nf(k) (c),Pn(x) = Σ -(x - c)k.k!k=0Show that for any other polynomial Qn # Pn of order n it holdsf(x) - Pn(x)limx →C : f(x) - Qn(x)= = 0.Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of fvia a polynomial of order n.

Answered: Image /qna-images/answer/0615d50f-8e44-4124-aca2-8620a7deef58.jpg

Answered: n f(k) (c), Pn(x) = Σ -(x - c)k. k! k=0…

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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Find the Taylor polynomial T3(x)
Let f(x) = 2xc + 1 C + 3x --9 8 Oc==3 4 Oc=21/20 Oc=1 Oc==3 Determine the value of c so that x = is a vertical asymptote of f(x).
The resistivity p of a given metal depends on the temperature t according to the equation p(t) = P20(-20) where a and p20 are constants for a given metal Find a third-degree Taylor polynomial at t = 20 that approximates the resistivity function. ¨p(t)=p20|1+ a(t−20)+ — a(t−201²+alt-201³] Ob. p(t)=p20[1+ alt-20)+ a²(t-20)²+a³(t-20)³] O.C. Oa. ¯p(t)=p20|1+a(t−20)+ — a²(t-201²+ = -a³(r-201³] ○ d. p(t) = P200³ (t-201³ O e. 1+a(t−20)+ 1⁄2a²(t−20)² + — a³(t−20)³] o(t) = P20[1 + alt a
Suppose that g(x) is a function that is continuous for all real values of x. Let's say that you want to compute the third-degree Taylor polynomial T3(x) centered at 8 for g(x), and you know that g(8) = 2, g' (8) = -5, g "(8) = 18, and g "(8) = 48. (In case you are having trouble reading the notation, I just gave you the value of g at 8, followed by the values of the first, second, and third derivatives of g at 8, in that order.) What is T3(x)?
Consider three real numbers a < c < b and a function f : (a, b) → R such that the (n + 1)-th derivative f(n+1) exists on (a,b) and is continuous. Let Pn denote the Taylor polynomial of order n of f around c i.e. Would you mind showing me the detail, please. Thank you so much
n f(k) (c), Pn(x) = Σ -(x - c)k. k! k=0 Show that for any other polynomial Qn # Pn of order n it holds f(x) - Pn(x) lim x →C : f(x) - Qn(x) = = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.

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n f(k) (c), Pn(x) = Σ -(x - c)k. k! k=0 Show that for any other polynomial Qn # Pn of order n it holds f(x) - Pn(x) lim x →C : f(x) - Qn(x) = = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.