Question 1

Which of the following is the general formula for the Taylor polynomial of degree $$n$$ for a function $$f(x)$$ centered at $$x = a$$?

Accepted Answer

$$p_n(x) = \sum_{k=0}^{n} \frac{f$$^{(k)}$$(a)}{k!} (x-a)^k$$

Question 2

Suppose $$f(x) is an $$even function. What can you say about the Maclaurin polynomial for $$f(x)$$?

Accepted Answer

It contains only even powers of $$x$$

Question 3

Given $$f(x) = e^x$$, what is the Maclaurin polynomial of degree 3?

Accepted Answer

$$p_3(x) = 1 + x$$ + \frac{$$x^2$$}{2} + \frac{$$x^3$$}{6}$$

Question 4

Which statement about the remainder term $$R_n(x) in $$Taylor's Theorem is correct?

Accepted Answer

$$R_n(x) = \frac{f$$^{(n+1)}$$(c)}{(n+1)!} (x-a$$)^{n+1}$$ for some c$ between a$ and x$

Question 5

If $$f(x) = \sin(x)$$, what is the Maclaurin polynomial of degree 5?

Accepted Answer

$$p_5(x) = x$$ - \frac{$$x^3$$}{6} + \frac{$$x^5$$}{120}$$

Question 6

A medical researcher wants to approximate $$\cos(0.1)$$ using a Maclaurin polynomial. Which is the best polynomial to use for a good approximation?

Accepted Answer

$$p_2(x) = 1$$ - \frac{$$x^2$$}{2}$$

Question 7

Which of the following is the Taylor polynomial of degree 2 for $$f(x) = e^x$$ centered at $$x = 2$$?

Accepted Answer

$$p_2(x$$) = $$e^2$$ + $$e^2(x-2$$) + \frac{$$e^2$$}{2}$$(x-2)^2$$

Question 8

A chemist is modeling the concentration of a reactant using $$f(x) = e$$^{-x}$$. What is the Maclaurin polynomial of degree 4 for f(x$$)$$?

Accepted Answer

$$p_4(x) = 1 - x$$ + \frac{$$x^2$$}{2} - \frac{$$x^3$$}{6} + \frac{$$x^4$$}{24}$$

Question 9

Which of the following is true about the Maclaurin polynomial for $$\sinh(x)$$?

Accepted Answer

It contains only odd powers of $$x$$

Question 10

A physicist wants to approximate $$\sin(\frac{\pi}{6})$$ using a Taylor polynomial of degree 4 centered at $$x = \frac{\pi}{4}. $$Which of the following is the correct polynomial?

Accepted Answer

$$p_4(x$$) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}(x-\frac{\pi}{4}) - \frac{1}{2\sqrt{2}}(x-\frac{\pi}{4}$$)^2$$ - \frac{1}{2\sqrt{2}}(x-\frac{\pi}{4}$$)^3$$ + \frac{1}{8\sqrt{2}}(x-\frac{\pi}{4}$$)^4$$

Question 11

Which of the following is the Maclaurin polynomial of degree 4 for $$\cos(x)$$?

Accepted Answer

$$p_4(x) = 1$$ - \frac{$$x^2$$}{2} + \frac{$$x^4$$}{24}$$

Question 12

A biologist is studying population growth modeled by $$f(x) = e$$^{2x}$$. What is the Maclaurin polynomial of degree 3 for f(x$$)$$?

Accepted Answer

$$p_3(x) = 1 + 2x$$ + $$2x^2$$ + \frac{4}{3}$$x^3$$

Question 13

Which of the following is the Taylor polynomial of degree 2 for $$f(x) = \ln(x)$$ centered at $$x = 1$$?

Accepted Answer

$$p_2(x) = (x-1$$) - \frac{1}{2}$$(x-1)^2$$

Question 14

A mathematician wants to approximate $$\sin(0.2)$$ using a Maclaurin polynomial. Which is the best polynomial to use for a good approximation?

Accepted Answer

$$p_3(x) = x$$ - \frac{$$x^3$$}{6}$$

Question 15

Which of the following is the Maclaurin polynomial of degree 4 for $$\cosh(x)$$?

Accepted Answer

$$p_4(x) = 1$$ + \frac{$$x^2$$}{2} + \frac{$$x^4$$}{24}$$

Question 16

A student is asked to use Taylor's Theorem to estimate the error when approximating $$e^x by $$its Maclaurin polynomial of degree 2 at $$x = 0.1. $$Which expression gives the error bound?

Accepted Answer

$$|R_2$(0.1)| \leq \frac{e$$^{c}$$}{6} (0.1)^3$ for some c$ between 0$ and 0.1$

Question 17

Which of the following is the Maclaurin polynomial of degree 3 for $$f(x) = \sin(x)$$?

Accepted Answer

$$p_3(x) = x$$ - \frac{$$x^3$$}{6}$$

Question 18

A computer scientist is using Newton's method to find a root of $$f(x) = x^2$ - 2. $$What is the iterative formula for Newton's method in this case?

Accepted Answer

$$x_{n+1} = x_n - \frac{x_n^2$ - 2}{2x_n}$$

Taylor and Maclaurin Polynomials and Series: Theory, Examples, and Applications

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