Calculus: Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillet

Chapter 11

Power Series - all with Video Answers

Educators

Section 1

Approximating Functions with Polynomials

Problem 1

Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function $f$. What matching conditions are satisfied by the polynomial?

Jack Chen

Jack Chen

Numerade Educator

Problem 2

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

Jack Chen

Numerade Educator

Problem 3

The first three Taylor polynomials for $f(x)=\sqrt{1+x}$ centered at 0 are $p_{0}=1, p_{1}=1+\frac{x}{2},$ and $p_{2}=1+\frac{x}{2}-\frac{x^{2}}{8} .$ Find three approximations to $\sqrt{1.1}$.

Hoan Nguyen

Numerade Educator

Problem 4

Suppose $f(0)=1, f^{\prime}(0)=2,$ and $f^{\prime \prime}(0)=-1 .$ Find the quadratic approximating polynomial for $f$ centered at 0 and use it to approximate $f(0.1)$.

Jack Chen

Numerade Educator

Problem 5

Suppose $f(0)=1, f^{\prime}(0)=0, f^{\prime \prime}(0)=2,$ and $f^{(3)}(0)=6 .$ Find the third-order Taylor polynomial for $f$ centered at 0 and use it to approximate $f(0.2)$.

Jack Chen

Numerade Educator

Problem 6

How is the remainder $R_{n}(x)$ in a Taylor polynomial defined?

Jack Chen

Numerade Educator

Problem 7

Suppose $f(2)=1, f^{\prime}(2)=1, f^{\prime \prime}(2)=0,$ and $f^{(3)}(2)=12.$ Find the third-order Taylor polynomial for $f$ centered at 2 and use this polynomial to estimate $f(1.9)$.

Jack Chen

Numerade Educator

Problem 8

Suppose you want to estimate $\sqrt{26}$ using a fourth-order Taylor polynomial centered at $x=a$ for $f(x)=\sqrt{x}$. Choose an appropriate value for the center $a$.

Jack Chen

Numerade Educator

Problem 9

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$\quad f(x)=8 x^{3 / 2}, a=1 ; \text { approximate } 8 \cdot 1.1^{3 / 2}$$

Jack Chen

Numerade Educator

Problem 10

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$f(x)=\frac{1}{x}, a=1 ; \text { approximate } \frac{1}{1.05}$$

Jack Chen

Numerade Educator

Problem 11

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$f(x)=e^{-2 x}, a=0 ; \text { approximate } e^{-0.2}$$

Jack Chen

Numerade Educator

Problem 12

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$f(x)=\sqrt{x}, a=4 ; \text { approximate } \sqrt{3.9}$$

Jack Chen

Numerade Educator

Problem 13

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$f(x)=(1+x)^{-1}, a=0 ; \text { approximate } \frac{1}{1.05}$$

Jack Chen

Numerade Educator

Problem 14

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$f(x)=\cos x, a=\frac{\pi}{4} ; \text { approximate } \cos (0.24 \pi)$$

Jack Chen

Numerade Educator

Problem 15

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$f(x)=x^{1 / 3}, a=8 ; \text { approximate } 7.5^{1 / 3}$$

Jack Chen

Numerade Educator

Problem 16

Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at $a$
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
$$f(x)=\tan ^{-1} x, a=0 ; \text { approximate } \tan ^{-1} 0.1$$

Jack Chen

Numerade Educator

Problem 17

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ centered at $a=0$ for $f(x)=\cos 6 x$.

Jack Chen

Numerade Educator

Problem 18

Find the Taylor polynomials $p_{1}, \ldots, p_{5}$ centered at $a=0$ for $f(x)=e^{-x}$.

Jack Chen

Numerade Educator

Problem 19

Find the Taylor polynomials $p_{3}$ and $p_{4}$ centered at $a=0$ for $f(x)=(1+x)^{-3}$.

Jack Chen

Numerade Educator

Problem 20

Find the Taylor polynomials $p_{4}$ and $p_{5}$ centered at $a=\pi / 6$ for $f(x)=\cos x$.

Jack Chen

Numerade Educator

Problem 21

Find the Taylor polynomials $p_{1}, p_{2},$ and $p_{3}$ centered at $a=1$ for $f(x)=x^{3}$.

Jack Chen

Numerade Educator

Problem 22

Find the Taylor polynomials $p_{3}$ and $p_{4}$ centered at $a=1$ for $f(x)=8 \sqrt{x}$.

Jack Chen

Numerade Educator

Problem 23

Find the Taylor polynomial $p_{3}$ centered at $a=e$ for $f(x)=\ln x$.

Jack Chen

Numerade Educator

Problem 24

Find the Taylor polynomial $p_{2}$ centered at $a=8$ for $f(x)=\sqrt[3]{x}$.

Jack Chen

Numerade Educator

Problem 25

Graphing Taylor polynomials
a. Find the nth-order Taylor polynomials for the following functions centered at the given point $a$, for $n=1$ and $n=2$.
b. Graph the Taylor polynomials and the function.
$$f(x)=(1+x)^{-1 / 2}, a=0$$

Jack Chen

Numerade Educator

Problem 26

Graphing Taylor polynomials
a. Find the nth-order Taylor polynomials for the following functions centered at the given point $a$, for $n=1$ and $n=2$.
b. Graph the Taylor polynomials and the function.
$$f(x)=\ln (1-x), a=0$$

Jack Chen

Numerade Educator

Problem 27

Graphing Taylor polynomials
a. Find the nth-order Taylor polynomials for the following functions centered at the given point $a$, for $n=1$ and $n=2$.
b. Graph the Taylor polynomials and the function.
$$f(x)=\sin x, a=\frac{\pi}{4}$$

Jack Chen

Numerade Educator

Problem 28

Graphing Taylor polynomials
a. Find the nth-order Taylor polynomials for the following functions centered at the given point $a$, for $n=1$ and $n=2$.
b. Graph the Taylor polynomials and the function.
$$f(x)=\sqrt{x}, a=9$$

Jack Chen

Numerade Educator

Problem 29

Approximations with Taylor polynomials
a. Use the given Taylor polynomial $p_{2}$ to approximate the given quantity.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
Approximate $\sqrt{1.05}$ using $f(x)=\sqrt{1+x}$ and $p_{2}(x)=1+x / 2-x^{2} / 8$.

Jack Chen

Numerade Educator

Problem 30

Approximations with Taylor polynomials
a. Use the given Taylor polynomial $p_{2}$ to approximate the given quantity.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
Approximate $1 / \sqrt{1.08}$ using $f(x)=1 / \sqrt{1+x}$ and $p_{2}(x)=1-x / 2+3 x^{2} / 8$.

Jack Chen

Numerade Educator

Problem 31

Approximations with Taylor polynomials
a. Use the given Taylor polynomial $p_{2}$ to approximate the given quantity.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
Approximate $e^{-0.15}$ using $f(x)=e^{-x}$ and $p_{2}(x)=1-x+x^{2} / 2$.

Jack Chen

Numerade Educator

Problem 32

Approximations with Taylor polynomials
a. Use the given Taylor polynomial $p_{2}$ to approximate the given quantity.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
Approximate $\ln 1.06$ using $f(x)=\ln (1+x)$ and $p_{2}(x)=x-x^{2} / 2$.

Jack Chen

Numerade Educator

Problem 33

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$e^{0.12}$$

Jack Chen

Numerade Educator

Problem 34

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$\cos (-0.2)$$

Jack Chen

Numerade Educator

Problem 35

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$\tan (-0.1)$$

Jack Chen

Numerade Educator

Problem 36

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$\ln 1.05$$

Jack Chen

Numerade Educator

Problem 37

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$\sqrt{1.06}$$

Jack Chen

Numerade Educator

Problem 38

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$\sqrt[4]{79}$$

Jack Chen

Numerade Educator

Problem 39

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$\sinh 0.5$$

Jack Chen

Numerade Educator

Problem 40

Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with $n=3$
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
$$\tanh 0.5$$

Jack Chen

Numerade Educator

Problem 41

Find the remainder $R_{n}$ for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of $n$.
$$f(x)=\sin x, a=0$$

Jack Chen

Numerade Educator

Problem 42

Find the remainder $R_{n}$ for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of $n$.
$$f(x)=\cos 2 x, a=0$$

Jack Chen

Numerade Educator

Problem 43

Find the remainder $R_{n}$ for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of $n$.
$$f(x)=e^{-x}, a=0$$

Jack Chen

Numerade Educator

Problem 44

Find the remainder $R_{n}$ for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of $n$.
$$f(x)=\cos x, a=\frac{\pi}{2}$$

Jack Chen

Numerade Educator

Problem 45

Find the remainder $R_{n}$ for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of $n$.
$$f(x)=\sin x, a=\frac{\pi}{2}$$

Jack Chen

Numerade Educator

Problem 46

Find the remainder $R_{n}$ for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of $n$.
$$f(x)=\frac{1}{1-x}, a=0$$

Jack Chen

Numerade Educator

Problem 47

Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
$$\sin 0.3, n=4$$

Jack Chen

Numerade Educator

Problem 48

Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
$$\cos 0.45, n=3$$

Jack Chen

Numerade Educator

Problem 49

Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
$$e^{0.25}, n=4$$

Jack Chen

Numerade Educator

Problem 50

Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
$$\tan 0.3, n=2$$

Jack Chen

Numerade Educator

Problem 51

Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
$$
e^{-0.5}, n=4
$$

Jack Chen

Numerade Educator

Problem 52

Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
$$\ln 1.04, n=3$$

Jack Chen

Numerade Educator

Problem 53

Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
$$\sin x=x-\frac{x^{3}}{6} \text {on } \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$$

Jack Chen

Numerade Educator

Problem 54

Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
$$\cos x=1-\frac{x^{2}}{2} \text {on } \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$$

Jack Chen

Numerade Educator

Problem 55

Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
$$e^{x}=1+x+\frac{x^{2}}{2} \text {on } \left[-\frac{1}{2}, \frac{1}{2}\right]$$

Jack Chen

Numerade Educator

Problem 56

Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
$$\tan x=x \text {on } \left[-\frac{\pi}{6}, \frac{\pi}{6}\right]$$

Jack Chen

Numerade Educator

Problem 57

Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
$$\ln (1+x)=x-\frac{x^{2}}{2} \text {on } [-0.2,0.2]$$

Jack Chen

Numerade Educator

Problem 58

Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
$$\sqrt{1+x}=1+\frac{x}{2} \text {on } [-0.1,0.1]$$

Jack Chen

Numerade Educator

Problem 59

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than $10^{-3} ?$ (The answer depends on your choice of a center.)
$$e^{-0.5}$$

Jack Chen

Numerade Educator

Problem 60

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than $10^{-3} ?$ (The answer depends on your choice of a center.)
$$\sin 0.2$$

Jack Chen

Numerade Educator

Problem 61

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than $10^{-3} ?$ (The answer depends on your choice of a center.)
$$\cos (-0.25)$$

Jack Chen

Numerade Educator

Problem 62

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than $10^{-3} ?$ (The answer depends on your choice of a center.)
$$\ln 0.85$$

Jack Chen

Numerade Educator

Problem 63

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than $10^{-3} ?$ (The answer depends on your choice of a center.)
$$\sqrt{1.06}$$

Jack Chen

Numerade Educator

Problem 64

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than $10^{-3} ?$ (The answer depends on your choice of a center.)
$$\frac{1}{\sqrt{0.85}}$$

Jack Chen

Numerade Educator

Problem 65

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Only even powers of $x$ appear in the Taylor polynomials for $f(x)=e^{-2 x}$ centered at 0.
b. Let $f(x)=x^{5}-1 .$ The Taylor polynomial for $f$ of order 10 centered at 0 is $f$ itself.
c. Only even powers of $x$ appear in the $n$th-order Taylor polynomial for $f(x)=\sqrt{1+x^{2}}$ centered at 0.
d. Suppose $f^{\text {- }}$ is continuous on an interval that contains $a$, where f has an inflection point at $a$. Then the second-order Taylor polynomial for $f$ at $a$ is linear.

Jack Chen

Numerade Educator

Problem 66

Taylor coefficients for $x=a$ Follow the procedure in the text to show that the $n$ th-order Taylor polynomial that matches $f$ and its derivatives up to order $n$ at $a$ has coefficients
$$c_{k}=\frac{f^{(k)}(a)}{k !}, \text { for } k=0,1,2, \ldots, n$$

Jack Chen

Numerade Educator

Problem 67

Matching functions with polynomials Match functions a-f with Taylor polynomials $A-F$ (all centered at 0 ). Give reasons for your choices.
a. $\sqrt{1+2 x}$
b. $\frac{1}{\sqrt{1+2 x}}$
c. $e^{2 x}$
d. $\frac{1}{1+2 x}$
e. $\frac{1}{(1+2 x)^{3}}$
f. $e^{-2 x}$

A. $p_{2}(x)=1+2 x+2 x^{2}$
B. $p_{2}(x)=1-6 x+24 x^{2}$
C. $p_{2}(x)=1+x-\frac{x^{2}}{2}$
D. $p_{2}(x)=1-2 x+4 x^{2}$
E. $p_{2}(x)=1-x+\frac{3}{2} x^{2}$
F. $p_{2}(x)=1-2 x+2 x^{2}$

Jack Chen

Numerade Educator

Problem 68

Dependence of errors on $x$ Consider $f(x)=\ln (1-x)$ and its Taylor polynomials given in Example $8 .$
a. Graph $y=\left|f(x)-p_{2}(x)\right|$ and $y=\left|f(x)-p_{3}(x)\right|$ on the interval $[-1 / 2,1 / 2]$ (two curves).
b. At what points of $[-1 / 2,1 / 2]$ is the error largest? Smallest?
c. Are these results consistent with the theoretical error bounds obtained in Example $8 ?$

Jack Chen

Numerade Educator

Problem 69

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=\sin x=x$$

Jack Chen

Numerade Educator

Problem 70

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=\tan x=x$$

Jack Chen

Numerade Educator

Problem 71

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=\cos x=1-\frac{x^{2}}{2}$$

Jack Chen

Numerade Educator

Problem 72

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=\tan ^{-1} x=x$$

Jack Chen

Numerade Educator

Problem 73

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=\sqrt{1+x}=1+\frac{x}{2}$$

Jack Chen

Numerade Educator

Problem 74

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=\ln (1+x)=x-\frac{x^{2}}{2}$$

Jack Chen

Numerade Educator

Problem 75

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=e^{x}=1+x$$

Jack Chen

Numerade Educator

Problem 76

Small argument approximations Consider the following common approximations when $x$ is near zero.
a. Estimate $f(0.1)$ and give a bound on the error in the approximation.
b. Estimate $f(0.2)$ and give a bound on the error in the approximation.
$$f(x)=\sin ^{-1} x=x$$

Jack Chen

Numerade Educator

Problem 77

Errors in approximations Suppose you approximate $f(x)=\sec x$ at the points $x=-0.2,-0.1,0.0,0.1,0.2$ using the Taylor polynomials $p_{2}(x)=1+x^{2} / 2$ and $p_{4}(x)=1+x^{2} / 2+5 x^{4} / 24 .$ Assume the exact value of sec $x$ is given by a calculator.
a. Complete the table showing the absolute errors in the approximations at each point. Show three significant digits.
b. In each error column, how do the errors vary with $x ?$ For what values of $x$ are the errors largest and smallest in magnitude?
$$\begin{array}{|c|c|c|}
\hline x & \left|\sec x-p_{2}(x)\right| & \left|\sec x-p_{4}(x)\right| \\
\hline-0.2 & & \\
\hline-0.1 & & \\
\hline 0.0 & & \\
\hline 0.1 & & \\
\hline 0.2 & & \\
\hline
\end{array}$$

Jack Chen

Numerade Educator

Problem 78

Errors in approximations Carry out the procedure described in Exercise 77 with the following functions and Taylor polynomials.
$$f(x)=\cos x, p_{2}(x)=1-\frac{x^{2}}{2}, p_{4}(x)=1-\frac{x^{2}}{2}+\frac{x^{4}}{24}$$

Jack Chen

Numerade Educator

Problem 79

Errors in approximations Carry out the procedure described in Exercise 77 with the following functions and Taylor polynomials.
$$f(x)=e^{-x}, p_{1}(x)=1-x, p_{2}(x)=1-x+\frac{x^{2}}{2}$$

Jack Chen

Numerade Educator

Problem 80

Best center point Suppose you wish to approximate $\cos (\pi / 12)$ using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at $\pi / 6 ?$ Use a calculator for numerical experiments and check for consistency with Theorem $11.2 .$ Does the answer depend on the order of the polynomial?

Jack Chen

Numerade Educator

Problem 81

Best center point Suppose you wish to approximate $\cos (\pi / 12)$ using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at $\pi / 6 ?$ Use a calculator for numerical experiments and check for consistency with Theorem $11.2 .$ Does the answer depend on the order of the polynomial?

Jack Chen

Numerade Educator

Problem 82

Proof of Taylor's Theorem There are several proofs of Taylor's Theorem, which lead to various forms of the remainder. The following proof is instructive because it leads to two different forms of the remainder and it relies on the Fundamental Theorem of Calculus, integration by parts, and the Mean Value Theorem for Integrals. Assume $f$ has at least $n+1$ continuous derivatives on an interval containing $a$.
a. Show that the Fundamental Theorem of Calculus can be written in the form
$$f(x)=f(a)+\int_{a}^{x} f^{\prime}(t) d t$$
b. Use integration by parts $\left(u=f^{\prime}(t), d v=d t\right)$ to show that \right.
$$f(x)=f(a)+(x-a) f^{\prime}(a)+\int_{a}^{x}(x-t) f^{\prime \prime}(t) d t$$
c. Show that $n$ integrations by parts give
$$\begin{aligned}
f(x)=& f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots \\
&+\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\underbrace{\int_{a}^{x} \frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t}_{R_{n}(x)}
\end{aligned}$$
d. Challenge: The result in part (c) has the form $f(x)=p_{n}(x)+R_{n}(x),$ where $p_{n}$ is the $m$ th-order Taylor polynomial and $R_{n}$ is a new form of the remainder, known as the integral form of the remainder. Use the Mean Value Theorem for Integrals (Section 5.4 ) to show that $R_{n}$ can be expressed in the form
$$R_{n}(x)=\frac{f^{(n+1)}(c)}{(n+1) !}(x-a)^{n+1}$$
where $c$ is between $a$ and $x$

Jack Chen

Numerade Educator

Problem 83

Tangent line is $p_{1}$ Let $f$ be differentiable at $x=a$.
a. Find the equation of the line tangent to the curve $y=f(x)$ at $(a, f(a))$.
b. Verify that the Taylor polynomial $p_{1}$ centered at $a$ describes the tangent line found in part (a).

Jack Chen

Numerade Educator

Problem 84

Local extreme points and inflection points Suppose $f$ has continuous first and second derivatives at $a$.
a. Show that if $f$ has a local maximum at $a$, then the Taylor polynomial $p_{2}$ centered at $a$ also has a local maximum at $a$.
b. Show that if $f$ has a local minimum at $a$, then the Taylor polynomial $p_{2}$ centered at $a$ also has a local minimum at $a$.
c. Is it true that if $f$ has an inflection point at $a$, then the Taylor polynomial $p_{2}$ centered at $a$ also has an inflection point at $a ?$
d. Are the converses in parts (a) and (b) true? If $p_{2}$ has a local extreme point at $a$, does $f$ have the same type of point at $a$ ?

Jack Chen

Numerade Educator

Problem 85

Approximating $\sin x$ Let $f(x)=\sin x,$ and let $p_{n}$ and $q_{n}$ be nth-order Taylor polynomials for $f$ centered at 0 and $\pi,$ respectively.
a. Find $p_{5}$ and $q_{5}$.
b. Graph $f, p_{5},$ and $q_{5}$ on the interval $[-\pi, 2 \pi] .$ On what interval is $p_{5}$ a better approximation to $f$ than $q_{5} ?$ On what interval is $q_{5}$ a better approximation to $f$ than $p_{5} ?$
c. Complete the following table showing the errors in the approximations given by $p_{5}$ and $q_{5}$ at selected points.
$$\begin{array}{|c|c|c|}
\hline x & \left|\sin x-p_{5}(x)\right| & \left|\sin x-q_{5}(x)\right| \\
\hline \pi / 4 & & \\
\hline \pi / 2 & & \\
\hline 3 \pi / 4 & & \\
\hline 5 \pi / 4 & & \\
\hline 7 \pi / 4 & & \\
\hline
\end{array}$$
d. At which points in the table is $p_{5}$ a better approximation to $f$ than $q_{5} ?$ At which points do $p_{5}$ and $q_{5}$ give equal approximations to $f ?$ Explain your observations.

Jack Chen

Numerade Educator

Problem 86

Approximating $\ln x$ Let $f(x)=\ln x,$ and let $p_{n}$ and $q_{n}$ be the nth-order Taylor polynomials for $f$ centered at 1 and $e$, respectively.
a. Find $p_{3}$ and $q_{3}$.
b. Graph $f, p_{3},$ and $q_{3}$ on the interval (0,4].
c. Complete the following table showing the errors in the approximations given by $p_{3}$ and $q_{3}$ at selected points.
$$\begin{array}{|c|c|c|}
\hline x & \left|\ln x-p_{3}(x)\right| & \left|\ln x-q_{3}(x)\right| \\
\hline 0.5 & & \\
\hline 1.0 & & \\
\hline 1.5 & & \\
\hline 2 & & \\
\hline 2.5 & & \\
\hline 3 & & \\
\hline 3.5 & & \\
\hline
\end{array}$$
d. At which points in the table is $p_{3}$ a better approximation to $f$ than $q_{3}$ ? Explain your observations.

Jack Chen

Numerade Educator

Problem 87

Approximating square roots Let $p_{1}$ and $q_{1}$ be the first-order Taylor polynomials for $f(x)=\sqrt{x},$ centered at 36 and $49,$ respectively.
a. Find $p_{1}$ and $q_{1}$.
b. Complete the following table showing the errors when using $p_{1}$ and $q_{1}$ to approximate $f(x)$ at $x=37,39,41,43,45,$ and 47 Use a calculator to obtain an exact value of $f(x)$.
$$\begin{array}{|c|c|c|}
\hline x & \left|\sqrt{x}-p_{1}(x)\right| & \left|\sqrt{x}-q_{1}(x)\right| \\
\hline 37 & & \\
\hline 39 & & \\
\hline 41 & & \\
\hline 43 & & \\
\hline 45 & & \\
\hline 47 & & \\
\hline
\end{array}$$
c. At which points in the table is $p_{1}$ a better approximation to $f$ than $q_{1}$ ? Explain this result.

Jack Chen

Numerade Educator

Problem 88

A different kind of approximation When approximating a function $f$ using a Taylor polynomial, we use information about $f$ and its derivative at one point. An alternative approach (called interpolation) uses information about $f$ at several different points. Suppose we wish to approximate $f(x)=\sin x$ on the interval $[0, \pi]$.
a. Write the (quadratic) Taylor polynomial $p_{2}$ for $f$ centered at $\pi / 2$.
b. Now consider a quadratic interpolating polynomial $q(x)=a x^{2}+b x+c .$ The coefficients $a, b,$ and $c$ are chosen such that the following conditions are satisfied:
$$q(0)=f(0), q\left(\frac{\pi}{2}\right)=f\left(\frac{\pi}{2}\right), \text { and } q(\pi)=f(\pi)$$
Show that $q(x)=-\frac{4}{\pi^{2}} x^{2}+\frac{4}{\pi} x$
c. Graph $f, p_{2},$ and $q$ on $[0, \pi]$
d. Find the error in approximating $f(x)=\sin x$ at the points $\frac{\pi}{4}$ $\frac{\pi}{2}, \frac{3 \pi}{4},$ and $\pi$ using $p_{2}$ and $q$.
e. Which function, $p_{2}$ or $q$, is a better approximation to $f$ on $[0, \pi] ?$ Explain.

Jack Chen

Numerade Educator