A- The goal of this part is to find an approximate value of In(1.1) by using the Taylor expansion. We consider the function f(t) = In(1 + t). Show by induction on n that, the nth derivative of f is given by: /() = (-1)^-1 (n – 1)! (1+t)" Write the nh Taylor expansion of f at a=0 with the remainder R.. Show that max f+)(t)| = n! tel0,0.1] 10-(+1) when r 0.1. Deduce that |RIS Deduce how many terms in Taylor expansion of f at a =0 do we need to approximate f(0.1) In(1.1) within 6 decimale places. The goal of this part is the calculation of limits of functions by using the Taylor expansion. By atilizing the MacLaurin-Young formula of order 2 of cos z and sin r determine the limits n+1 I sinz - lim -40 2-2 cos z COs z e- lim Z-40

Q: 3. Find the 5th Taylor polynomial p5 for f(x) = cos x at xo = 0. What is the error of this…

A: Given,                 f(x)=cos(x), at x0=0. To find the 5th Taylor polynomial p5  for f(x)…

Q: Find the Fourier series expansion of the function x+² in [-7, 7].

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Q: K Use the Taylor polynomial of degree four for the function f(x) = 5x below at x=0 to approximate…

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Q: 2. Let f(8) = 202, 0<8 <5; f(e) = f(0 + 5). a) Find the Fourier coefficients; and b) hence writef(0)…

A: Given function is f(θ)=2θ2, 0&lt;θ&lt;5;  f(θ)=f(θ+5) Here, period is T=5 Since,Fourier series of…

Q: Consider the function f : [-, π] → R : x → 1 + 2x + sin(2x) and extend it periodically to all of R.…

A: As per the question we are given the following functions : f:[−π,π]→R,f(x)=1+2x+sin⁡(2x)And we have…

Q: Let f(x) = x – n, -2n < x < 2n; f(x) = f(x + 4n). a) Sketch f(x) for -2n < x < 67. b) Find the…

A: The following formulae are used to find the Fourier coefficients of the function f(x) on the…

Q: Find the Taylor polynomial of order 3 generated by fat a. f(x)= In (x + 1), a=5 O A. P3(x) = In 4 –…

A: To find the Taylor polynomial for given function.

Q: rom the Fourier expansion of f(x) = {(1– x)? 0<x<1 t

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Q: (3x2-2xy+3y2)dx = 4xydy Find the general solution by determining the type of differential equation.

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Q: Consider the function: f(x) = ln(r) Determine the quadratic Taylor Polynomial about the point zo =…

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Q: Use the fact that e0.3 <e <3 and 1

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Q: If the first five terms of the Taylor expansion for f(x) about x = 0 are 2-3x + === x² + 7 then…

A: we are given the Taylor series expansion for a function f(x) and we need to identify f'''(0)

Q: Let C z-1=3 and1 = §, = dz. Then

A: The solution is given as

Q: a) Which of the following is the correct definition for the nth Taylor Polynomial for the function f…

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Q: 47. Let Tn be the Taylor polynomial for f(x) = In x at a = 1, and let c > 1. Show that |c – 1|a+1 n…

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Q: Use a Taylor polynomial centered at x = 0 to estimate In(1.20) to within 0.01. a) 0.2 b) 00.18 c)…

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Q: 1) Find the n"-order Taylor polynomials of the given function centered at the given point for n = 2…

A: Solve the following

Q: If you use the second Taylor polynomial of f(x)=√x centered at x = 1 to estimate √1.1 as √1.1=1+…

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Q: Find the n-th Taylor polynomials for 1 about x = 3 and express it in f(x) = x+ 2 sigma notation.

A: Taylor polynomial of nth degree of f(x) centered at 'a' :          f(x) = f(a)…

Q: 1. (a) Use Taylor's theorem to find the fourth degree expansion of In r about x = 1. (b) Using the…

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Q: Given the function f(x) = 3x² - 2x on the interval -2<* ≤2 (a) Find the Fourier series expansion (b)…

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Q: Let f(2)= Calculate the Taylor expansion of f about the point z = 4i, in the region |z-4i|< 8. Write…

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Q: ii. Compute f (x) = In(1 + 2x) at xị+1 = 1.1 by first, second, third, and fourth degree Taylor…

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Q: Determine the Taylor series expansion about a = 1 of the function defined by f (x) = log3(2 – 1).

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Q: sin (7(N+t))=(-1)"|1-(2x-1) for 0s1s1

A: fx=sinπN+t=-1N1-2x-12 ,0≤t≤1,N∈I Sine function is periodic with period of 2π.…

Q: 1. Consider the function f(x) = -1 0 1 f(x + 4) -2 <x<-1 -1 ≤ x ≤ 1 1 < x≤2

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Q: a) y tan-¹x b) y = -x Find the MACLAURIN EXPANSION of the 2 functions above. (Use at least 5 terms)

A: Maclaurin series of f is given by, ∑k=0∞fk(0)k!xk =f(0) +f'(0)x + f''(0)2!x2 + . . . + fn(0)n!xn + .…

Q: a. Using MATLAB, find the fourier series of [t(n²-t²)]/12 - π <t<π

A: a) [t(?^2-t^2)]/12     -?&lt;t&lt;? Here is a MATLAB code to find the Fourier series of the function…

Q: f(x) = {0

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Q: Write out the Taylor Polynomial about a = 1 of degree 4 and corresponding remainder for f(x) =…

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Q: Let f(8) = 202, 0<0 < 5 ; f(8) = f(8 + 5). %3D a) Find the Fourier coefficients; and b) hence…

A: Given function is           f(θ)=2θ2, 0&lt;θ&lt;52l=5,   l=52 Also…

Q: Given the periodic function over the period [-π/2, π/2] f(x) = x|x|

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