University:
San Diego State University
Course:
MATH 151 | Calculus II
Academic year:

2023
Views:

254

Pages:

1
Author:

cicwyr4xc

Workshop #28 1. Consider the function f (x) = sin x. (a) Find the first four non-zero terms of the Taylor series for f (x) centered at a = π2 . (b) Write the Taylor series for f (x) centered at a = π 2 in summation notation. (c) Let T4 (x) be the polynomial we get if we remove all terms with degree higher than 4 in the Taylor series. Graph T4 (x) and f (x) on your graphing calculator. For what values of x would you say T4 (x) is a good estimate of f (x)? (d) Now graph T6 (x) on the same set of axes as f (x). For what values of x would you say T6 (x) is a good estimate of f (x)? 2. Consider the function f (x) = ln x. (a) Find the first five non-zero terms of the Taylor series for f (x) centered at a = 1. (b) Write the Taylor series for f (x) centered at a = 1 in summation notation. (c) Use T2 (x) to estimate ln 1.1. (d) Plug ln(1.1) into your calculator. To how many decimal places was your estimate correct? (e) If you want your estimate of ln(1.1) to be correct up to 6 decimal places, how many terms of the Taylor series would you need? 3. (a) Find the degree three Taylor polynomial for f (x) = √ (b) Use this to estimate 3 0.5. 4. Find the area of the region bounded by y = √ 3 x, centered at a = 1. √ x − 1 and y = x − 1. 5. Determine whether the following integrals converge or diverge. If they converge, compute them. Z ∞ 1 dx (a) (2x + 1)3 1 Z 1 1 dx (b) 1 (2x + 1)3 −2 Z −1 2 1 (c) dx 3 −1 (2x + 1) Z −1 1 (d) dx 3 −∞ (2x + 1)