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: 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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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)?
Given the Taylor polynomial for f(x) at x = 3, find f(³) (3): T₁(x) = 1 + x −3+2(x − 3)² + 2(x-3)³ - 3(x − 3)¹ f(3) (3) = HHH
Find a basis for the space of 2 by 3 matrices whose nullspace contains (2, 1, 1 ).
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
Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other (both are growing larger as more sand is dumped). How fast is the base diameter increasing when the base is 18 feet long? Round your answer to three decimal places. ft/min
Let fe C[a, b] and ro € [a, b]. In this problem, we derive a difference formula to approxi- mate f"(ro). (a) Let f(x) = P3(x) + R(x), where P3(x) is the fourth Taylor Polynomial of f about zo. Find out the expressions of P3(1) and R(æ). (b) Assume that for small h > 0, zo ±h e [a, b]. Compute f(xo + h) + f(zo – h) in terms of the Taylor Polynomial P3 obtained in part (a). (c) Using the result in part (b), derive a difference formula to approximate f"(ro). Find out the leading order of h in the truncation error. (d) Assume that M := max f() (z) is given. Consider the effect of round-off error in the computation. Assume that the round-off error in each term can be bounded by a given number e > 0. Show that the total error of approximating f"(x0) using the derived Pe[a,b] formula satisfies: Mh? |Round off plus Truncation errors| < 12 (e) Find the optimal step size h, (in terms of M and e) such that this minimizes the error bound obtained in part (d).

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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.