Question

Math 163 April 28th.

1) Consider the function ?(?) = ? 2? 3?. Write the first three non-zero terms of the following series, and find a series formula:

a. the Maclaurin series of ?(?).

b. the Taylor series of ?(?) centered at ? = −2.

2) Consider the function ?(?) = ? arctan(3?). Write the first three non-zero terms of the following series, and find a series formula:

a. the Maclaurin series of ?(?).

b. the Taylor series of ?(?) centered at ? = 1/3. (write terms only, a formula is quite difficult)

3) Consider the function ℎ(?) = ? sin(2?). Write the first three non-zero terms of the following series, and find a series formula:

a. the Maclaurin series of ?(?).

b. the Taylor series of ?(?) centered at ? = ?.

4) Consider the function ?(?) = cos(2?).

a. Write the first three non-zero terms of the Maclaurin series, and find a series formula.

b. Evaluate the limit lim?→0 1−cos (2?) 3? 2 without using L’Hospital’s rule

5) Consider the function ?(?) = sin(?). Find the Taylor series formula when centered at ? = ?/3.

6) We know from integral calculus that by using the substitution method:

∫ 2? cos(? 2 ) ?? = sin(? 2 ) + ?

Prove this antiderivative is still correct when using Maclaurin series representations in place of the functions.

7) Evaluate and simplify the series formula as much as possible:

∫ ? ? − 1 ? 2 ??

EXTRA CREDIT (1 point for each problem)

For problems 1 – 5, use desmos (or other graphing software/calculator) to plot the function and any series that are in the problems.

Answer 1