MAT 302 Assignmrnent Problem 9.7

MAT 302 Assignmrnent Problem 9.7 | Borough of Manhattan Community College

9.7 Taylor Polynomials And Approximations

Question 1

Compute T2(x)T2(x) at x=1x=1 for y=exy=ex and use a calculator to compute the error |ex−T2(x)||ex−T2(x)| at x=1.4x=1.4.

Question 2

Calculate the Taylor polynomials T2(x)T2(x) and T3(x)T3(x) centered at x=πx=π for f(x)=sin(x)f(x)=sin(x).

T2(x)T2(x) must be of the form

A+B(x−π)+C(x−π)2

Question 3

(1 point) Write the Taylor polynomial T5(x)T5(x) for the function f(x)=cos(x)f(x)=cos(x) centered at x=0.

Question 4

(1 point) What is the minimal degree Taylor polynomial about x=0x=0 that you need to calculate cos(1)cos⁡(1) to 3 decimal places?

degree =  6

To 6 decimal places?

degree =  8

Question 5

(1 point) The function f(x)f(x) is approximated near x=0x=0 by the second degree Taylor polynomial P2(x)=7x−1−2x2P2(x)=7x−1−2x2.

Give values:

Question 6

(1 point) Let T2(x)T2(x) be the Taylor polynomial of degree 22 for the function f(x)=cos(x)f(x)=cos⁡(x) at a=0a=0. Suppose you approximate f(x)f(x) by T2(x)T2(x). If |x|≤1|x|≤1, what is the bound for your error of your estimate?

Hint: use the alternating series approximation.

Question 7

(1 point) Use an appropriate local quadratic approximation to approximate tan59.2∘tan⁡59.2∘, and compare the result to that produced directly by your calculating utility.

Enter the local quadratic approximation of tan59.2∘tan⁡59.2∘.

Question 8

(1 point) Find the Taylor polynomial of degree 3 for sin(x)sin⁡(x), for xx near 0:

P3(x)= 

Approximate sin(x)sin⁡(x) with P3(x)P3(x) to simplify the ratio:

Using this, conclude the limit: