Calculus II, Spring 2022 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 04

1. Section 10.7: 29, 33, 37, 40, 43, 47, 51, 55, 60.

Hint for problem 51: x − 2 = x − 5 + 3 = 3(1 + \frac{x − 5}{3}).

Remark on problem 55: Take the sentence ”converges to sin x” for granted for now. part (c): first six terms means coefficients of 1, x, ⋯, x^{5}.

2. Section 10.7: Find a power series that converges on (1, 3) and diverges otherwise. Do the same for (1, 3], [1, 3) and [1, 3], respectively.

3. Section 10.8: Problems 5, 7, 15, 23, 29, 35.

Remark for problem 5, 7: Taylor polynomial of order n generated by f at a is simply the first n + 1 terms of the Taylor series generated by f at a. See the definition on page 642. Do problem 5 and 7 for n = 3.

Hint for problem 5: Another way of finding the answer (without performing all the

derivatives) is to write x = x − 2 + 2 = 2(1 + \frac{x − 2}{2}) and the formula for geometric series.

Remark for problem 23: We know that f(x) = ∑_{n=0}^{3} b_{n}(x − 2)^{n} for some c_{n}’s (for example, one can conclude this by repeated division by (x − 2)). Nevertheless, it is enough to assume f(x) can be written in this form. Using a similar procedure as in page 6 of Lecture 08 note, one can show that the final answer is the same as f(x) without explicitly knowing the values of c_{n}.

4. Section 10.8: Let

f(x) = \{ \begin{matrix}0, & x = 0 \\ e^{−1/x^{2}}, & x ≠ 0 \end{matrix}

It is known that f^{(n)}(0) = 0 for all n. Verify this for f′(0) and f′′(0) (go through the details).