DTaylor series for the function f(x)=5cosx, at x = is:Taylor Series. (k) (a)00Σk=0 k!..... +OB.Oc-(x-a)=f(a) + f'(a) (x-a) +OA^f(x)=5+ ²(x-x)²52n!OD.:-5+(x-a)" +....f(x)=-5---(x-x)²f(x)=5-Moving to the next question prevents changes to this answer.f" (a)2!(x-a) ² +flll(a)3!(x-a) ³+Question 4 of 5 >

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Answered: Taylor series for the function…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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TAYLOR SERIES Approximate f(7) of the function f(x) = sin(x) centered at 0, accurate to 3 significant digits. Round-off your answers to 4 decimal places.
1. (a) Find the Fourier series of the function (x) = 1 - |x), - 2 < x< 2 (b) Hence prove that 1 + 32 52 +..... 72 8.
11. (a) Find the Fourier series of the 27-periodic function given on the interval -<< π by f(x) = sinx. (b) Plot several partial sums to illustrate the convergence of the Fourier series.
= f(x) = { 2. Let 0 -2
e(x-3)* - 1 (x-3)² 1) The continuous function fis defined by f(x) = of all orders at x = 3. = 3. (a) Write the first four nonzero terms and the general term of the Taylor series for g(x) = e(x-3)* about x = (Hint: Use the Maclaurin series for e.) for x3 and f(3) = 0. The function fhas derivatives (b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for fabout x = 3. (c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b). (d) Use the Taylor series for fabout x = 3 to determine whether the graph of fhas any points of inflection.
5. (a) Find the Fourier Series of the function (x) = 0 if -n < x < 0 (x) = nx- x if 0 < x < n. (b) Use the series obtained in part (a) for an appropriate x to show 1+ 22 32 42 6
1 g(x) = (7 +x)3 g(x) = 2 5 „n (-1)"(n+1)(n + 2) 2.(7)"+3 n=0 Part 3 Use your answers above to now express the function as a power series (centered at x = 0). x2 h(x) = (7 + x)3 h(x) = n=0
f(x)= x²/1+x³, power series representation is?
The function f is defined by the power series (-1)" x²n f(x) = Σ n=0 (2n+1)! for all real numbers x. Use the first and second derivative test by finding f'(x) and f"(x). Determine whether f has a local maximum, a local minimum, or neither at x=0. x4 1+1-1 5! = 1 −²+ 3! 7! +. + (-1)"x²n (2n+1)!
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Taylor series for the function f(x)=5cosx, at x= is: Taylor Series (k) (a) 8 Σ k=0 ..... + OA OB. Oc - (x − a) * =ƒ (a) +ƒ'(a) (x − a) + OD. f(n) (a) n! )=5+ f(x) = -5+ = (x-x)² 5 -5- f(x)=5- (x-a)" +.... f"l(a) 2! (x-a) ² + flll (a) 3! (x-a) ³+