Question 1

Which of the following is the linear approximation (Taylor polynomial of degree 1) for $$f(x) = 	an x$$ centered at $$a = \pi$$?

Accepted Answer

$$p_1(x) = x$$ - \pi$$

Question 2

Find the quadratic approximation (Taylor polynomial of degree 2) at $$x = 0$$ for $$f(x) = \sqrt{64 - x^2$}.$$

Accepted Answer

$$p_2(x) = 8$$ - \dfrac{$$x^2$$}{16}$$

Question 3

What is the Taylor polynomial of order 3 for $$f(x) = \ln(x+1)$$ centered at $$a = 4$$?

Accepted Answer

$$P_3(x$$) = \ln 5 + \dfrac{x-4}{5} - \dfrac{$$(x-4)^2$$}{50} + \dfrac{$$(x-4)^3$$}{375}$$

Question 4

Using the Taylor polynomial of degree 3 at $$x = 16$$, approximate $$\sqrt{16.07} to $$four decimal places.

Accepted Answer

$$4.0087$$

Question 5

Find the radius of convergence for the series $$\sum\limits_{n=0}^{\infty} (x-4)^n.$$

Accepted Answer

$$1$$

Question 6

What is the radius of convergence for the series $$\sum\limits_{n=0}^{\infty} \dfrac{(x-7)^n}{n5^6$n}$$?

Accepted Answer

$$6$$

Question 7

Determine the interval of convergence for the series $$\sum\limits_{n=1}^{\infty} \dfrac{(x-3)^n}{(5n)!}.$$

Accepted Answer

$$-\infty \u003c x \u003c \infty$$

Question 8

Given $$f(x) = \dfrac{1}{1-x} = \sum\limits_{k=0}^{\infty} x^k$$ for $$|x| \u003c 1$$, what is the power series representation for $$f(10x) = \dfrac{1}{1-10x}$$?

Accepted Answer

$$\sum\limits_{k=0}^{\infty} (10x)^k$$

Question 9

Find the power series representation for $$f(x) = e$$^{9x}$$ using the Maclaurin series for e^x$.

Accepted Answer

$$\sum\limits_{k=0}^{\infty} \dfrac{9^k}{k!} x^k$$

Question 10

By differentiating term by term the expansion of $$f(x) = \dfrac{1}{(1+x)^2$}$$, what is the resulting series for $$f(x) = -\dfrac{2}{(1+x)^3$}$$?

Accepted Answer

$$\dfrac{d}{dx}(1 - 2x + 3x^2$ - 4x^3$ + \ldots) = -2(1 - 3x + 6x^2$ - \ldots)$$

Question 11

Find the first four nonzero terms in the Maclaurin series for $$f(x) = e$$^{-5x}$$.

Accepted Answer

$$1 - 5x + \dfrac{25x^2$}{2!} - \dfrac{125x^3$}{3!}$$

Question 12

What is the Taylor series generated by $$f(x) = e^x at$$ $$x = 8$$?

Accepted Answer

$$\sum\limits_{n=0}^{\infty} \dfrac{e^8$ (x-8)^n}{n!}$$

Question 13

Use power series operations to find the Taylor series at $$x = 0$$ for $$f(x) = x^6$ e^x.$$

Accepted Answer

$$\sum\limits_{n=0}^{\infty} \dfrac{x$$^{n+6}$$}{n!}$$

Question 14

Find the first four terms of the binomial series for $$(1 + x$$)^{3/7}$$.

Accepted Answer

$$1 + \dfrac{3}{7}x + \dfrac{3}{49}x^2$ + \dfrac{1}{343}x^3$

Question 15

A chemist is modeling the concentration of a reactant using the function $$f(x) = e$$^{-5x}$$, where x$ is time in hours. Using the first four terms of the Maclaurin series, estimate the concentration after x = 0.1$ hours.

Accepted Answer

$$1 - 0.5 + 0.025 - 0.00208 \approx 0.5229$$

Question 16

Suppose a population grows according to $$P(t) = e$$^{9t}$$, where t$ is in years. Using the first three terms of the Maclaurin series, estimate the population at t = 0.1$ years.

Accepted Answer

$$1 + 0.9 + 0.405 = 2.305$$

Question 17

Which of the following best describes the process of finding the Taylor series for $$f(x) = e^x$$ centered at $$x = a$$?

Accepted Answer

$$f(x) = \sum\limits_{n=0}^{\infty} \dfrac{e^a (x-a)^n}{n!}$$

Question 18

A medical researcher uses the series $$\sum\limits_{n=0}^{\infty} (x-4)^n to $$model the effect of a drug, where $$x is $$the dosage. For which values of $$x$$ does the model converge?

Accepted Answer

$$3 \u003c x \u003c 5$$

Chapter 11 Review – Calculus: Taylor Series, Approximations, and Power Series

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