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Calculus 6th

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

Chapter 10

Approximating Functions Using Series

Educators


Problem 1

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

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Problem 2

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

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Problem 3

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

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Problem 4

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

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Problem 5

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

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

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

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Problem 10

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

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Problem 11

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

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Problem 12

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

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Problem 13

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

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Problem 14

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

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Problem 15

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

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Problem 16

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

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Problem 17

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 $f(x)$

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Problem 18

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) $\quad f(0)$
(b) $f^{\prime}(0)$
(c) $\quad f^{\prime \prime}(0)$
(d) $\quad f^{\prime \prime \prime}(0)$

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Problem 19

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

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Problem 20

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

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Problem 21

(a) Based on your observations in Problems $19-20$ make a conjecture about Taylor approximations in the case when $f$ is itself a polynomial.
(b) Show that your conjecture is true.

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Problem 22

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

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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.
$$\begin{array}{c|c|c|c|c|c}
\hline f(0) & f^{\prime}(0) & f^{\prime \prime}(0) & f^{\prime \prime \prime}(0) & f^{(4)}(0) & f^{(5)}(0) \\
\hline-3 & 5 & -2 & 0 & -1 & 4 \\
\hline
\end{array}$$

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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 $n > 0, f^{(n)}(0)=-(-2)^{n}$

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Problem 25

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?
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?
(GRAPH CAN'T COPY)

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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?
(GRAPH CAN'T COPY)

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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?
(GRAPH CAN'T COPY)

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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?
(GRAPH CAN'T COPY)

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Problem 29

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

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Problem 30

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 $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$

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Problem 31

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) $\lim _{h \rightarrow 0} \frac{e^{h}-1-h-\frac{h^{2}}{2}}{h^{3}}$

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Problem 32

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

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Problem 33

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.
(FIGURES CAN'T COPY)

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Problem 34

Derive the formulas given in the box on page 543 for the coefficients of the Taylor polynomial approximating a function $f$ for $x$ near $a$

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Problem 35

(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 $h(x) ?$

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Problem 36

(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 $f^{\prime \prime}(0) g(0)+$ $f(0) g^{\prime \prime}(0)=0$

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Problem 37

(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 $f(x)=e^{-2 x} ?$

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Problem 38

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

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Problem 39

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.

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Problem 40

When we model the motion of a pendulum, we 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.

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Problem 41

(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

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Problem 42

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.

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Problem 43

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

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Problem 44

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

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Problem 45

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

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Problem 46

Decide if the statements in Problems 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 $f(x)=g(x)$

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Problem 47

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

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Problem 48

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

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Problem 49

Decide if the statements in Problems 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 $x=0$

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Problem 50

Decide if the statements in Problems 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 $x$

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Problem 51

Decide if the statements in Problems 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 $x=a$

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Problem 52

Decide if the statements in Problems 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 ).

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Problem 53

Decide if the statements in Problems 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 ).

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