Calculus Single Variable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 10

Approximating Functions Using Series - all with Video Answers

Educators

Section 1

Taylor Polynomials

Problem 1

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\frac{1}{1-x}, \quad n=3,5,7$$

Tyler M.

Tyler M.

Numerade Educator

Problem 2

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\frac{1}{1+x}, \quad n=4,6,8$$

Tyler M.

Numerade Educator

Problem 3

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\sqrt{1+x}, \quad n=2,3,4$$

Tyler M.

Numerade Educator

Problem 4

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\sqrt[3]{1-x}, \quad n=2,3,4$$

Tyler M.

Numerade Educator

Problem 5

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\cos x, \quad n=2,4,6$$

Tyler M.

Numerade Educator

Problem 6

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\ln (1+x), \quad n=5,7,9$$

Tyler M.

Numerade Educator

Problem 7

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\arctan x, \quad n=3,4$$

Tyler M.

Numerade Educator

Problem 8

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\tan x, \quad n=3,4$$

Tyler M.

Numerade Educator

Problem 9

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$\frac{1}{\sqrt{1+x}}, \quad n=2,3,4$$

Tyler M.

Numerade Educator

Problem 10

Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant.)
$$(1+x)^{p}, \quad n=2,3,4$$

Tyler M.

Numerade Educator

Problem 11

Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.
$$\sqrt{1-x}, \quad a=0, \quad n=3$$

Tyler M.

Numerade Educator

Problem 12

Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.
$$e^{x}, \quad a=1, \quad n=4$$

Tyler M.

Numerade Educator

Problem 13

Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.
$$\frac{1}{1+x}, \quad a=2, \quad n=4$$

Justin F.

Numerade Educator

Problem 14

Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.
$$\cos x, \quad a=\pi / 2, \quad n=4$$

Justin F.

Numerade Educator

Problem 15

Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.
$$\sin x, \quad a=-\pi / 4, \quad n=3$$

Justin F.

Numerade Educator

Problem 16

Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.
$$\ln \left(x^{2}\right), \quad a=1, \quad n=4$$

Justin F.

Numerade Educator

Problem 17

The function $f(x)$ is approximated near $x=0$ by the third-degree Taylor polynomial
\[
P_{3}(x)=2-x-x^{2} / 3+2 x^{3}
\]
Give the value of
(a) $f(0)$
(b) $f^{\prime}(0)$
(c) $f^{\prime \prime}(0)$
(d) $f^{\prime \prime \prime}(0)$

Justin F.

Numerade Educator

Problem 18

Find the second-degree Taylor polynomial for $f(x)=$ $4 x^{2}-7 x+2$ about $x=0 .$ What do you notice?

Justin F.

Numerade Educator

Problem 19

Find the third-degree Taylor polynomial for $f(x)=$ $x^{3}+7 x^{2}-5 x+1$ about $x=0 .$ What do you notice?

Justin F.

Numerade Educator

Problem 20

(a) Based on your observations in Problems $18-19$, make a conjecture about Taylor approximations in the case when $f$ is itself a polynomial.
(b) Show that your conjecture is true.

Justin F.

Numerade Educator

Problem 21

The Taylor polynomial of degree 7 of $f(x)$ is given by
\[
P_{7}(x)=1-\frac{x}{3}+\frac{5 x^{2}}{7}+8 x^{3}-\frac{x^{5}}{11}+8 x^{7}
\]
Find the Taylor polynomial of degree 3 of $f(x)$.

Justin F.

Numerade Educator

Problem 22

Find the value of $f^{(5)}(1)$ if $f(x)$ is approximated near $x=1$ by the Taylor polynomial
\[
p(x)=\sum_{n=0}^{10} \frac{(x-1)^{n}}{n !}
\]

Justin F.

Numerade Educator

Problem 23

Find a simplified formula for $P_{5}(x),$ the fifth-degree Taylor polynomial approximating $f$ near $x=0$.
Use the values in the table.
$$\begin{array}{c|c|c|c|c|c}
\hline f(0) & f^{\prime}(0) & f^{\prime \prime}(0) & f^{\prime \prime \prime}(0) & f^{(4)}(0) & f^{(5)}(0) \\
\hline-3 & 5 & -2 & 0 & -1 & 4 \\
\hline
\end{array}$$

Mutahar M.

Numerade Educator

Problem 24

Find a simplified formula for $P_{5}(x),$ the fifth-degree Taylor polynomial approximating $f$ near $x=0$.
Let $f(0)=-1$ and, for $n>0, f^{(n)}(0)=-(-2)^{n}$

Justin F.

Numerade Educator

Problem 25

(a) From a graph of $f(x)=\ln (2 x+4)$ decide if the tangent line approximation to $f$ at $x=0$ has a positive or negative slope.
(b) Find the tangent line approximation.
(c) Find the second and third Taylor approximations to $f$ about $x=0$

Linda H.

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Problem 26

Suppose $P_{2}(x)=a+b x+c x^{2}$ is the second-degree Taylor polynomial for the function $f$ about $x=0 .$ What can you say about the signs of $a, b,$ and $c$ if $f$ has the graph given below?

Justin F.

Numerade Educator

Problem 27

Suppose $P_{2}(x)=a+b x+c x^{2}$ is the second-degree Taylor polynomial for the function $f$ about $x=0 .$ What can you say about the signs of $a, b,$ and $c$ if $f$ has the graph given below?

Justin F.

Numerade Educator

Problem 28

Suppose $P_{2}(x)=a+b x+c x^{2}$ is the second-degree Taylor polynomial for the function $f$ about $x=0 .$ What can you say about the signs of $a, b,$ and $c$ if $f$ has the graph given below?

Justin F.

Numerade Educator

Problem 29

Suppose $P_{2}(x)=a+b x+c x^{2}$ is the second-degree Taylor polynomial for the function $f$ about $x=0 .$ What can you say about the signs of $a, b,$ and $c$ if $f$ has the graph given below?

Justin F.

Numerade Educator

Problem 30

Use the third degree Taylor polynomial $P_{3}(x)=4+2(x-1)-2(x-1)^{2}+(x-1)^{3} / 2$ of $f(x)$ about $x=1$ to find the given value, or explain why you can't.
$$f(1)$$

Justin F.

Numerade Educator

Problem 31

Use the third degree Taylor polynomial $P_{3}(x)=4+2(x-1)-2(x-1)^{2}+(x-1)^{3} / 2$ of $f(x)$ about $x=1$ to find the given value, or explain why you can't.
$$f^{\prime \prime \prime}(1) \quad 3$$

Justin F.

Numerade Educator

Problem 32

Use the third degree Taylor polynomial $P_{3}(x)=4+2(x-1)-2(x-1)^{2}+(x-1)^{3} / 2$ of $f(x)$ about $x=1$ to find the given value, or explain why you can't.
$$f^{\prime \prime}(0)$$

Justin F.

Numerade Educator

Problem 33

Use the third degree Taylor polynomial $P_{3}(x)=4+2(x-1)-2(x-1)^{2}+(x-1)^{3} / 2$ of $f(x)$ about $x=1$ to find the given value, or explain why you can't.
$$f^{(4)}(1)$$

Justin F.

Numerade Educator

Problem 34

Use a Taylor polynomial with the derivatives given to make the best possible estimate of the value.
$$f(1.1), \text { given that } f(1)=3, f^{\prime}(1)=2, f^{\prime \prime}(1)=-4$$

Justin F.

Numerade Educator

Problem 35

Use a Taylor polynomial with the derivatives given to make the best possible estimate of the value.
$$\begin{aligned}
&g(1.8), \text { given that } g(2)=-3, g^{\prime}(2)=-2, g^{\prime \prime}(2)=0\\
&g^{\prime \prime \prime}(2)=12
\end{aligned}$$

Justin F.

Numerade Educator

Problem 36

Use a Taylor polynomial with the derivatives given to make the best possible estimate of the value.
$$h^{\prime}(0.1), \text { given that } h(0)=6, h^{\prime}(0)=2, h^{\prime \prime}(0)=-4$$

Justin F.

Numerade Educator

Problem 37

Use the Taylor approximation for $x$ near 0 ,
\[
\sin x \approx x-\frac{x^{3}}{3 !}
\]
to explain why $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

Justin F.

Numerade Educator

Problem 38

Use the fourth-degree Taylor approximation for $x$ near 0
\[
\cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}
\]
to explain why $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$

Justin F.

Numerade Educator

Problem 39

Use a fourth-degree Taylor approximation for $e^{h},$ for $h$ near $0,$ to evaluate the following limits. Would your answer be different if you used a Taylor polynomial of higher degree?
(a) $\lim _{h \rightarrow 0} \frac{e^{h}-1-h}{h^{2}}$
(b) $\lim _{h \rightarrow 0} \frac{e^{h}-1-h-\frac{h^{2}}{2}}{h^{3}}$

Justin F.

Numerade Educator

Problem 40

If $f(2)=g(2)=h(2)=0,$ and $f^{\prime}(2)=h^{\prime}(2)=0$
$g^{\prime}(2)=22,$ and $f^{\prime \prime}(2)=3, g^{\prime \prime}(2)=5, h^{\prime \prime}(2)=7,$ cal-
culate the following limits. Explain your reasoning.
(a) $\lim _{x \rightarrow 2} \frac{f(x)}{h(x)}$
(b) $\lim _{x \rightarrow 2} \frac{f(x)}{g(x)}$

Justin F.

Numerade Educator

Problem 41

One of the two sets of functions, $f_{1}, f_{2}, f_{3},$ or $g_{1}, g_{2}$ $g_{3},$ is graphed in Figure $10.8 ;$ the other set is graphed in Figure $10.9 .$ Points $A$ and $B$ each have $x=0 .$ Taylor polynomials of degree 2 approximating these functions near $x=0$ are as follows:
$$\begin{array}{ll}
f_{1}(x) \approx 2+x+2 x^{2} & g_{1}(x) \approx 1+x+2 x^{2} \\
f_{2}(x) \approx 2+x-x^{2} & g_{2}(x) \approx 1+x+x^{2} \\
f_{3}(x) \approx 2+x+x^{2} & g_{3}(x) \approx 1-x+x^{2}
\end{array}$$
(a) Which group of functions, the $f$ s or the $g$ s, is represented by each figure?
(b) What are the coordinates of the points $A$ and $B ?$
(c) Match each function with the graphs (I)- (III) in the appropriate figure.

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Problem 42

Derive the formulas given in the box on page 520 for the coefficients of the Taylor polynomial approximating a function $f$ for $x$ near $a$.

Justin F.

Numerade Educator

Problem 43

(a) Find and multiply the Taylor polynomials of degree 1 near $x=0$ for the two functions $f(x)=$ $1 /(1-x)$ and $g(x)=1 /(1-2 x)$
(b) Find the Taylor polynomial of degree 2 near $x=0$ for the function $h(x)=f(x) g(x)$
(c) Is the product of the Taylor polynomials for $f(x)$ and $g(x)$ equal to the Taylor polynomial for the function $h(x) ?$

Justin F.

Numerade Educator

Problem 44

(a) Find and multiply the Taylor polynomials of degree 1 near $x=0$ for the two functions $f(x)$ and $g(x)$
(b) Find the Taylor polynomial of degree 2 near $x=0$ for the function $h(x)=f(x) g(x)$
(c) Show that the product of the Taylor polynomials for $f(x)$ and $g(x)$ and the Taylor polynomial for the function $h(x)$ are the same if $f^{\prime \prime}(0) g(0)+$ $f(0) g^{\prime \prime}(0)=0$

Justin F.

Numerade Educator

Problem 45

(a) Find the Taylor polynomial approximation of degree 4 about $x=0$ for the function $f(x)=e^{x^{2}}$.
(b) Compare this result to the Taylor polynomial approximation of degree 2 for the function $f(x)=e^{x}$ about $x=0 .$ What do you notice?
(c) Use your observation in part (b) to write out the Taylor polynomial approximation of degree 20 for the function in part (a).
(d) What is the Taylor polynomial approximation of degree 5 for the function $f(x)=e^{-2 x} ?$

Justin F.

Numerade Educator

Problem 46

The integral $\int_{0}^{1}(\sin t / t) d t$ is difficult to approximate using, for example, left Riemann sums or the trapezoid rule because the integrand $(\sin t) / t$ is not defined at $t=0 .$ However, this integral converges; its value is $0.94608 \ldots .$ Estimate the integral using Taylor polynomials for sin $t$ about $t=0$ of
(a) Degree 3
(b) Degree 5

Justin F.

Numerade Educator

Problem 47

Consider the equations $\sin x=0.2$ and $x-\frac{x^{3}}{3 !}=0.2$
(a) How many solutions does each equation have?
(b) Which of the solutions of the two equations are approximately equal? Explain.

Justin F.

Numerade Educator

Problem 48

When we model the motion of a pendulum, we sometimes replace the differential equation
\[
\frac{d^{2} \theta}{d t^{2}}=-\frac{g}{l} \sin \theta \quad \text { by } \quad \frac{d^{2} \theta}{d t^{2}}=-\frac{g}{l} \theta
\]
where $\theta$ is the angle between the pendulum and the vertical. Explain why, and under what circumstances, it is reasonable to make this replacement.

Justin F.

Numerade Educator

Problem 49

(a) Using a graph, explain why the following equation has a solution at $x=0$ and another just to the right of $x=0$
\[
\cos x=1-0.1 x
\]
(b) Replace cos $x$ by its second-degree Taylor polynomial near 0 and solve the equation. Your answers are approximations to the solutions to the original equation at or near 0.

Justin F.

Numerade Educator

Problem 50

Let $P_{n}(x)$ be the Taylor polynomial of degree $n$ approximating $f(x)$ near $x=0$ and $Q_{n}(x)$ the Taylor polynomial of degree $n$ approximating $f^{\prime}(x)$ near $x=0 .$ Show that $Q_{n-1}(x)=P_{n}^{\prime}(x)$.

Justin F.

Numerade Educator

Problem 51

Let $P_{n}(x)$ be the Taylor polynomial of degree $n$ approximating $f(x)$ near $x=0$ and $Q_{n}(x)$ the Taylor polynomial of degree $n$ approximating its antiderivative $F(x)$ with $F(0)=1 .$ Show that $Q_{n+1}(x)=1+\int_{0}^{x} P_{n}(t) d t$.

Justin F.

Numerade Educator

Problem 52

Explain what is wrong with the statement.
If $f(x)=\ln (2+x),$ then the second-degree Taylor polynomial approximating $f(x)$ near $x=0$ has a negative constant term.

Justin F.

Numerade Educator

Problem 53

Explain what is wrong with the statement.
Let $f(x)=\frac{1}{1-x} .$ The coefficient of the $x$ term of the Taylor polynomial of degree 3 approximating $f(x)$ near $x=0$ is -1.

Justin F.

Numerade Educator

Problem 54

Explain what is wrong with the statement.
If the Taylor polynomial of degree 3 approximating $f$ about $x=0$ is
\[
P_{3}(x)=\frac{x}{2}+\frac{x^{2}}{3}+\frac{x^{3}}{4}
\]
then the Taylor polynomial of degree 3 approximating $f$ about $x=1$ is
\[
P_{3}(x)=\frac{(x-1)}{2}+\frac{(x-1)^{2}}{3}+\frac{(x-1)^{3}}{4}
\]

Justin F.

Numerade Educator

Problem 55

Give an example of:
A function $f(x)$ for which every Taylor polynomial approximation near $x=0$ involves only odd powers of $x$.

Justin F.

Numerade Educator

Problem 56

Give an example of:
A third-degree Taylor polynomial near $x=1$ approximating a function $f(x)$ with $f^{\prime}(1)=3$.

Justin F.

Numerade Educator

Problem 57

Give an example of:
Two different functions having the same Taylor polynomial of degree 1 about $x=0$.

Justin F.

Numerade Educator

Problem 58

Give an example of:
A function whose Taylor polynomial of degree 1 about $x=0$ is closer to the values of the function for some values of $x$ than its Taylor polynomial of degree 2 about that point.

Justin F.

Numerade Educator

Problem 59

Decide if the statements are true or false. Give an explanation for your answer.
If $f(x)$ and $g(x)$ have the same Taylor polynomial of degree 2 near $x=0,$ then $f(x)=g(x)$.

Justin F.

Numerade Educator

Problem 60

Decide if the statements are true or false. Give an explanation for your answer.
Using $\sin \theta \approx \theta-\theta^{3} / 3 !$ with $\theta=1^{\circ},$ we have $\sin \left(1^{\circ}\right) \approx$ $1-1^{3} / 6=5 / 6$

Justin F.

Numerade Educator

Problem 61

Decide if the statements are true or false. Give an explanation for your answer.
The Taylor polynomial of degree 2 for $e^{x}$ near $x=5$ is $1+(x-5)+(x-5)^{2} / 2$

Justin F.

Numerade Educator

Problem 62

Decide if the statements are true or false. Give an explanation for your answer.
If the Taylor polynomial of degree 2 for $f(x)$ near $x=0$ is $P_{2}(x)=1+x-x^{2},$ then $f(x)$ is concave up near $x=0$

Justin F.

Numerade Educator

Problem 63

Decide if the statements are true or false. Give an explanation for your answer.
The quadratic approximation to $f(x)$ for $x$ near 0 is better than the linear approximation for all values of $x$.

Justin F.

Numerade Educator

Problem 64

Decide if the statements are true or false. Give an explanation for your answer.
A Taylor polynomial for $f$ near $x=a$ touches the graph of $f$ only at $x=a$.

Justin F.

Numerade Educator

Problem 65

Decide if the statements are true or false. Give an explanation for your answer.
The linear approximation to $f(x)$ near $x=-1$ shows that if $f(-1)=g(-1)$ and $f^{\prime}(-1)<g^{\prime}(-1),$ then $f(x)<g(x)$ for all $x$ sufficiently close to -1 (but not equal to -1 ).

Justin F.

Numerade Educator

Problem 66

Decide if the statements are true or false. Give an explanation for your answer.
The quadratic approximation to $f(x)$ near $x=-1$ shows that if $f(-1)=g(-1), f^{\prime}(-1)=g^{\prime}(-1),$ and $f^{\prime \prime}(-1)<g^{\prime \prime}(-1),$ then $f(x)<g(x)$ for all $x$ sufficiently close to -1 (but not equal to -1 ).

Justin F.

Numerade Educator