(1) Find the Taylor series about 0 for the following functions: (a) ce (b) sin(x), (c)...
(2) Show that sin(x) is the sum of its Taylor series. (3) Find the first three nonzero terms of the Taylor series about 0 for the following functions (a) cos(x2) (b) e (c) tan(x)
2. Find the Taylor series about x = 0 for x ^ 2 * cos(x ^ 2) . Also, find an expression for the general term of the series if the index starts with k = 0 0. (Hint: First find the Taylor series for cos x ^ 2 2. Find the Taylor series about x = 0 for x?cos(x?). Also, find an expression for the general term of the series if the index starts with k = 0. (Hint:...
Compute the first three non-zero terms of the Taylor series for the functions: Q.1 [10 Marks] Compute the first three non-zero terms of the Taylor series for the functions: (a) (i) f(x)-In( 1 ) about a-0 where Ir < 1 (Hint: In(it)-In(1+z)-In(1-r)) (ii) From your result in (i) find ËIn(쁩) dt Page: 1 of3 MAT1841 Assignment 2 2019 Continuous Mathematics for Computer Science 3 +3+4-10 (c) h(z) = exp (sin r) about a = 픔 Q.1 [10 Marks] Compute the...
For each of the following functions indicate the matching Taylor Series centered at r=0. 1) sin(2) 2) cos(2) 3) 4) e 5) 1.2 6) D 7) 12:22 8) - In(1 - 1) 9) e--- 10) S* cos(t)dt Taylor Series Choices: a) § 3 b) (-1)=-17 c) Š(-1)" N=0 no NEO d) nr-1 e) Σα" f) 2.2 no n=0 g) 2nx2n-2 h) (-1)" (an+1)+(2n) 4+1 i) (-1)n-1 nel n=0 n=0 j) (-1)" (2n+1)! 2+1 k) § 21 k) 2ne2n-1 1) (-1)"?"...
2. (New ways to find Taylor series) It's not always easy to write down Taylor series representations by computing all the successive derivatives of a function as follows. (a) Find, by evaluating derivatives at 0, the first three nonzero terms in the Taylor series about 0 for the function g(x) -sin a2 in the text or class such as e", sin , and cos a (b) Use Taylor series expansions already es to find an infinite series representation expansion for...
-a" (a) Find the Taylor series for sinx about x 0, and prove that it converges to sinx uniformly on any bounded interval [-N,N (b) Find the Taylor expansion of sinx about xt/6. Hence show how to annrmximate D. -a" (a) Find the Taylor series for sinx about x 0, and prove that it converges to sinx uniformly on any bounded interval [-N,N (b) Find the Taylor expansion of sinx about xt/6. Hence show how to annrmximate D.
Use substitution to find the Taylor series at x = 0 of the function 16 sin(-x). What is the general expression for the nth term in the Taylor series at x = 0 for 16 sin (-x)? 00 (Type an exact answer.) no
1. (Taylor Polynomial for cos(ax)) For f(x)cos(ar) do the following. (a) Find the Taylor polynomials T(x) about 0 for f(x) for n 1,2,3,4,5 (b) Based on the pattern in part (a), if n is an even number what is the relation between Tn (x) and TR+1()? (c) You might want to approximate cos(az) for all in 0 xS /2 by a Taylor polynomial about 0. Use the Taylor polynomial of order 3 to approximate f(0.25) when a -2, i.e. f(x)...
need answer for #2, which is about arctan #1 Suppose we want to estimate sin (0.1) using the Taylor series of sin x at x = 0. (a) Find roughly a number of nonzero terms which are required to compute sin(0.1) to n decimals (you do not need to find the minimum number.) (b) Find such a number more accurately, i.e. find a smaller one, using any mathematical tech niques. You may want to use the fact related to Stirling's...
2 1. The Taylor series for a function f about x =0 is given by k=1 Ikitt (a) Find f(")(). Show the work that leads to your answer. (b) Use the ratio test to find the radius of convergence of the Taylor series for f about x=0. c) Find the interval of convergence of the Taylor series of f. (a) Use the second-degree Taylor polynomial for f about x = 0 to approximate s(4)