiate between the uses of vectors in mathematics vs. physics. This website... The American Mathematical Association of Two-Year Colleges (AMATYC) has compiled a collection of mathematics resources related to various subjects and disciplines. ?Math Across the Community College Curriculum? is the... Math in the "real world" happens all the time, and it can involve everything from buying a car to following a simple (or complex) recipe. The "Math in Daily Life" site offers up a series of interesting ways to get... This website provides a syllabus for a course that uses "writing to reinforce the learning and understanding of mathematical concepts, while also using mathematics and statistics as a topic for the essays." Students...
Mathematics For Class XI 1st Edition (Paperback) Price: Rs.401 Mathematics for Class XI is a reference book by R. D. Sharma for students of class XI following the syllabus issued by the Central Board of Secondary Education. Summary Of The Book Mathematics for Class XI is a reference book aimed at helping students of Class XI in their efforts to prepare for class exams as well as to provide a good foundation in the preparation for competitive exams. The chapters in this book provide a description of concepts covered in Class XI, along with a multitude of problems for each chapter. The book has an algorithmic approach and comes with clear explanations of the theory supplemented by illustrations, making concepts easy to understand and remember. The book also contains concise summaries of concepts and formulae at the end of each chapter, making it a good book to have handy for revisions just before the exam. For students looking to learn by practice, this book is a great source for problems. It comes with many exercises, and the problems listed are graded and come with the solved answers. The book also contains a number of exercises with unsolved problems that the student may use to further sharpen his/her skills. Mathematics for Class XI addresses the entire syllabus for Class XI. Some topics covered in the book include trigonometric equations, functions, ratios, and graphs of trigonometric functions, sine and cosine formulae and their applications, mathematical induction, and transformation formulae. With detailed explanations of the theory and a wealth of solved and unsolved problems, this book makes for a reliable source book for Class XI students to help in their efforts of doing well in their competitive exams. About R. D. Sharma R. D. Sharma is an Indian author and teacher. He is a widely published author of Mathematics textbooks and reference books. Some other books by R. D. Sharma include Mathematics for Class XII, Mathematics for Class 12 (Set of 2 Volumes), Objective Mathematics For IIT-JEE, AIEEE and All Other Engineering Entrance Examinations, and Mathematics Class-VII. R. D. Sharma has a Doctorate in Mathematics, and completed his Bachelor's and Master's degrees, with Honours and double gold medals, from the University of Rajasthan. He is presently with the Directorate of Technical Education, Delhi, serving as the Head of Department, Science and Humanities. Most Helpful Reviews (5 of 21) The 'certified buyer' badge indicates that this user has purchased this product on flipkart.com. R.D Sharma is a MATH-EMATITION If you are looking for a book that can make you a class topper in mathematics, then this is one of the best books you will ever come across(don't get your hopes high. Just read on). If you are someone who loves to challenge yourself, then this the book you would love(provided you have enough time to take up the challenge as this book has a heck lot of sums.But don't panic). I always had a problem in mathematics. But believe me I am getting 90% plus in my exams because of this book(and my school book.But this book played a bigger role). If you are someone who is like a math wiz, then you could call this - a book to stop and look at. But if you are someone who needs a little help and you are currently not having anyone to help you. Then I would suggest that you either continue learning using your school book or search the web for the book that is suitable for your needs. This book is truly awesome but only if you have someone to help you(like me. I have a math tuition teacher who helps me) or if you are a math wiz who wants to challenge yourself to sums of a different level(don't worry, the sums will still revolve around your school syllabus). If the criteria(as I mentioned above) is met, then YOU WILL BE TRULY SUCCESSFUL BY USING THIS WONDERFUL BOOK. NOTE: This book has many sums. So be prepared to solve them. Remember that if you can solve all the sums in the book then you will be a certified XIth standard student MATH-EMATITION. Remember that the above certificate will be held by you for only one year, after which you will again have to prepare for your XIIth standard. The 'certified buyer' badge indicates that this user has purchased this product on flipkart.com. Mathematics by R.D.Sharma for class IX, X, XI and XII R.D.Sharma's Sir's books are always helpful.The way Dr.R.D.Sharma starts off with introducing the chapter and explaining the examples so clearly and the exercise followed by the examples are really useful.The exercise problems are slightly different from the examples but going through the examples would be of great use to solve the exercise ones. If we work out every exercise problems, understanding the concept becomes very easy. I am working as a lecturer in Engineering College and even today Dr.R.D.Sharma's CBSE text book are of great use to me and ultimately benefits the students. For those who are finding it difficult with the basics or with class XI and XII math, R.D.Sharma's books are sufficient to score better and foremost important understand better. The 'certified buyer' badge indicates that this user has purchased this product on flipkart.com. Look no further,'cause you're staring at the BEST. R.D. Sharma is undoubtedly the best book for 11th grade mathematics that you could find anywhere. I started using R.D. SHARMA in class 10 and to be honest,I have been blown away ever since. If you want to build confidence in the subject of mathematics,look no further,because what you see in front of you is probably the best book available in the market. It's CBSE based,though but that's not much of an issue since ISC and CBSE Boards do not differ much in their Math syllabi,so to say. Moreover if you solve every problem that is there in the book,be sure that you certainly will be able to ace JEE or any other competitive exam whatsoever. It spans about some 1200 odd pages and trust me you're gonna get sums that will take you for a ride. The key features of the book are : 1. Algorithmic approach 2. Stress on concepts 3. Tremendously huge number of practice sums(unsolved) as well as hundreds of solved examples from a myriad of topics. HOWEVER,if you are really serious about acing your mathematics exam or simply enhancing your knowledge and challenging yourself,you shall have to devote a considerable amount of time because not only does this book house a number of sums,believe me some of them are so difficult to get thorugh it will take you days before you can finally brag in fornt of your geeky acquaintances that you were able to solve them while they couldn't . So if you are hard working and are ready to relentlessly toil to get a really good score in mathematics,look no further because I got on my doubts whether you shall find some other book to suit your needs better . Last but not the least I would like to applaud the amazing service that flipkart again provided me by way of their quick delivery of my order. I hope they keep up the good work. The 'certified buyer' badge indicates that this user has purchased this product on flipkart.com. The best book of mathematics for class 11 Firstly, the book is awesome... books written by dr r.d. sharma proves extremely useful for the students. This book contains a large number of well graded solved examples and many illustrative examples and problems are also included in it. The book has some unique features which fulfills the needs of students. the book is a fine blend of NCERT , CBSE and other competitive-examination based problems. students of different boards would also find it extremely helpful. Lastly, i would like to say that every student of class 11 shall go through it.With daily practice, this book will surely help us in achieving our target. The book is an elixir of our career and so without thinking any more , we shall flipkart it!
GCSE Mathematics in a key requirement for many job roles as it demonstrates an understanding of numbers and an ability to work with them. And yet many people find numbers intimidating. With this in mind, this home study course has been designed to guide you through the subject in logical, easy-to- follow steps, building on your understanding as your confidence grows. The GCSE places great emphasis on "doing mathematics" and relating this wherever possible, to everyday life. Certain techniques and formulae need to be learnt, but the emphasis on "doing" means that you should work carefully through all the examples and exercises in order to be able to solve problems effectively. Mathematics is essentially a holistic subject and, as such, should be taught in this way with connections being made between the sections on Number and algebra, Shape, space and measures and Handling data as demanded in the National Curriculum. For example, Number underpins the whole of mathematics. The Maths GCSE distance learning course gently guides the student through basic mathematical skills, progressing onto more advanced material as the student's skills and abilities develop. A reasonable level of proficiency in arithmetical skills is assumed. Each lesson begins with a set of clearly stated objectives and an explanation of its place in the overall programme of study. Effective learning is encouraged through frequent activities and self-assessment questions. There are thirteen tutor-marked assignments and a practice exam paper. The Syllabus: Our Maths GCSE course prepares students for AQA GCSE Mathematics syllabus 4365 for exams in 2012 and later years. The course covers an intermediate and higher tier level. The course covers the intermediate level with an optional unit included to also cover the additional higher tier so you have a choice of which tier to complete. We have chosen this syllabus as the most suited to distance learning. Assessment is by two examination papers. Paper 1is 40% of the total marks. The paper is a Written Paper (Non-calculator). The exam is 1 Hour, 15 Minutes for the Intermediate Tier & 1 Hour 30 Minutes for the Higher Tier. Paper 2 is 60% of the total marks. The paper is a Written Paper (With calculator). The exam is 1 Hour, 45 Minutes for the Foundations Tier & 2 Hours for the Higher Tier. Every student receives a personal tutor with whom they need to keep in regular contact. The essential relationship between student and tutor begins with an introductory letter and a telephone call to help decide on a study plan. All tutors are fully qualified teachers holding either a Postgraduate Certificate in Education or a degree in education. So when you enrol on a course you can be assured of the assistance of a professionally qualified subject specific tutor. We don't set a limit on how often you can contact your tutor. The advice and encouragement our tutors provide are an important part of your learning experience. You can contact your tutor by Phone, Post or Email. There are tutor-marked assignments (TMA's) in every course. These help the student consolidate the learning he or she has done and prepare for examinations. All tutors have a local rate telephone number such as a 0845 number. This means that when you phone your tutor it will only cost you the price of a local call. Whilst you are still actively submitting assignments, you should expect to be in contact with your tutor regularly. Remember, to get the most out of your tutor, you will need to tell them when you need assistance. Examination Centres: There are currently over 300 testing centres across the country, so finding one close by is not very difficult. In December/January we contact you to start arranging your examination dates. We always try to give whatever help we can with examination information, but the entering of students for examinations and any dealings with the examination centres are entirely the responsibility of the student. Examination fees need to be paid direct to your chosen exam centre. Prices for exams will vary depending on the exam centre's administration costs. You can find out further information about the Private Candidate process and locate suitable exam centres from the AQA Private Candidate Page. Studying from Abroad: It is possible to study GCSE Courses from anywhere in the world. However some home study courses are easier to study from abroad than others and some restrictions to the type of tuition available apply. If you wish to study a GCSE then you will need to sit the examinations in the UK. Please remember that most GCSE's have several examinations which may well be spread over several weeks developing believed
The following computer-generated description may contain errors and does not represent the quality of the book: Although Algebra naturally follows Arithmetic la a course of scientific studies, yet the change from numbers to a system U-i of reasoning entirely conducted by letters and signs is rather abrupt and not unfrequently discourages the pupil. In this work it has been the intention to form a connecting iisk between Arithmetic and Algebra, to unite and blend, as ras possible, the reasoning on numbers with the more abstruse method of analysis. The Algebra of M.Bourdon has been closely followed. Indeed, it has been a part of the plan, to furnish an introduction to that admirable treatise, which is justly considered, both in Europe and this country, as the best work on the subject of which it treats, that has yet appeared. This work, however, even in its abridged form, is too voluminous for schools, and the reasoning is too elaborate and metaphysical for beginners. It has been thought that a work which should so far modify the system of M.Bourdon as to bring it within the scope of our common schools, by giving to it a more practical and tanV) form, could not fail to be useful. Such is the object of Lthe Elementary Algebra. It is hoped it may advance the cause of education, and prove a useful introduction to a full course of mathematical studies. QThe author has thought best to add in the present edition, a few pages in the form of an Introduction, and also a Treatise on Logarithms. Habtfobs, September 1838.
Algebra is one of the main branches in mathematics. The book series of elementary algebra exercises includes useful problems in most topics in basic algebra. About the book Description Algebra is one of the main branches in mathematics. The book series of elementary algebra exercises includes useful problems in most topics in basic algebra. The problems have a wide variation in difficulty, which is indicated by the number of stars. Content Author Biographies Preface Real numbers Equations Inequalities About the Author Mr. Wenlong Wang Mr. Wenlong Wang is a retired mathematics educator in China. He has been working on algebra and geometry problems for many years, and has taught many students in the past few years. He is an expert and a senior researcher in mathematics education. Professor Hao Wang Professor Hao Wang is a faculty member in the Department of Mathematical & Statistical Sciences at the University of Alberta, an advisory board member of Centre for Mathematical Biology, an associate editor for International Journal of Numerical Analysis & Modeling – Series B, an editor for Nonlinear Dynamics and Systems Theory (an international journal of research and surveys), an editor for a special issue of The Canadian Applied Mathematics Quarterly, and an associate faculty member for Faculty of 1000 Biology. Dr. Wang has strong interests in interdisciplinary research of mathematical biology. His research group is working on areas as diverse as modeling stoichiometry-based ecological interactions, microbiology, infectious diseases, predator-prey interactions, habitat destruction and biodiversity, risk assessment of oil sands pollution. Mathematical models include ordinary differential equations, delay differential equations, partial differential equations, stochastic differential equations, integral differential/difference equations
Contoh Soal Matematika Tentang Deret
Rent Book Buy New Book Used Book We're Sorry Sold Out eBook We're Sorry Not Available More New and Used from Private Sellers Starting at $7Spectrum Data Analysis & Probability Grades 68 helps young learners improve and strengthen their math skills, such as ratios, graph interpretation, and measures of central tendency. The best-selling Spectrum series provides standards-based exercises developed to supplement and solidify the skills students learn in school. Each full-color title includes an answer key.
How do mathematical equations translate into physical form? What formulas are needed to create a work of art? Students in A "Sine" of the Times chose their inspiration, developed designs... More > using oscillation graphs, and wrote reflections about how they learned to see math as well as themselves as mathematicians. With patterns as unique as their creators, this publication will inspire students and teachers to find creative applications for math in their own classroomsThe Calculus - Book 2 (63 pages) is a study of the calculus at GCE Advanced Level (approximately year 14). It follows on from The Calculus _ Book 1, which covers calculus from GCSE Level (year 12) up... More > to GCE Advanced Subsidiary Level (year 13). It contains clearly explained teaching text, worked examples (in graduated order of difficulty), followed by exercises (with fully worked answers).< Less 'Calculus...the way to do it' (140 pages) is a study of the calculus from basics up to GCE Advanced Level (years 12, 13 and 14). Section 1 is appropriate for students of GCSE Additional Mathematics... More > (year 12) and GCE Advanced Subsidiary Level (year 13). Section 2 is appropriate for GCE Advanced Level (year 14). It contains clearly explained teaching text, worked examples (in graduated order of difficulty), followed by exercises (with fully worked answers).< Less Trigonometry book (98 pages) covers material from GCSE Level (approximately age 16 years) to GCE Advanced Level (approximately age 18 years) - teaching text, worked examples, exercises (with full... More > worked answers)- ideal for examination revision. It is available also from our website at and on iPad from the iBookstore.< Less This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high... More > schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed. A brief tutorial on using Gnuplot to graph trigonometric functions is included.< Less
Developmental Arithmetic (3-0-3). Credit not applicable toward degrees. Required of students whose ACT Mathematics Main score is less than 15 or COMPASS Math score of 30 or less. Fundamental topics in arithmetic, geometry, and pre-algebra. 099 Developmental Algebra (3-0-3). Credit not applicable toward degrees. Required of students whose ACT Mathematics Main score is at least 15 but less than 19 or COMPASS Math score of 31 to 58. Fundamental topics in algebra for students with insufficient knowledge of high school level mathematics. PR: ACT Mathematics Main score of 15 or grade of "S" in MATH 098. 109 Algebra (3-0-3). Real numbers, exponents, roots and radicals; polynomials, first and second degree equations and inequalities; functions and graphs. PR: ACT Mathematics main score of 19 or grade of "S" in MATH 099. 211 Informal Geometry (3-0-3). Theorems are motivated by using experiences with physical objects or pictures and most of them are stated without proof. Point approach is used with space as the set of all points; review elementary geometry, measurement, observation, intuition and inductive reasoning, distance, coordinate systems, convexitivity, separation, angles, and polygons. No field credit for math majors/minors. PR: MATH 101 or higher. 220 Calculus I (4-0-4). A study of elements of plane analytical geometry, including polar coordinates, the derivative of a function with applications, integrals and applications, differentiation of transcendental functions, and methods of integration. PR: MATH 109 and MATH 110, or GNET 116, or ACT Mathematics main score of 26 or COMPASS Trigonometry score of 46 or above. 250 Discrete Mathematics (3-0-3). Treats a variety of themes in discrete mathematics: logic and proof, to develop students' ability to think abstractly; induction and recursion, the use of smaller cases to solve larger cases of problems; combinatorics, mathematics of counting and arranging objects; algorithms and their analysis, the sequence of instructions; discrete structures, e.g., graphs, trees, sets; and mathematical models, applying one theory to many different problems. PR: MATH 109 and MATH 110 or GNET 116. 290 Topics in Mathematics (1-4 hours credit). Formal course in diverse areas of mathematics. Course may be repeated for different topics. Specific topics will be announced and indicated by subtitle on the student transcript. PR: Consent of instructor. 400 Introduction to Topology (3-0-3). A study of set theory; topological spaces, cartesian products, connectedness; separation axioms; convergences; compactness. Special attention will be given to the interpretation of the above ideas in terms of the real line and other metric spaces. PR: MATH 240. 490 Topics in Mathematics (1-4 hours credit per semester). Advanced formal courses in diverse areas of mathematics. Courses may be repeated for different topics. Specific topics will be announced and indicated by subtitle on transcript. PR: Consent of instructor.
From the syllabus: The goal of this course is to provide a solid foundation in the concepts and computational tools of linear algebra, especially matrices, vector spaces, and linear transformations. The course will emphasize problem-solving, conceptual understanding, and geometric intuition. Compared to the calculus sequence, this course involves increased abstraction and mathematical depth, and thus it will help students develop skills understanding proofs and writing mathematics. The primary web site for this course was ClassWiki, which was also a collaborative writing project for the class. Unfortunately, ClassWiki is not currently available.
illustrates the mathematical concepts that a game developer needs to develop 3D computer graphics and game engines at the professional level. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more advanced topics in 3D programming such as illumination and visibility determination. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series. The third edition expands upon topics that include projections, shadows, physics, cloth simulation, and numerical methods. All of the illustrations have been updated, and the shader code has been updated to the latest high-level shading language specifications. {"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":43.26,"ASIN":"1435458869","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":34.79,"ASIN":"1133776574","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":61.12,"ASIN":"1568814135","isPreorder":0}],"shippingId":"1435458869::rqFD6kTHCdAqDWiHcAo3GafzestolalIzTZHUz9o3caDbylw%2FdbEAL6xtABFdY0prv%2BO%2BaK6URQ7dVZHZVPgY1ee10qSlxY1zVLCLOa6VPd5WpdxSEtMlg%3D%3D,1133776574::KGFmOkqWcaIHdWYkoePiLbzl8yGrEtsMHqroYt4rrI90Mw4pilQ9kRzmbaMzn1gO21jYSW637ccuNAGIap9hN3rnmYN4NzZ6Ms99JhLENl18fldEF3ZmrQ%3D%3D,1568814135::YE7Ml%2FOHUBP2w2BCDd55peD7jpuwIVTi1p5l8j2bdHrwKUdu466Mew2t7AUtQeGSMETbnEjTJNFQTEXvKzhE7Oql%2Fde3vtJocdjXwXzrKjSuDFmvWQaTxMore AboutMost Helpful Customer Reviews As a word of warning, do not purchase this book expecting it to teach you math fundamentals. If you do not have a background of at least algebra and trigonometry (and preferably a bit of calculus), you owe it to yourself to pick up another book and brush up on these fundamentals. While there are a few appendices covering a handful of topics, they are less about explaining the topic and more of reference pages. Mathematics for 3D Game Programming and Computer Graphics is an excellent reference book for anyone doing 3D work. The topics are very to the point and few pages are wasted explaining basic math principles (hence the warning about having a decent math background). The book probably won't teach anyone who doesn't know they underlying principles but will be your go-to reference for any algorithm you implement. The book starts with the reviews of the requisite vector, matrix, transformation (including rotations by quaternions) and basic geometry for a view frustum, but quickly dives into more advanced topics. Ray tracing is covered for all areas of use, from light maps to reflections. The lighting chapter covers texturing using several map types as well as lighting models with a very enjoyable discussion of specular reflection models. Solid chapters on culling using bounding volumes and portal systems, shadowing and curve algorithms round out the first half of the book. The second half is devoted to the mathematics of physics, with chapters on basic collision detection, linear and rotational physics. The simulation of fluids and cloth (one of the more difficult physical models to accurately compute in a game) gets it's own chapter and it's a highlight for anyone implementing character clothing animation or a realistic water volume.Read more › This book is just what I have been looking for: something that presents and cogently explains the math that is most useful for implementing 2d and 3d computer graphics. If the Kindle edition did not have the problems it has, I would give it 5 stars. However, it gets a poor rating for two reasons. One, the diagrams are too small! Other Kindle documents allow the reader to scale images, but not this one. Two, and this is just INEXCUSABLE: The Kindle edition, but not the print edition, has errors that make the equations and proofs worthless. I can't quote examples exactly because special characters don't show up properly, but here's a description of three examples: The book expects one to have significant prior knowledge in mathematics and physics. Not a good book for beginners and is not that valuable for experts. The book has 16 chapters and spends on average 20 to 30 pages per each very complex topic - how can you explain fluid dynamics in 20 pages? A reference of formulas? To a degree. It does not hurt owning the book to look up a formula if you forgot one. I found that I could cross-reference some of the problems with material on the Internet where you can find much better explanations. P.S. there is no programming in the book with the exception of a few OpenGL listings which I did not find interesting. I believe this book is targeted towards a lack of intro level computer graphics texts amongst colleges. I highly recommend this book because it is way cheaper than the texts I bought for my class ($140) AND it has the same contents if not more. It goes over everything I need to learn in my class and Eric is very easy to contact if you need extra explanation which is just unheard of. Great book. It has plenty of examples and explanation for each concept covered. I've used it as a reference on a number of projects. Some of the topics are geared more for advanced programmers, but intermediate skill level is really all that's necessary to understand them. From the first to the last page first, this book is very well written and full of examples which really make sense. From the beautiful math to the implementation examples: a masterpiece. If you are want to understand what is inside an 3D graphic engine (for gaming or not), this THE BOOK! Congratulations Eric Lengyel.
Paramount ChemistryAlgebra 1 or elementary algebra is very important to know as it is required for higher math courses, like geometry and algebra 2. Algebra 1 can be challenging if concepts in prealgebra were not fully understood. However some concepts in algebra 1 can be challenging as well.
Study and Research Facilities Library The University of Manchester Library is one of the largest and best-equipped in the country containing over 5 million printed items. It also provides access to the largest collection of electronic resources of any UK academic library, including more than 500,000 electronic books, over 25,000 unique electronic journals, and several hundred other databases. Most of these can be accessed on and off campus by staff and students. For further information please consult the guides provided on the Library's subject pages. Recent electronic resources made available to the School include the SIAM ebook collection, comprising almost 400 books from a leading source of knowledge for the world's applied mathematics and computational science communities, and the Encyclopaedia of Mathematics, the most up-to-date and comprehensive English-language graduate-level reference work in the field of mathematics today. Computing facilities The School has excellent computing facilities, ranging from PCs to UNIX state-of-the-art workstations with access to the powerful supercomputers at Research Computing Services. Software available for use by students includes Matlab 7, Mathematica 5, Maple 10, S-Plus 7, Minitab 14, SAS 9.1, SPSS 13, tecplot 10, GSharp, and the NAG f77 and f90 libraries (mark 21). All students have access to these facilities in a number of clusters distributed around the School. Postgraduate students, in addition, have access to the above hardware and software on machines located in their rooms. Students and staff of the School can also access to various PC clusters located across campus. There are several large public clusters located in the John Ryland's library, and a large public cluster (open 23 hours a day) located in Manchester Computing on the ground floor of the Kilburn building; these clusters run Windows XP. Study Rooms The School is located in the purpose-built and centrally-located Alan Turing Building. The School's space within the Building has been designed to meet the needs of students and staff. There are a number of study rooms specifically for our undergraduates. In addition to these study areas, the School has an Undergraduate Common Room where can relax and spend social time together. G.211 The Undergraduate Work Room Off to the left through the Undergraduate Common Room is the Undergraduate Work Room. Students often like to work together and we see this as a natural place where students can gather to do mathematics. Notice boards providing course information will be positioned in the room and there will also be pigeonholes where messages and mail for undergraduate will be placed. You should check these pigeonholes regularly in case there is something for you. In this room there are two areas where groups can work round a blackboard which will be partitioned for most of the time and these will be used for some first year supervisions. When not booked for supervisions they can be used as additional work space. Careers information will be placed in this room (or possibly in the Undergraduate Common Room). G.101 The Undergraduate Quiet Study Room Off to the right through the Common Room is the Undergraduate Quiet Study Room. This is intended as a quiet area where students who wish to work on their own can do so without interruption. In this room there is a collection of undergraduate text books which are available for reference. These are books which were collected together during the move over the summer. It may be possible to improve this collection if it is well used. Students are asked not to remove the books from the room. This cluster is on the right of the atrium just before the Undergraduate Common Room and is available for use at any time when the building is open so long as it is not in use by a class. The far end can be partitioned off for use by small classes and this area is used for some postgraduate teaching.
Note: This page requires Internet Explorer for Windows and MathPlayer to properly display and demonstrate MathPlayer's new accessibility features. To learn more about obtaining Internet Explorer and MathPlayer, please visit the MathPlayer Download page. MathPlayer contains Design Science's math-to-speech technology. We started working on this technology many years ago. Based on feedback and continuous work, we have greatly expanded math-to-speech capability so that we are the clear leaders in producing understandable speech from math. Our technology delivers different speech based on the user's needs and content area of the page. We also support 15 different languages and have plans to add even more languages to MathPlayer in the future. If you are using a learning disability tool that highlights words as it speaks them, it will probably highlight the math also. See if your assistive technology vendor is listed on our AT Support page as supporting synchronized highlighting of math. MathPlayer 3 incorporates the ability to generate braille from MathML so that it can be sent to a refreshable braille display. Check our AT Support page to see if your assistive technology vendor supports braille output. While you can often use the left and right arrow keys to move back and forth by character to hear the words that MathPlayer spoke, we know better navigation is very important. We are actively developing navigation capabilities as part of joint IES grant with ETS. We expect navigation to be incorporated into a future release. You can make MathPlayer speak the math embedded in a web page two ways: Right-click on an equation and choose the Speak Expression command. Use a screen reader product that will read the entire web page and invoke MathPlayer to speak the math. Just as a test, right-click on the equation below and choose Speak Expression: α2+β2≤1 [MathML Equation -- requires MathPlayer] In order for this demonstration to work, you must have a MathPlayer-compatible text-to-speech engine installed on your computer. If you don't have such an engine, MathPlayer will display a dialog directing you to this page. Please follow the instructions for installing a text-to-speech engine in the next section. Users with low vision may also benefit from MathPlayer's MathZoom feature; simply click on the expression and an enlarged version will appear -- click again to close it. If you are running Windows XP or later, an additional download of a text-to-speech engine should not be required. If need be, you can download a free text-to-speech engine from Microsoft. First download and install Microsoft Reader and then download and install the text-to-speech engine mentioned on that page. The installer will suggest that you "activate" Microsoft Reader -- this is not necessary for MathPlayer. You can change the voice that it is used to speak the math, along with the rate and volume of the speech using Window's Speech Control Panel. Select the "Text to speech" tab to see your speech options. Many people prefer the female voice for speaking math. Higher quality text-to-speech engines can be purchased from other vendors. For example, AT&T Natural Voices are compatible with MathPlayer's "speak expression" command and can be purchased at one of the sites listed at the NextUp website. Other speech engines should work if they support Microsoft's SAPI 5 interface. It is very common for textbooks and technical papers to embed snippets of math such as x2EamaaCaaaleqabaGaaGOmaaaaaaa@37C3@ , tDamaaDaaaleaacaaIXaaabaaaaaaa@37BE@ or sin⁡(θ)aci4CaiaacMgacaGGUbGaaiikaiabeI7aXjaacMcaaaa@3BC4@[MathML Equation -- requires MathPlayer] inline. Even short equations and inequalities such as α2+β2≤aeqySde2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqOSdi2aaWbaaSqabeaacaaIYaaaaOGaeyizImQaaGymaaaa@3E55@[MathML Equation -- requires MathPlayer] are common. In fact, upwards of 90% of all math expressions in technical papers are short, inline expressions. All of these examples were written in Microsoft Word and MathType and exported to MathML using MathType's "MathPage" technology. MathPage technology was added to MathType in version 5.0. No special work is needed to author the expressions to make them accessible. Any product that exports MathML will produce pages that MathPlayer can speak.
Get FREE access to NCERT Solutions Model solutions Explanatory videos Share solutions with friends Get NCERT Solutions for your class: Select Class Select class from the list Select Subject Select subject from the list NCERT solutions are extremely useful for students as this segment includes detailed answers to the questions mentioned in NCERT textbooks. These solutions help students get the underlying concepts of each chapter (in respective subjects) and they can compare their answers with the same. How will NCERT Solutions help you study better? FREE online questions and model answers for all classes Complete and customized answers for each question in consonance with NCERT textbooks Extensively researched solutions created by subject matter experts Answers have been framed in lucid and easy to understand language Videos and animations to support the solutions of difficult questions in Mathematics Answers enhanced with graphs and illustrations for better comprehension of concepts
There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture. This book explains why these problems exist, why they matter, what... Motivating readers by making maths easier to learn, this work includes complete past exam papers and student-friendly worked solutions which build up to practice questions, for all round exam preparation
Beginning Algebra 9780321784919 ISBN: 032178491X Edition: 6th Revise Pub Date: 2011 Publisher: Pearson Education Summary: Written by Elayn Martin-Gay, Beginning Algebra was released in 2011 in this 6th revised format. Complete with plenty of new material compared to earlier versions, you can find out more about getting to grips with algebra and learning how to start in the right way. Published by Pearson Education, this text book is available at the cheapest prices. Buy Beginning Algebra online now and take advantage of the lowest price...s on previously owned text books. Sell back your copy now if you already own it and make the most of a good deal. Try Valore Books for Beginning Algebra deals now. Martin-Gay, Elayn is the author of Beginning Algebra, published 2011 under ISBN 9780321784919 and 032178491X. Eight hundred twenty four Beginning Algebra textbooks are available for sale on ValoreBooks.com, three hundred used from the cheapest price of $39.88, or buy new starting at $76.90.[read more]80321785282 shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less]
Exercises I've discovered that simple exercises are exceptionally useful during a seminar to complete a student's understanding, so you'll find a set at the end of each chapter. Most exercises are designed to be easy enough that they can be finished in a reasonable amount of time in a classroom situation while the instructor observes, making sure that all the students are absorbing the material. Some exercises are more advanced to prevent boredom for experienced students. The majority are designed to be solved in a short time and test and polish your knowledge. Some are more challenging, but none present major challenges. (Presumably, you'll find those on your own – or more likely they'll find you
Find out more about our school Mathematics At the Lake Champlain Waldorf School, we teach math to build the traditional skills of algebra, geometry and calculus. After all, we want you to have the keys that will let you go anywhere in the worlds of math and science. But we have something more in mind, too. We think everyone can glimpse the beauty and rigor of higher mathematics, and be a better person and thinker for it. High school math at LCWS is taught in the block system and in ongoing daily track classes. This combination allows you to dive into new math concepts and stay with them for a while, exploring all the implications of an idea. You will learn about math applications, play with math, and get to know the personalities behind the great discoveries. At the same time, in your track classes, you get daily practice and skill-building that lets you move ahead with confidence through algebra, geometry, advanced algebra, pre‐calculus, and calculus. Algebra I Track Class This course begins with a thorough review of basic arithmetic, including decimals, fractions and use of formulas. Students study simple linear equations, work with two linear equations, and use equations to solve word problems (integer, simple area, and perimeter, age, and money problems). Polynomials include a thorough review of the laws of exponents, operations with monomials, recognizing like terms and adding, subtracting and multiplying polynomials, and factoring. Students review ratios, proportions, irrational numbers, and simplifying square roots. Towards the end of the year, students study negative and fractional exponents, functions, the slope of a line, and the basic trigonometric definitions. We also derive the quadratic formula. Algebra II Track Class This skills course begins with a review of linear equations, including solving, graphing, and finding the slope of a line. Students review solving quadratic equations several different ways: factoring, completing the square, and using the quadratic formula. Students study the conic sections: parabolas, circles, hyperbolas, and ellipses. They also study functions and graph many rational functions. We review solving systems of equations and use this to solve word problems, such as area and projectile problems, different types of numbers, logarithms, and growth problems, and then study the number e and imaginary numbers, the trigonometric functions, and statistics, which includes calculating standard deviation and statistical probability. Permutations and Combinations 9th Grade Main Lesson Students learn how to calculate the number of permutations and combinations, such as finding the odds of winning the lottery, the chances of winning with a certain hand in card games, and the odds against a sports team that has already lost two games in a series, and apply this to various word problems. This leads to a study of Pascal's Triangle, Pascal's biography and other contributions to mathematics, and the history of probability theory. Geometry 9th Grade Main Lesson Students build on their work with platonic solids by drawing two dimensional plane projections of them, utilizing the graphic science of "descriptive geometry." Plates are developed that show several 2-D projections of the 3-D shapes. Distances and visibility of lines and points are determined from the plates. This graphic science develops the students' visualization skills, analytic thinking, and precision with drafting. Trigonometry 10th Grade Main Lesson Trigonometry comes from Greek and means triangle measurement. That is the way this branch of mathematics began and that is what we will be studying in this course, although Trigonometry grew far beyond its roots and is used in many interesting and important ways in science, mathematics, and engineering. These applications are called Analytical Trigonometry and are studied in more advanced math courses. Surveying 10th Grade Main Lesson Surveying is defined as the collection of data relating to the recording of surface and near surface features of the earth and generally their representation as a map, plan or digital model of the those features. In this block we will explore a brief history of surveying, the effects of technology on the field, concepts in trigonometry, working in horizontal and vertical planes, a good deal of fieldwork and also a good deal of time in the laboratory doing calculations. This block will also include a good deal of group work. You will work with a group for the entire block on your fieldwork and calculations. How well you work within your group will determine how much material we cover and a good portion of your grade. Projective Geometry 11th Grade Main Lesson Block We begin with points, lines, and planes familiar to us from Euclidean geometry. We treat each of these elements as equally primitive. We remove the idea of measurement. What is left of a line when you remove a quantity of length? We use drawings to guide our imagination. We strive to imagine these elements and their relationships as wholes rather than as measured parts. What begins to emerge is an exact mathematics of qualities. We notice that the web of inter-relationships holds no matter how much we wiggle things. We discover that the Euclidean idea of infinity transforms into something much different in projective space. We discover that projective space is full of unexpected forms similar to forms we find in the natural world. We discover a totally consistent symmetry without the exceptions needed in Euclidean space. What kind of a science could this mathematics reveal? Calculus 12th Grade Main Lesson This class will explore the applications of calculus to various practical engineering, biological, economic, and geometric problems. The course is intended to prepare a student for the possibility of entering a math-intensive college track such as engineering, pre-medicine, physics, or mathematics. Senior Mathematics 12th Grade Main Lesson Senior Mathematics In this skills course, we begin with reviewing trigonometry and graphing functions. Other units throughout the year include computer programming, pre-calculus topics, and solving interest rate and growth problems.
About: Fast Fourier Transforms Metadata Name: Fast Fourier Transforms ID: col10550 Language: English (en) Summary: This book uses an index map, a polynomial decomposition, an operator factorization, and a conversion to a filter to develop a very general and efficient description of fast algorithms to calculate the discrete Fourier transform (DFT). The work of Winograd is outlined, chapters by Selesnick, Pueschel, and Johnson are included, and computer programs are provided.
Saxon Math Homeschool Curriculum for Grades 7-12 If you have a math student who likes to work a variety of problems and would be bored with twenty-five of the same type of problems, then your student will like Saxon Math. If your student needs to work twenty-five problems of the same type to learn a concept, this might not be the best program for you unless you adapt the problem sections. Our daughters who chose Saxon for their math did quite well in math courses at college. $5 Flat Rate Standard Shipping On Any Pre-Paid Order to Continental US We think the Solution Manuals are a must for Algebra ½ and up! Not just the answers are included; the problems are worked for you! They come with the homeschool kits for Saxon Math 54 to Saxon Math 87. Instruction CDs If you need help with Saxon Math or want offer additional instruction, then we can highly recommend that you purchase an instruction CD. Two instruction CD options are available. The Saxon Homeschool Teacher Lesson and Test CDs offer instruction for each lesson. The instructor shows how to work every problem in the book. In the Saxon Teacher CDs, a teacher with extensive Saxon teaching experience goes over a lesson and works every practice problem problem set, and test problem. The Saxon Teacher instruction is secular, and there are several different instructors. The DIVE CDs have a lecture for each lesson and then the instructor, Dr. Shormann, works several problems that are similar to those in the Saxon textbook. DIVE CDs are taught by Dr. Shormann with a Christian perspective. The Algebra ½ Kit 3rd Edition covers all topics normally taught in pre-algebra, as well as additional topics from geometry and discrete mathematics. It is recommended for seventh-graders who plan to take first-year algebra in the eighth grade, or for eighth-graders who plan to take first-year algebra in the ninth grade. The Saxon Algebra ½ Homeschool Kit 3rd Edition contains one of each of the following items. Saxon Algebra ½ Answer Key and Tests3rd Edition The Saxon Algebra 1, Third Edition is made up of five instructional components: Introduction of the New Increment, Examples with Complete Solutions, Practice of the Increment, Daily Problem Set, and Cumulative Tests. Saxon Algebra 1 3rd Edition Answer Key and Tests The new Saxon Geometry Kit is available now. For those of you who want a separate geometry from Saxon, this should work. In the meantime, Saxon Algebra 1 Third Edition and Algebra 2 Third Edition still include geometry. If you are required by law to keep a log, please log the lessons on geometry separately as geometry. Saxon Geometry First Edition Homeschool Kit With Solutions Manual Publisher: Saxon Homeschool. ISBN-13: 9781600329760 This kit includes one of each of the following: Geometry textbook (9781602773059) Homeschool Testing Book with cumulative tests, answer forms, and answer key (9781600329777), and Saxon Geometry Solutions Manual (978160275619). List $130.55 Price $104.99 Saxon Geometry, 1st Edition Homeschool Testing Book Publisher: Saxon Homeschool. ISBN-13: 9781600329777 Unlike the Homeschool Packets in other Saxon Math curriculum, the Saxon Geometry Homeschool Testing book has the test forms and the answers for those tests. There are no answers to problems in the book. It makes sense. There are complete solutions in the Solutions Manual which is included in the kit and is not sold separately.
Algebra: Word Problems Help and Practice Problems Find study help on linear applications for algebra. Use the links below to select the specific area of linear applications you're looking for help with. Each guide comes complete with an explanation, example problems, and practice problems with solutions to help you learn linear applications for algebra. Now that you have read through all the chapters in this book, you are ready to take the test. Like the introductory test, the test has 50 questions, which cover the same topics presented in this book. The questions are similar to the Math Word Problems Practice Quiz This test has 30 multiple-choice questions and is in the same format as the introductory test (Introductory Math Word Problems Practice Quiz). Each question corresponds ... Algebra Word Problems Basic algebra problems ask you to solve equations in which one or more elements are unknown. The unknown quantities are represented by variables, which are letters of the alphabet, such as x or y. The ...
Arithmetic resources Find the Highest Common Factor and Lowest Common Multiple of pairs of natural numbers, including Euclid's algorithm. This Quick Reference leafet has been contributed to the mathcentre Community Project by Leslie Fletcher and reviewed by Lesley Wright, Liverpool John Moores University. This leaflet provides information on symbols and notation commonly used in mathematics. It shows the meaning of a symbol and, where necessary, an example and an indication of how the symbol would be said. For further information from mathcentre resources, a search phrase is given. This Quick Reference leaflet is contributed to the mathcentre Community Project by Janette Matthews and reviewed by Tony Croft, University of Loughborough. Computer-aided assessment of maths, stats and numeracy from GCSE to undergraduate level 2. These resources have been made available under a Creative Common licence by Martin Greenhow and Abdulrahman Kamavi, Brunel University.
Statistics Online Courses At Middlesex County College, Statistics I and II are offered as Internet classes. For Statistics I (Math 123), the pre-requisite is Algebra II. For Statistics II, the pre-requisite is Statistics I. Students should be able to read and follow math directions, have a TI - 83 or 84 calculator, have computer internet access and have Excel installed on their computer. Students must have good organizational skills. Successful students are able to budget their time appropriately. The amount of time spent for Statistics weekly should mimic the amount of time that would be spent both in class and outside of class studying if the class was taken in a traditional format. For both classes, students use the College WebCT site to obtain weekly lessons. The lessons are designed to supplement what is presented in the text book. Each lesson summarizes the major points of the topic. In addition, screen clips from both Excel and the TI are presented. Students should print out the lessons, read them and then do the weekly problems. Students may be required to send their solution to selected problems to the instructor. In Math 123, students will take about 6 quizzes on line. For each quiz, there is a limited amount of time that the quiz will be available. Students will complete 2 projects, either mailing or dropping them off to the instructor. In addition, there will be 3 tests and a final exam, which are taken in the MCC testing center. In Math 124, students will compete a 2 part project. The project will either be mailed or hand-delivered to the instructor. In addition, there will be 4 tests and a final exam, which are taken in the MCC testing office. If a student is unable to take the tests in the testing office, alternate arrangements must be made, where the test is taken under appropriate supervision. For example, a student attending another college might take the tests at that college's testing office. For further information, contact This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
Book summary "Admirably meets the topology requirements for the pregraduate training of research mathematicians."--American Mathematical Monthly Crucial to modern mathematics, topology is equally essential to many other disciplines, from quantum mechanics to sociology. This stimulating introduction employs the language of point set topology to define and discuss topological groups. The text examines set-theoretic topology and its applications in function spaces, as well as homotopy and the fundamental group. This new theoretical knowledge is applied to concrete problems, such as the calculation of the fundamental group of the circle and a proof of the fundamental theorem of algebra. The abstract development concludes with the classification of topological groups by equivalence under local isomorphism. Throughout this text, a sustained geometric development functions as a single thread of reasoning that unifies the topological course. Well-chosen exercises, along with a selection of problems in each chapter that contain interesting applications and further theory, help solidify students' working knowledge of topology and its applications.
Summary: These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs. New to the Third Edition are exercises that provide guided practice for the textbook's Problem-Solving Strategies, focusing in particular on working symbolically
Summary: Participants explore the base-ten structure of the number system, consider how that structure is exploited in multi-digit computational procedures, and examine how basic concepts of whole numbers reappear when working with decimals. Paperback 2nd Edition text. CLEAN PAGES NO HIGHLIGHITNG/WRITING Used in book Used tems may have stickers on cover. May have writing/underlining or highlighting, and varying degrees of wear.. Ships fas...show moret. Ships fast. Expedited shipping 2-4 business days; Standard shipping 7-14 business days. Ships from USA! ...show less $33.09 +$3.99 s/h New TextbookBarn Woodland Hills, CA 0133733130 Premium Books are Like New or Brand New books direct from the publisher sometimes at a discount. These books are not available for expedited shipping and may take up to 14 business days to...show more receive. ...show less $34
Cotati AlgebraBuilding confidence in the concepts introduced here and the ways to express them is critical. Understanding Precalculus and Trig concepts need not be daunting. They can be explained in simple terms and there are ways to remember key concepts
Mathematics/Computer Science Goals Goal I: Learn to apply precise, logical reasoning to problem solving Learning Objectives for Goal I (a) Students should be able to perform complex tasks; explore subtlety; discern patterns, coherence, and significance; undertake intellectually demanding mathematical reasoning; and reason rigorously in mathematical arguments in order to solve complex problems. Learning Objectives for Goal II (b) Students should be able to undertake independent work, develop new ideas, and discover new mathematics. (c) Students should be able to state problems carefully, articulate assumptions, and apply appropriate strategies. (d) Students should possess personal motivation and enthusiasm for studying and applying mathematics; and attitudes of mind and analytical skills required for efficient use, appreciation, and understanding of mathematics. Goal III: Read and communicate mathematics with understanding and clarity. Learning Objectives for Goal III (e) Students should be able to read, write, and speak mathematically; read and understand technically-based materials; contribute effectively to group efforts; communicate mathematics clearly in ways appropriate to career goals; conduct research and make oral and written presentations on various topics; locate, analyze, synthesize, and evaluate information; create and document algorithms; think creatively at a level commensurate with career goals; and make effective use of the library. (f) Students should possess skill in expository mathematical writing, have a disposition for questioning, and be aware of the ethical issues in mathematics. Goal IV: Nature of Mathematics Learning Objectives for Goal IV: (g) Students should possess an understanding of the breadth of the mathematical sciences and their deep interconnecting principles; substantial knowledge of a discipline that makes significant use of mathematics (understanding of interplay among applications, problem-solving, and theory; understanding and appreciation of connections between different areas of mathematics and with other disciplines; awareness of the abstract nature of theoretical mathematics and the ability to write proofs; awareness of historical and contemporary contexts in which mathematics is practiced; understanding of the fundamental dichotomy of mathematics as an object of study and a tool for application; and critical perspectives on inherent limitations of the discipline). Goal V: Mathematical Modeling Learning Objectives for Goal V: (h) Students should be able to apply mathematics to a broad spectrum of complex problems and issues (formulate and solve problems; undertake some real-world mathematical modeling project; solve multi-step problems; recognize and express mathematical ideas embedded in other contexts; use the computer for simulation and visualization of mathematical ideas and processes; and use the process by which mathematical and scientific facts and principles are applied to serve society). Goal VI: Content Specific Goals Learning Objectives for Goal VI: (i) Students should understand theory and applications of calculus and the basic techniques of discrete mathematics and abstract algebra. (j) Students should be able to write computer programs in a high level language using appropriate data structures (or to use appropriate software) to solve mathematical problems. Mathematics Department Goals and Learning Objectives: As outlined by the Mathematical Association of America (MAA), "Mathematical Association of America's Committee on the Undergraduate Program in Mathematics (CUPM)." World of Opportunity Wizard Wow! Did you know Daemen has a program that concentrates on developing your leadership skills?
Working at the University of Nottingham, Dr. Rebecca Taylor and her colleagues have created this very fine set of resources designed to assist teachers who seek to utilize mathematics in the service of teaching... Created by Anne Cavagnaro and Ida Ponder of Columbia College, this course will help students develop critical thinking skills, differentiate between scientific and pseudoscientific data, apply statistics in real world... The American Mathematical Association of Two-Year Colleges (AMATYC) has compiled a collection of mathematics resources related to various subjects and disciplines. ?Math Across the Community College Curriculum? is the... The Economics Challenge is a economics competition for high school students living in the twelfth district -- Alaska, Arizona, California, Hawaii, Idaho, Oregon, Utah, and Washington. The competition strives to... A discussion of the application of physics (particle physics, quantum theory, and statistical mechanics) to finance theory, with econophysicists attempting to break ground in Wall Street, and the possible underlying...
Repetition, drills and application ensure mastery of computational skills with Lifepac Math. Students will progress to higher-level cognitive reasoning and analysis as their problem solving ability increases. Perfect for students who flourish in a self-paced, individualized learning format, each consumable LIFEPAC combines lessons, exercises, projects, reviews and tests. Product: Lifepac Math: Grade 9: Unit 10: Quadratic and Algebraic Equations Vendor: Alpha Omega Binding Type: Paperback Media Type: Book Minimum Grade: 9th Grade Maximum Grade: 9th Grade Weight: 0.4 pounds Length: 11 inches Width: 8.5 inches Height: 0.125 inches Vendor Part Number: MAT 0910 Subject: Algebra, Calculus & Trig Math: Grade 9: Unit 10: Quadratic and Algebraic Equations. Average Rating Parent Rating Comments I would not recommend Switched on Schoolhouse. We have tried it two different years with different children and were not successful either time. It has not been a good match for my family.
Programs of Study MATH-1040 - Business Math This course is for the student who needs specific math skills to address financial problems and/or applications. Students will learn mathematics as it relates to retail, payroll, financial analysis, interest earned, and money management. Students may use a calculator and computer to solve a variety of applications.
Created by Lang Moore for the Connected Curriculum Project, the purpose of this module is to provide an introduction to the elementary complex transcendental functions -- the exponential, sine, and cosine functions.... This lesson from Illuminations teaches students the features of sine and cosine graphs. Students use uncooked spaghetti to demonstrate the properties of the unit circle, which they will then represent with graphs. It is... This lesson will help students understand the law of cosines and the Pythagorean theorem. The material has students find out whether a triangle is acute, obtuse, or right; determine a formula for the law of cosines to...Created by David Smith for the Connected Curriculum Project, the purpose of this module is to study data that may be modeled by sinusoidal functions; in particular, to determine average level, period, frequency,...
Math 112 General Information The main website for Math 112 can be found through here. Click on "Math 112 Fall 10" and log in with your UW-NetID. This page has all the information you will need on a day-to-day basis, including the syllabus, daily schedule, and links to the homework assignments (which you will complete online). You should bookmark this page. This website (the one you are reading now) has extra information that is specific to the two sections that I am teaching, Sections 14 and 15. You should read both websites very carefully. If you have any questions about policy, or have a conflict with one of the exam dates, please come and talk to me during the first two weeks of the semester (before Sept. 10 for exam conflicts). Textbooks There are two texbooks for this course. The first is College Algebra Worktext by Schroeder which you should buy at the bookstore and bring to class every day. This text is like a workbook with many examples and room to work them out in the book. I recommend taking notes in the book instead of in a separate notebook. The second book is College Algebra by Stitz and Zeager. This book is available entirely online. (There is a link on the Moodle page.) You do not need to print out this book. Rather, you should use it as a reference as you are working on homework and studying. Homework There will be a lot of homework in this class. I cannot stress enough how important it is for you to do all of the homework and to put effort into it. You cannot pass this class by just memorizing a bunch of formulas and equations. You need to first understand the ideas and then get a feel for the processes that are used to solve problems. This "feel" is only acquired to doing lots and lots of problems, i.e. the homework. If you successfully complete and understand all of the homework, you will do very well in this course. Homework will be assigned, completed, and graded via WeBWorK. You can find links directly to the individual assignments on the main Math 112 website (Moodle). This is a online service where you will login to see your homework, submit your answers, and see immediately if they are correct or incorrect. You will have 5 chances to submit answers for most problems, so if your answer is incorrect, you can try again with no penalty. Additionally, after the assignment is due, you will have access to all of the correct answers. For more information, see the WeBWorK section below. Some homework assignments require you to write your answer on paper and give it to me (usually the graphing problems). Please print the "handin" sheets and use these - it makes the grading a lot easier! The handin sheets are on Moodle, below the links to the homework assignments. Quizzes We will have short weekly quizzes in class on Mondays. They will cover the material that was presented in class the week before. The goal of the quizzes is not to punish you if you do not understand the material yet. Instead, they have two purposes. First, they indicate to both you and me, how well you understand the material. It is one thing to solve homework problems while you have your text and notes available, and quite a different thing to be able to solve a test problem with no resources. The quizzes serve as an "in between" so that you can see where you are weak before the test comes along. The second purpose is learn how to write down the solution to a problem in a neat, orderly, readable, and logical way. You will be expected to do this on the exams! However, since you only need to submit your answers for the homework, we cannot check if you are doing it there. Participation A portion of your grade will be from participation. To get the points, you simply need to present two problems on the board during the course of the semester. Exams There will be three midterm exams during the semester, in addition to the final exam. The midterms are evening exams. Please mark the following dates on your calendar so that you do not schedule something else that conflicts. It is guaranteed that 80% of exam problems are slight modifications of homework problems. But remember that you will need to give your solutions in a neat, orderly, readable, and logical way to get full credit! Grades This is a total of 1000 possible points. Your letter grade will be assigned based on the following scale: A: 900-1000 B: 750-900 C: 600-750 D: 500-600 F: 0-600 There will be no make-up quizzes or homework in this class. Instead, I will keep only your 10 highest scores and 30 highest homework scores. This means that roughly 2 quizzes and 7 homeworks will be dropped. WeBWorK Some things to note about using WeBWorK: You do not have to do the whole assignment at the same time. You can do the problems individually whenever you want. Be sure to take advantage of the "Preview Answer" button. This will convert your answer into "readable math" so that you can check that what you typed in is what you really meant to type in. Some of the numbers and variables in the homework problems are randomized. This means that you will not have exactly the same questions as your friend. If you prefer to work away from the computer, you can print a copy of your homework problems. Just click on the "Download hardcopy of of this homework set" link at the top of the problem list. The answer fields are case-sensitive, so make sure your variables and function names are in the correct case. Make sure you read the instructions for each problem. They may contain information about the correct formatting that you need to use. There is an "Email Instructor" button below every question. After you click on this, you can type a message and send an email to me. (It will also go to Mike Schroeder - the course coordinator.) Be sure to describe clearly the issues you are having so that I can help you. Describe in detail what you have attempted and where you think the error might be. If I receive a message that just says "HELP!!!" or "I don't know how to do this problem." then I will not be able to help you. The homework assignments are all due at midnight, but plan ahead! I will not be able to answer your question if you send me an email at 11:30 the night the homework is due. In fact, I am generally busy on MWF evenings, so if you send me an email after 4 pm, I might not see it until the next day. "I think I have the right answer, but WeBWorK says that it is wrong!!" Did you use the preview button? You should do this before you submit an answer to make sure that you typed what you really meant to type. Having parenthesis in a different place will very often change the answer! Re-read the question. For some questions, the answer needs to be in a specific format or form. Or, you may have just mis-read the question. Still not right? These same homework questions have been used for many semesters, so there is a 99% chance that WeBWorK is right and you are wrong. Go back to your notes to make sure you are approaching the problem correctly. If you still can't get it right, either use the "email instructor" button or talk to me during office hours or class.
More About This Textbook Overview The goal of this series is to provide readers with a strong foundation in Algebra. Each book is designed to develop readers' critical thinking and problem-solving capabilities and prepare readers for subsequent Algebra courses as well as "service" math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. Algebra and Problem Solving. Functions, Linear Functions, and Inequalities. Systems of Linear Equations and Inequalities. Polynomials, Polynomial Functions, and Factoring. Rational Expressions, Functions, and Equations. Radicals, Radical Functions, and Rational Exponents. Quadratic Equations and Functions. Exponential and Logarithmic Functions. Conic Sections and Nonlinear Systems of Equations. Sequences, Series, and the Binomial Theorem. For anyone interested in introductory and intermediate algebra and for the combined introductory and intermediate algebra. Related Subjects Meet the Author Bob Blitzer is a native of Manhattan and received a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English literature) from the City College of New York. His unusual combination of academic interests led him toward a Master of Arts in mathematics from the University of Miami and a doctorate in behavioral sciences from Nova University. Bob's love for teaching mathematics was nourished for nearly 30 years at Miami Dade College, where he received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College and an endowed chair based on excellence in the classroom. In addition to his Developmental Algebra Series, Bob has written textbooks covering college algebra, algebra and trigonometry, precalculus, and liberal arts mathematics, all published by Pearson Education. When not secluded in his Northern California writer's cabin, Bob can be found hiking the beaches and trails of Point Reyes National Seashore, and tending to the chores required by his beloved entourage of horses, chickens, and irritable roost
Precalculus - With 2 CDS - 4th edition Summary: Bob Blitzer's background in mathematics and behavioral sciences, along with his commitment to teaching, inspired him to develop a precalculus series that gets students engaged and keeps them engaged. Presenting the full scope of mathematics is just the first step. Blitzer draws students in with applications that use math to solve real-life problems. and Precalculus, all published by Pearson Prentice Hall35 +$3.99 s/h Good Nettextstore Lincoln, NE 2009 Hardcover8.57 +$3.99 s/h Good SellBackYourBook Aurora, IL 0321559843 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!! $9.709321559845
Geometry Book Description: A complete introduction to geometry Investigate the world of planes and solids with this revised edition of Geometry. Students go beyond the basics of geometry. Chapter openers and colorful photos invite students into the text. They explore geometric solids, triangles, the Pythagorean Theorem, quadratic equations, length, area, and volume. Throughout, Geometry presents short, lively lessons students can grasp easily. Illustrated examples abound. Features include Estimation Activities, Algebra Review, and Geometry in Your Life. Calculator Practice exercises make use of the special features of graphing calculators. Best of all, students learn to apply geometry to situations in their own lives.Overall, this high-interest, low-readability text makes it easy for you to engage students who struggle with reading, language, or a learning disability.Lexile Level 820 Reading Level 4-5 Interest Level 7-12
Visualizing Algebra Patterns and Problems Beginner Join 19,645 Students In collaboration with We're still building the full experience Approx. 4 months (work at your own pace) Class Summary Throughout this course, we will use algebra to quantify and describe the world around us. Have you ever wondered how many songs can fit onto your flash drive? By the end of the course, you'll have stronger skills for modeling problems, analyzing patterns, and using algebra to arrive at conclusions. What Will I Learn? Understand and develop the language, models, and tools for math reasoning. Reason with numbers and units to solve problems. Identify, analyze, extend, and create patterns and functions. Make connections about relations and functions through multiple representations: graphs, charts, tables, equations. What Should I Know? Almost nothing! Students should know their multiplication tables (1-12) and be proficient with adding and subtracting integers (for example, -7 - 16). A pencil and paper will be a student's best aid rather than a calculator. Syllabus Unit 1 - Review of Foundations Unit 2 - Equations and Inequalities Unit 3 - Graphs of Linear Equations Unit 4 - Systems of Equations Unit 5 - Exponents and Polynomials Unit 6 - Factoring Unit 7 - Rational Expressions Unit 8 - Rational Exponents and Radicals Unit 9 - Solving Quadratic Equations Unit 10 - Quadratic Equations and Applications Unit 11 - Conic Sections Unit 12 - Inverses, Exponentials, and Logarithms Course Instructors Sandra DeSousa Instructor Sandra DeSousa is an Adjunct Instructor of Mathematics at San Jose State University and San Jose City College. She has been teaching developmental math for the last 6 years (in both small classrooms and large lecture halls). Teaching algebra to her struggling students is her favorite because she loves when they "get it." She is always trying new and different teaching methods to help her students succeed and is very excited to work with Udacity. Chris Saden Course Developer After graduating from Emory in 2008, Chris dabbled in college admissions for a year, which led him to teach high school mathematics in Oakland. He radiates a love for learning and believes everyone deserves a great education. In 2012, Chris joined Udacity to reach thousands of students and share his joy of problem solving with the world.
The Word Problems Guide demystifies the most challenging of mathquestions on the GMAT-the word problem. This book equips students with broad, powerful strategies, as well as specific tools, for tackling GMAT word problems in all their various guises. Manhattan GMAT's Foundations of Math book provides a refresher of the basic math concepts tested on the GMAT. Designed to be user-friendly for all students, this book provides easy-to-follow explanations of fundamental math concepts and step-by-step application of these concepts to example problems. With nine chapters and over 600 practice problems, this book is an invaluable resource to any student who wants to cement their understanding and build their basic math skills for the GMAT. The Official Guide to the GMAT, 13th Edition will be the first guide on the market that contains official sample questions from the new Integrated Reasoning section that will be part of the GMAT exam in June 2012. Because this component is more interactive than other core components in the GMAT exam (Verbal, Quantitative, and Analytical Writing), the Integrated Reasoning practice questions will be housed on a companion website that readers will be able to access after purchasing the book. The Integrated Reasoning and Essay Guide demystifies the brand-new Integrated Reasoning section, which demands a blend of mathematical and verbal proficiency. This book equips students with the core strategies and tools for tackling difficult Integrated Reasoning questions. It also provides effective techniques for approaching the essay portion of the GMAT.
It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The... see more It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated as it is the basis of all mathematical modeling used in applications found in all disciplines.Traditionally, the study of algebra is separated into a two parts, Elementary and Intermediate Algebra. This textbook by John Redden, Elementary Algebra, is the first part written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course.John Redden's Elementary Algebra takes the best of the traditional, practice-driven algebra texts and combines it with modern amenities to influence learning, like online/inline video solutions, as well as, other media driven features that only a free online text can deliver. Using the online text in conjunction with a printed version of the text could promote greater understanding (at a lower cost than most algebra texts).From the traditional standpoint, John employs an early and often approach to real world applications, laying the foundation for students to translate problems described in words into mathematical equations. It also clearly lays out the steps required to build the skills needed to solve these equations and interpret the results. With robust and diverse exercise sets, students have the opportunity to solve plenty of practice problems. Elementary Algebra has applications incorporated into each and every exercise set. To do this John makes use of the classic "translating English sentences into mathematical statements" subsections in chapter 1 and as the text introduces new key terms.A more modernized element; embedded video examples, are present, but the importance of practice with pencil and paper is consistently stressed. This text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today.In addition, textual notation is introduced as a means to communicate solutions electronically throughout the text. While it is important to obtain the skills to solve problems correctly, it is just as important to communicate those solutions with others effectively in the modern era of instant communications.While algebra is one of the most diversely applied subjects, students often find it to be one of the more difficult hurdles in their education. With this in mind, John wrote Elementary Algebra from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success. This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960.... see more This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960. Robinson's modern infinitesimal approach puts the intuitive ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners to understand than the more common approach via limits.The First Edition of this book was published in 1976, and a revised Second Edition was published in 1986, both by Prindle, Weber & Schmidt. The book is now out of print and the copyright has been returned to the author. ״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a... see more ״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a surprisingly deep subject. We assume that students have some familiarity with basic set theory, and calculus. But very little of this nature will be needed. To a great extent the book is self-contained. It requires only a certain amount of mathematical maturity. And, hopefully, the student's level of mathematical maturity will increase as the course progresses. Before the course is over students will be introduced to the symbolic programming language Maple which is an excellent tool for exploring number theoretic questions.״ Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty... see more Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages.The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional.forall x is an introduction to sentential logic and first-order predicate logic with identity, logical systems that... see more forall x is an introduction to sentential logic and first-order predicate logic with identity, logical systems that significantly influenced twentieth-century analytic philosophy. After working through the material in this book, a student should be able to understand most quantified expressions that arise in their philosophical reading. This books treats symbolization, formal semantics, and proof theory for each language. The discussion of formal semantics is more direct than in many introductory texts. Although forall x does not contain proofs of soundness and completeness, it lays the groundwork for understanding why these are things that need to be proven.In formal logic, sentences and arguments are translated into mathematical languages with well-defined properties. If all goes well, properties of the argument that were hard to discern become clearer. This text describes two formal languages which have been of special importance to philosophers: truth-functional sentential logic and quantified predicate logic. The book covers translation, formal semantics, and proof theory for both languages. This can be used as the textbook for a semester long course in logic, for a unit on logic, or for self-directed study. Each chapter contains practice exercises; solutions to selected exercises appear in an appendix. The author is an assistant professor of philosophy at the University at Albany, SUNY. The materials here form a textbook for a course in mathematical probability and statistics for computer science students.״Why... see more The materials here form a textbook for a course in mathematical probability and statistics for computer science students.״Why is this course different from all other courses?״ * Computer science examples are used throughout, in areas such as: computer networks; data and text mining; computer security; remote sensing; computer performance evaluation; software engineering; data management; etc. * The R statistical/data manipulation language is used throughout. Since this is a computer science audience, a greater sophitication in programming can be assumed. It is recommended that my R tutorial, R for Programmers, be used as a supplement. * Throughout the units, mathematical theory and applications are interwoven, with a strong emphasis on modeling: What do probabilistic models really mean, in real-life terms? How does one choose a model? How do we assess the practical usefulness of models? * There is considerable discussion of the intuition involving probabilistic concepts. However, all models and so on are described precisely in terms of random variables and distributions.For topical coverage, see the book's detailed table of contents. A textbook that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation,... see more A textbook that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who (1) have had a previous course in prealgebra, (2) wish to meet the prerequisite of a higher level course such as elementary algebra, and (3) need to review fundamental mathematical concepts and techniques
Students investigate limits and continuity of functions. In this limits and continuity of functions lesson, students find the limit as a function approaches a given value. Students find the domain of functionsNinth graders investigate the functional relationship of different environmental phenomena. In this math lesson, 9th graders create models of various natural disasters. They use logarithmic and exponential functions to interpret population growth. Design an experiment to model a leaky faucet and determine the amount of water wasted due to the leak. Middle schoolers graph and write an equation for a line of best fit. They use their derived equation to make predictions about the amount of water that whould be wasted from one leak over a long period of time or the amount wasted by serveral leaks during a specific time period. In this functions instructional activity, students identify the domain and range of a function. They find the slope of a line. Students determine the limits of an equation and note the discontinuous and continuous functions. This three-page instructional activity contains 14 multiple-choice problems. This Mean Value Theorem and Rolle's Theorem worksheet is very thorough in explaining the two Theorums and showing the formulas. There are six prractice problems for classwork, and eight additional problems for homework. Ninth graders examine the different systems of government politics and power. As they study they'll record their reflections in their journals and in return share their insights with their classmates. For mastery, 9th graders prepare and perform a skit for their classmates. Are your calculus pupils aware that they are standing on the shoulders of giants? This lesson plan provides a big picture view of the connection between differential and integral calculus and throws in a bit of history, as well. Note: The calculus controversy paper is not included but one can find a number of good resources on the Internet regarding the development of calculus and the role of Newton and Leibnez. Students identify the different properties of a quadratic formula. In this algebra lesson, students take a quiz and discuss the problems form the quiz after it is submitted. They identify the important concepts dealing with quadratics and how it is related to the history of math. In this graphs and derivatives learning exercise, students graph the derivative of a function. They find the intervals and identify the maximum and minimum. This one page learning exercise contains three multi-step problems. students emulate a real world problem and model it with mathematics working in a small group. Observe and facilitate discussion as the groups use various strategies, such as (graphs, brainstorming, tables, and dialogue. In this complex function activity students identify the derivative of given functions and explore the rules of differentiation. This two-page activity contains explanations and examples, followed by ten problems. Young scholars record and graph data. In this graphing lesson plan, students plant seeds and, once the seeds sprout, record the change in height of the plants for several days. They make graphs to display to data and then predict how the future growth may appear on the graph. They write stories about the life cycle of a planet that corresponds to a graph they are given. In this limits worksheet, students evaluate limits and graph given expressions. They classify graphs as removable discontinuities, jump discontinuities, or asymptotes. this two-page worksheet contains seven multi-step problems.
Designed to support both teachers and university-based tutors in mentoring pre-service and newly qualified mathematics teachers at both primary and secondary levels, this essential new book offers straightforward practical advice that is based on practice, underpinned by research, and geared specifically towards this challenging subject area. Addressing... more... Mathematicians is a remarkable collection of ninety-two photographic portraits, featuring some of the most amazing mathematicians of our time. Acclaimed photographer Mariana Cook captures the exuberant and colorful personalities of these brilliant thinkers and the superb images are accompanied by brief autobiographical texts written by each mathematician.... more... This unique monograph building bridges among a number of different areas of mathematics such as algebra, topology, and category theory. The author uses various tools to develop new applications of classical concepts. Detailed proofs are given for all major theorems, about half of which are completely new. Sheaves of Algebras over Boolean Spaces will... more... Professional development is often determined by black and white thinking. Either issues are considered as being good or bad, or statements like teachers should or teachers must are transported. However, it is easily forgotten from which perspective the judgment is taken, surely it is not the teacher's one. Profoundly respecting and cherishing the... more...
Matrices -- Matrices on the Calculator Summary: A teacher's guide to working with matrices on the calculator. This starts with a lecture. You have to show them how to do matrices on the calculator. They should be able to… Enter several matrices at once Add several matrices Subtract Multiply Find inverses All of this is explained step-by-step in the "Conceptual Explanations." There are two things I stress. First, whenever I enter a matrix, I always check it. For instance, after I enter matrix [A], I go back to the home screen and go [A][Enter], and the calculator displays matrix [A] to me, so I can make sure I typed it right. One small mistype will ruin a whole problem, and it's really easy to do! Second, the calculator is very smart about interpreting equations. After you enter three matrices, you can just type [A][B]-1-[C] and it will multiply [A] by the inverse of [B] and then subtract [C]. Homework "Homework—Calculators." Depending on how things go, they may be able to finish this in class and have no homework
Dynamical Systems with Applications using Maple Book Description: This work covers material for an introductory course in the theory of dynamical systems. There is a short tutorial in MAPLE to facilitate the understanding of the theory. The text is divided into two parts: continuous systems using differential equations and discrete dynamical systems. Differential equations are used to model examples taken from various topics such as mechanical systems, interacting species, electronic circuits, chemical reactions, and meterology. The second part of the text deals with real and complex dynamical systems. Examples are taken from population modelling, nonlinear optics, and materials science. Linear algebra and real and complex analysis are prerequisites
THINKING MATHEMATICALLY - 2nd edition Summary: Thinking Mathematically is perfect for anyone who wants to develop their powers to think mathematically, whether at school, at university or just out of interest. This book is invaluable for anyone who wishes to promote mathematical thinking in others or for anyone who has always wondered what lies at the core of mathematics. Thinking Mathematically reveals the processes at the heart of mathematics and demonstrates how to encourage and develop them. Extremely practic...show moreal, it involves the reader in questions so that subsequent discussions speak to immediate experience47.78 +$3.99 s/h Acceptable Revival Books Haslingden, 2010 Paperback Fair Ex Library book with usual stamps and stickers. Small remainder mark to the page edges. Previously water damaged, slight waving to the page edges. A few small marks to the page...show more

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