Taylor and Maclaurin Series

Question 1

What are all x values for which the series \sum_{n=1}^∞\frac{\left(-1\right)^n}{n}\left(x+\frac{3}{2}\right)^n converge?

Accepted Answer

-\frac{5}{2}<x\le-\frac{1}{2}

Question 2

Which function has the same Maclaurin series as its derivative?

Accepted Answer

ex

Question 3

What type of function can be approximated by a Taylor polynomial?

Accepted Answer

Only differentiable functions

Question 4

What is the approximation of the value of sin 1 obtained by using the fifth-degree Taylor polynomial about x=0 for sinx?

Accepted Answer

1-\frac{1}{6}+\frac{1}{120}

Question 5

True or False: Every Taylor series converges to its function for all x.

Accepted Answer

False

Question 6

Mrs. Bolden Trivia: True or False: Mrs. Bolden is a published author.

Accepted Answer

True

Question 7

What is the coefficient of x^2 in the Taylor series for \frac{1}{\left(1+x^2\right)} about x=0?

Accepted Answer

3

Question 8

What is the general form of a Taylor polynomial centered at a?

Accepted Answer

\sum_{ }^{ }\frac{f^n\left(a\right)}{n!}\left(x-a\right)^n

Question 9

Find the Lagrange error bound using the third-degree Taylor polynomial to approximate f(x) on [0,1] if: f^4x=e^{\sin x}

Accepted Answer

0.097

Question 10

Which of the following is the Maclaurin series for sin(x)?

Accepted Answer

\sum_{ }^{ }\frac{\left(-1\right)^nx^{2n+1}}{\left(2n+1\right)!}

Question 11

Let P(x)=3x^2-5x^3+7x^4+3x^5 be the fifth-degree Taylor polynomial for function f about x=0. What is f'''(0)?

Accepted Answer

-30

Question 12

What is the interval of convergence for the power series \sum_{n=1}^∞\frac{\left(x-3\right)^n}{n\cdot2^n} ?

Accepted Answer

1\le x<5

Question 13

The power series \sum_{n=0}^∞a_n\left(x-3\right)^n converges at x=5. Which of the following is true?

Accepted Answer

The series converges at x=2

Question 14

Mrs Bolden Trivia: Where was Mrs. Bolden born and raised?

Accepted Answer

Baton Rouge, Louisiana

Question 15

Given f(x)=ex what is the Maclaurin polynomial of degree 3?

Accepted Answer

1+x+\frac{x^2}{2!}+\frac{x^3}{3!}

Question 16

The Maclaurin series for function f is given by f\left(x\right)=\sum_{n=0}^∞\left(-\frac{x}{4}\right)^nWhat is f(3)?

Accepted Answer

\frac{4}{7}

Question 17

What does the remainder term in a Taylor series measure?

Accepted Answer

The error of approximation

Question 18

What point is a Maclaurin polynomial centered at?

Accepted Answer

0

Question 19

What's the degree-4 Maclaurin approximation for cos(x)?

Accepted Answer

1-\frac{x^2}{2!}+\frac{x^4}{4!}

Question 20

If \sum_{n=0}^∞a_nx^n is a Taylor polynomial that converges to f(x) for all real x, then f'(1)=

Accepted Answer

\sum_{n=0}^∞na_n^{ }

Questions 20