Question 1

Which of the following best describes the Taylor series of a function $$f(x)$$ centered at $$x=a$$?

Accepted Answer

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

Question 2

Suppose $$f(x) = e^x. $$What is the third-degree Taylor polynomial for $$f(x)$$ centered at $$x=0$$?

Accepted Answer

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

Question 3

Which of the following is the Taylor series for $$\sin(x)$$ centered at $$x=0$$?

Accepted Answer

$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x$$^{2n+1}$$

Question 4

A medical device uses the Taylor polynomial of $$\ln(1+x) at$$ $$x=0 to $$approximate values for small $$x. $$What is the second-degree Taylor polynomial for $$\ln(1+x)$$?

Accepted Answer

$$x - \frac{x^2$}{2}$$

Question 5

Which of the following statements about Taylor series is TRUE?

Accepted Answer

A Taylor series may not converge to the original function for all $$x.$$

Question 6

A chemist uses the Taylor expansion of $$e^x to $$estimate $$e^{0.1}. $$Using the second-degree Taylor polynomial centered at $$x=0$$, what is the approximation?

Accepted Answer

$$1 + 0.1 + \frac{(0.1)^2$}{2}$$

Question 7

Which of the following is the Maclaurin series for $$\cos(x)$$?

Accepted Answer

$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x$$^{2n}$$

Question 8

A physicist models a pendulum using the Taylor expansion of $$\sin(x)$$ for small angles. What is the first nonzero term in the Taylor series for $$\sin(x) at$$ $$x=0$$?

Accepted Answer

$$x$$

Question 9

Which of the following is the remainder (error) term $$R_n(x) in $$Taylor's theorem for a function $$f(x)$$ centered at $$x=a$$?

Accepted Answer

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

Question 10

Suppose $$f(x) = \arctan(x). $$What is the general term of its Maclaurin series?

Accepted Answer

$$(-1)^n \frac{x$$^{2n+1}$$}{2n+1}$$

Question 11

A biologist uses the Taylor series for $$\ln(1+x) to $$estimate $$\ln(1.05). $$Using the third-degree Taylor polynomial centered at $$x=0$$, what is the approximation?

Accepted Answer

$$0.05 - \frac{(0.05)^2$}{2} + \frac{(0.05)^3$}{3}$$

Question 12

Which of the following functions is equal to its Taylor series centered at $$x=0$$ for all real $$x$$?

Accepted Answer

$$e^x$$

Question 13

A Taylor series is used to approximate a function near a point. Which of the following best explains why higher-degree Taylor polynomials provide better approximations near the center?

Accepted Answer

They include more terms from the function's derivatives, capturing more local behavior.

Question 14

Suppose $$f(x) = \frac{1}{1-x}. $$What is the radius of convergence of its Maclaurin series?

Accepted Answer

1

Question 15

A Taylor series is used to model the spread of a disease by approximating $$e^{kt}$$ for small $$t. If$$ $$k=0.2$$ and $$t=0.1$$, what is the second-degree Taylor polynomial approximation for $$e^{kt} at$$ $$t=0$$?

Accepted Answer

$$1 + 0.2 	imes 0.1 + \frac{(0.2 	imes 0.1)^2$}{2}$$

Question 16

Which of the following is NOT a necessary condition for a function $$f(x) to $$have a Taylor series centered at $$x=a$$?

Accepted Answer

$$f(x)$$ must be periodic.

Taylor Series: Concepts and Applications

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