It is known that the Maclaurin series of function f(x) = tan x is tan x = x + x³ +²25 +.... Then the fifth derivative of the function g(x) = (1 + x²) tan x at 0 is g(5)(0) = O a) 84 b) 64 O c) 72 O d) 96 O e) 56 O f) 48

Q: (-1)^x2" (2n)! for all x €R. cos x = Let f(x) = cos(v). Determine the power series representation…

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Q: n as f • 27 (²) 2n n = 1 Evaluate | where f(x) => x" by identifying it as the value of a derivative…

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Q: Which of the following information is correct for the sequence 3 7 dx 1 dx 1 dx dx {an}n=l X (a) The…

A: Given: The given sequence is ann=1∞=13∫13dxx,15∫15dxx,17∫17dxx,19∫19dxx.....

Q: Evaluate fr.co x cos(x³) dx as an infinite series.

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Q: Example 3:- Show that the trigonometric series of f (x) = 3x for x € (-n, 1) given by 00 (-1)" sin…

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Q: What's the numerical approximation to the integral c COS Type your answer... √xdx? Use trapezoidal…

A: Trapezoidal rule

Q: ind the hal f CoSine range Seriesfor in the Intervat Hence that the function (X-H 32 1=1 (2n-1)2 12…

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Q: 3. Determine the half-range sine series for the function defined by: x. 0<x< f) =

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Q: 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e2x, a = 0 b)…

A: To find the Taylor polynomials of orders 0, 1, 2, and 3 generated by .

Q: (b) Prove that the Fourier series for f(x) is: 1 f(x) = ²0 (si -20 (sinx + sin3x +sin5x + ...) TU

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Q: 4. Find (a) the Fourier cosine series, (b) the Fourier sine series. Sketch f(x) and its two periodic…

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Q: A Fourier series is an infinite series given by Cn sin(nx) n=1 where Cn is called the coefficients…

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Q: Use series to evaluate lim 3x3 (e -3/x' - 1) X00 lim 3x° (e -3/x -1) =D X00 (Simplify your answer.)

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Q: Question 4. Let f be the function defined by the series f(z) = E for all z such that -1 < z < 1.…

A: Let , f(x) = ∑n=1∞xnn  for all x such that , -1 &lt; x &lt; 1. Here , an = xnn We know that , Ratio…

Q: Find the Fourier sine series for f(x) = on 2)=7 On - -L≤ x ≤L.

A: f(x) = x on –L ≤ x ≤ LFind the Fourier sine series for f(x).

Q: 6. Find the sine series of Sx, f(x) 0 <x < #/2 - x, T/2 < x < T

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Q: Show that the Fourier series of frx)= x² in El, 1] is x2 (-19^ 4 CosnTx, and hence show that n2 n=}…

A: Given function is fx=x2 in -1,1. We have to show that the Fourier series of fx=x2 in -1,1 is…

Q: 9. If a, =1.57, a, -(-1) -11. b, --)" f(t) and period f(t) =f(t+4) up to n=2 is then the Fourier…

A: Fourier series of the function

Q: (8) a) Using the Maclaurin series for f(x )= cos x find the first four nonzero terms of cos( x' ).…

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Q: Siven the following: 00 5(-1)"x2n '(-1)"x²n cos(x) = for all x E R (2n)! n=0 ) Find a power series…

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Q: e graph of the

A: Here, in part b we have to sketch the graph of the function to which fourier sine series converges…

Q: Question 4. Let f be the function defined by the series f(r) = for all z such that -1 < r < 1. Which…

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Q: a) With explanation, use the already-known Maclaurin series for e“ to obtain the Maclau- rin series…

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Q: Find the coefficients of the Fourier cosine series for f(x).

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Q: 20. Using = 00 1-x n=0 a) Find a power series for c) Find a power series for 1 1+x¹ - 4x³ (1+x¹) ²…

A: The given problem is to find the power series of the given functions with using the given power…

Q: 1) Solve for y" + (cos x)y = 0, given that cos x can be represented 1 2! + + 4! as a series of .…

A: Solution : Given, y"+cos(x)y=0 Let the expanded form of y as, y=∑n=0∞Cnxn By differentiating…

Q: b) The following function f(x) is periodic with period T = 27. Sketch the function over -47≤x≤ 47…

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Q: Prove that 2e 5 2 sin (Tx/4)+1/2 dx ≤ 2e-1 1+2√2 (i) Show that zloga 7/4 dx - (log) (log x)-3/4 (ii)…

A: All the parts are solved properly with detailed explanation.Hope you got your answer. If any query…