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

Find the first three non-Zero terms in the Taylor series
centered at 0 for the function.
f(x)=sin(3x)

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

Find a power series representation for the function:
?(?)=?^(2)arctan(?3)
?(?)=(?/(2−?))^3
Express the antiderivative as a power series;
∫?/(1+t^(3) ??
∫arctan(?)/(?)??

1. Consider the function f(x) = 2x^2 - 7x + 9
a) Find the second-degree Taylor series for f(x) centered at x =
0. Show all work.
b) Find the second-degree Taylor series for f(x) centered at x =
1. Write it as a power series centered around x = 1, and then
distribute all terms. What do you notice?

Hello,
Find the first three nonzero terms of the Maclaurin Series for
each function and the values of x for which the series converges
absolutely.
(cos(x))log(1+x)

A) Find the first 4 nonzero terms of the Taylor series for the
given function centered at a = pi/2
B) Write the power series using summation notation
f(x) = sinx

Calculus, Taylor series Consider the function f(x) = sin(x) x .
1. Compute limx→0 f(x) using l’Hˆopital’s rule. 2. Use Taylor’s
remainder theorem to get the same result: (a) Write down P1(x), the
first-order Taylor polynomial for sin(x) centered at a = 0. (b)
Write down an upper bound on the absolute value of the remainder
R1(x) = sin(x) − P1(x), using your knowledge about the derivatives
of sin(x). (c) Express f(x) as f(x) = P1(x) x + R1(x) x...

Question #11
Find:
The first 3 nonzero terms a Taylor series (at a = 0) of f1(x) =
sin x, f2(x) = x cos x, f3(x) = x/(2 + 2x^2 ), and f4(x) = x −
(x^3/2) − x^5
No explanation is necessary for finding: sin x, cos x, and 1/(1
− x).)

4. Consider the function
?(?) = ? + 1 ? − 2
Defined for ? > 0. First, show that this function has one
global minimum at ? = 1.
Then, take the Taylor series expansion about the point ? = 1,
writing ? = 1 + ?. Compute the terms out to ? 3 .

1. Find T5(x): Taylor polynomial of degree 5 of the function
f(x)=cos(x) at a=0.
T5(x)=
Using the Taylor Remainder Theorem, find all values of x
for which this approximation is within 0.00054 of the right answer.
Assume for simplicity that we limit ourselves to |x|≤1.
|x|≤ =
2. Use the appropriate substitutions to write down the first
four nonzero terms of the Maclaurin series for the binomial:
(1+7x)^1/4
The first nonzero term is:
The second nonzero term is:
The third...

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