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    Number Theory

    Riemann Sums, Taylor Polynomials, Taylor Residuals

    Please see the attached file for the fully formatted problems. 1. ? Calculate the Taylor Polynomial and the Taylor residual for the function . ? Prove that as , for all . ? Find the Taylor series of f. ? What is the radius of convergence for the Taylor series? Justify your answer. 2. ? Let f:[0,1] be a bo

    Linear Equations and their Solutions

    From the given polynomials, identify the polynomials of degree one. a. 311y - 5 - 43y b. (11y2)1/4 + 14 c. 10 + 19x - 2x2 d. 2 + 15x e. 52y4 + 7x + 2 f. (68)1y1 g. x3 + 3x - 9 Solve the following: i. -2x = 3x + 4 ii. 3x/4 = 6 iii. y/6 + 1 = 9 iv. 6 = -2x/4 v. Find f(1) for f(x) = 4x3 - 3x2 - x + 2

    Polynomials

    A. Solve the following questions involving fundamental operations on polynomials a.Find p(x) + 4q(x) given p(x)=4x4 + 10x3 - 2x2 + 13 and q(x) = 2x4+ 5x2 - 3 b. Find P(-1/2) if P(x) = 2x4 + x3 + 12 c. Simplify: (-4 + x2 + 2x3) - (-6 - x + 3x3) - (-6y3 + y2) d. Add: (2x2 + 6y2 + 4z2 + 3xy + yz + zx) + (4x2 + 3y2

    Euler Totient Function (Six Problems)

    For this problem it helps to know that: 3x7x13 = 273 (a) Define the Euler Totient function, (SYMBOL) For (b) to (f) please see attached. (PLEASE SEE ATTACHMENT FOR COMPLETE PROBLEM AND PROPER SYMBOLS)

    Euclid's Algorithm for Greatest Common Divisor

    1. (a) Use the Euclidcan Algorithm to find the greatest common divisor of 13 and 21 (b) Is 13 invertible in Z21? If so, find the reciprocal. (c) Suppose x and yare integers, what is the minimum positive value for 13x+21y? Determine all posible values of (x,y) for which the minimum is obtained. (PLEASE SEE ATTACHMENT FOR

    Factor Positive Integrers into Primes

    Factor into primes the following positive integers: (a) 25 (b) 4200 (c) 10(to the exponent)10 (d) 19 (e) 1 *Please see attachment for proper citation and complete instructions

    A Discussion On Prime Factorization

    Find two numbers that have a product of 81 and also have a sum of 30 (use prime factorization for the product) Please see attachment for the formatted question.

    Polynomial Example Problems

    Ace manufacturing has determined that the Cost of Labor for producing x transmissions is: 0.3x(squared) + 400x+550 dollars While the Cost of Materials is: 0.1x(squared) + 50x+800 dollars · Write a polynomial that represents the total cost of Materilas and Labor for producing x transmissions. · Evaluate the total cost pol

    Properties of complex numbers.

    Given A=(-1/2+isqrt(3)/2)^n 1. Show that A is real for any natural n 2. Show that for n=3K where K is a natural number, A=2

    Complex Numbers Questions

    1. The equation X^5 - 2X^4 - X^3 + 6X - 4 = 0 has a repeated root at X=1 and a root at X-2. By a process of division and solving a quadratic equation, find all the roots and hence write down all the factors of X^5 - 2X^4 - X^3 + 6X - 4 2. Given that cosX= (e^jx + e^-jx)/2

    Taylor Polynomials Using Substitution

    F(x) = ln5 + ln(1-1/5x) Using substitution in one of the standard Taylor series, find the Taylor series about f for 0. Give all terms up to the term in x^3.

    Game Theory : Two-Player Card Game

    In the two-player game of Two Stacks, a deck of cards (with the joker added, for a total of 53 cards) is randomly divided into two piles. The two players take turns removing cards from one pile or the other. On a player's turn, that player may remove any positive number of cards from a single pile. The object of the game is to r

    Synthetic Division, Long Division, Polynomial and Asymptotes

    I am having trouble with the following problems. Can someone please help me and explain the process? P. 295 1. # 16 2. # 28 3. # 36 P.309 4. # 78 5. # 80 6. # 82 P.334 7. #28 8 #44 P.364 9. # 36 10. # 46(Use Grapher only) See the attached file.

    Proof : Polynomials

    Problem: Let f(x) and g(x) be nonzero polynomials in R[x] and assume that the leading coefficient of one of them is a unit. Show that f(x)g(x) doesn't equal 0 and that deg[f(x)g(x)] = deg(f(x)) + deg(g(x))

    Number Theory

    Find all the integers such that when the final digit is deleted the new integer divides the original one. Can you generalize this to deleting other digits?

    Explaining Problem Solving

    A. Find 12/25 divide by 1/5 in two different ways. Explain your methods. b. Explain, using a diagram, how the following problems illustrate the two interpretations of division: partitive division ans measurement division. i. A road crew repaves 1 1/2 miles of road each day. How long will it take the crew to pave a 3/4 mile

    Working with Prime factorization

    Rewrite in the simplest form. State the GCF(greatest common factor) of numerator and denominator in each case. 1. 34/85 2. 123123/567567 Find 5/9+7/12 using three different denominators. Give your answers as mixed numbers in lowest terms. State the LCM (least common multiple)of the denominators.

    Prime number proof

    Prove that if p is a prime number, then p divides , for all n≥p. Here [r] denotes the greatest integer ≤ r , for any real number r. Does this result generalize to a result about instead of p ?

    Kernel equivalence and natural mapping

    For a mapping %:A -> B, let == denote the kernel equivalence of %, and let *:A -> A== denote the natural mapping. Define $:A== -> B by $([a]) = %(a) for every equivalence class [a] in A==. 1. Show that $ is well defined and one-to-one, and that $ is onto if % is onto. Furthermore, show that % = $*, so that % is the composite

    Approximation of Integrals and Taylor Polynomials

    Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square] dx, as x from 0 to 1 ), (a) Verify that ( LIMIT (n/ln(n)) [INTEGRAL (ln square (x)dx) - (ln square (n))] as n approach to the infinity ) = 1 (b) Compute LIMIT ((n square)/

    Polynomial Function Characteristic Examples

    Here is what the problem asks for: Give an example of a polynomial function f of degree 5 such that the only real roots of f(x) are -2,1,6 and f(2)=32. Show that your example works and leave f(x) in factored form.

    Approximation with Taylor Polynomials: Example Problems

    Note:% just a symbol Let I be an open interval containing the point x.(x not), and suppose that the function f:I->R has a continuous third derivative with f'''(x)>0 for all x in I. Prove that if x.+h is in I, there is a unique number % = %(h) in the interval (0,1) such that f(x.+h) = f(x.) + f'(x.)h + f"(x.+%h)(h^2)/2