) what is the nt Taylor polynomial of a function •po tlx) at x= c? 2) Find the thind Taylor polynomial Pe(x) f f(xl > cos x at X=r. 3) using 4) stale Taylor's theorem, giving B(x) above, estimate cos (r-.i). Taylor's Theorem, show that the emror in 5) Using your eshimele Using the explicit formula fn the ertor. your estima te in 3) for cos(r - 1) is less thaa . 0001.

Q: Consider the function f (x) = x + cos x a) Find the x-value(s) (if any) for which f(x) has local…

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Q: Exercise 2: Let f(x) = sec2(x), and g(x) = tan2(x). Prove f'(x) = g' (x). Thus, find a C E R such…

A: Given,

Q: Consider the function f(r) whose second derivative is f(z)- 8z+10 sin(r). If/@)- 4 and f'0)-4, what…

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Q: Use mathematical induction to prove that cos(x+n) = (-1)" cosx where n≥le J*.

A: Mathematical induction is defined as a mathematical technique that is used for the expression,…

Q: ■ Prove that the function x) = { win = ²+ = = 0 x X 0, x f(x) satisfies the intermediate value…

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Q: 8) Consider the function f(z) given by f(2)= 24-2 cos 20z² + 1. (a) Show that f(z) has zeros at eie,…

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Q: Discuss the difference between harmonic and conjugate harmonic function. Also find the analytic…

A: The given problem is to discuss the difference between harmonic and conjugate harmonic function and…

Q: 12. Let be the function given by f(x) = cos 3x+ and let P(x) be the fourth-degree Taylor 6…

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Q: Determine the pole ac the residue at A(z) = Ite² and ach pole Sinz +ZcOSZ

A: Step 1: Step 2: Step 3: Step 4:

Q: 2) Consider the function f(x) = -x + sin(-x) on the interval [0,1]. a) Approximate a root of f using…

A: Given function is              rewrite the above equation as               compare this with…

Q: Let Tn(x) denote the Chebyshev polynomial ofdegree n and defineUn−1(x) = 1/nT'n(x)for n = 1, 2, . .…

A: Definition:

Q: a) The function f: x 1→ a +b cos x, is defined for 0 <x< 2n. Given that f(0) = 10 and that f ( n)=1,…

A: Hera function is given and we need to find the value of a b and the range of function

Q: k ak ak a-1,a +1 (-1)* 1/½(-1)*+1 kak 4), a +1 a- (ak + b)(n) (ak+b)(n+1) a(n+1)

A: ∆ak= ak+1-ak      = ak(a-1) therefore anti difference of ak=ak/(a-1)

Q: What is the range of f(r) = cos(x)? i) [–1, 1] ii) [0, 1] iii) (-∞, ∞) O i O i

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Q: Q1:/Verify that the given function is harmonic and find it is harmonic conjugates so that f(z) = u +…

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Q: 2. Prove u(x, y) = e cosx is harmonic, then find its harmonic conjugate and construct the most…

A: (.)   Given ,                 u(x,y) = eycosx (.)  A real valued function  f(x,y)  is harmonic  if…

Q: 6. Find the Taylor polynomial T,(x) for the function f(x)= cosx at a="/ it to estimate cos(29780)

A:

Q: sin 7Z „Find the residue of f (z)at z = 4 (z – 4)? Let f (z) =

A: Note:

Q: 1. L{sc" sin wt} L{e* (acost+6 sin t)} -kt cosh 2t +-sinh 2t 2 3. L 2.

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Q: Q1:/Verify that the given function is harmonic and find it is harmonic conjugates so that f(z) = u +…

A: Harmonic conjugate of the function so that function is analytic

Q: (b) Let n, m be positive integers. Define a function (2zm + 3)"e-3z sin?(z) h(z) = zm+2n cos3 (z)…

A: We will use definition of argument principle

Q: Sotve caleulate the thind ordere Tayloru polynemial about (o,0)

A: