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Calculus: Early Transcendental Functions

Robert Smith, Roland Minton

Chapter 3

Applications of Differentiation - all with Video Answers

Educators

Section 1

Linear Approximations and Newton's Method

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Problem 1

In exercises $1-6,$ find the linear approximation to $f(x)$ at $x=x_{0}$. Use the linear approximation to estimate the given number.
$$
f(x)=\sqrt{x}, x_{0}=1, \sqrt{1.2}
$$

Sarah Parrigin
Sarah Parrigin
Numerade Educator
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Problem 2

Find the linear approximation to $f(x)$ at $x=x_{0}$. Use the linear approximation to estimate the given number.
$$
f(x)=(x+1)^{1 / 3}, x_{0}=0, \sqrt[3]{1.2}
$$

Sarah Parrigin
Sarah Parrigin
Numerade Educator
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Problem 3

Find the linear approximation to $f(x)$ at $x=x_{0}$. Use the linear approximation to estimate the given number.
$$
f(x)=\sqrt{2 x+9}, x_{0}=0, \sqrt{8.8}
$$

Sarah Parrigin
Sarah Parrigin
Numerade Educator
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Problem 4

Find the linear approximation to $f(x)$ at $x=x_{0}$. Use the linear approximation to estimate the given number.
$$
f(x)=2 / x, x_{0}=1,2 / 0.99
$$

Sarah Parrigin
Sarah Parrigin
Numerade Educator
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Problem 5

Find the linear approximation to $f(x)$ at $x=x_{0}$. Use the linear approximation to estimate the given number.
$$
f(x)=\sin 3 x, x_{0}=0, \sin (0.3)
$$

Sarah Parrigin
Sarah Parrigin
Numerade Educator
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Problem 6

Find the linear approximation to $f(x)$ at $x=x_{0}$. Use the linear approximation to estimate the given number.
$$
f(x)=\sin x, x_{0}=\pi, \sin (3.0)
$$

Sarah Parrigin
Sarah Parrigin
Numerade Educator
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Problem 7

In exercises 7 and 8 , use linear approximations to estimate the quantity.
(a) $\sqrt[4]{16.04}$
(b) $\sqrt[4]{16.08}$
(c) $\sqrt[4]{16.16}$

Sarah Parrigin
Sarah Parrigin
Numerade Educator
09:36

Problem 8

Use linear approximations to estimate the quantity.
(a) $\sin (0.1)$
(b) $\sin (1.0)$
(c) $\sin \left(\frac{9}{4}\right)$

Grace Muhihu
Grace Muhihu
Numerade Educator
04:20

Problem 9

In exercises 9-12, use linear interpolation to estimate the desired quantity.
A company estimates that $f(x)$ thousand software games can be sold at the price of $\$ x$ as given in the table.
$$\begin{array}{|l|l|l|l|}\hline x & 20 & 30 & 40 \\\hline f(x) & 18 & 14 & 12 \\\hline\end{array}$$

Grace Muhihu
Grace Muhihu
Numerade Educator
04:22

Problem 10

Use linear interpolation to estimate the desired quantity.
A vending company estimates that $f(x)$ cans of soft drink can be sold in a day if the temperature is $x^{\circ} \mathrm{F}$ as given in the table.
$$\begin{array}{|l|l|r|l|}\hline x & 60 & 80 & 100 \\\hline f(x) & 84 & 120 & 168 \\\hline\end{array}$$
Estimate the number of cans that can be sold at (a) $72^{\circ}$ and (b) $94^{\circ}$.

Grace Muhihu
Grace Muhihu
Numerade Educator
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Problem 11

Use linear interpolation to estimate the desired quantity.
An animation director enters the position $f(t)$ of a character's head after $t$ frames of the movie as given in the table.
$$\begin{array}{|l|l|l|l|}\hline t & 200 & 220 & 240 \\\hline f(t) & 128 & 142 & 136 \\\hline\end{array}$$
If the computer software uses interpolation to determine the intermediate positions, determine the position of the head at frame numbers (a) 208 and (b) 232 .

Grace Muhihu
Grace Muhihu
Numerade Educator
03:38

Problem 12

Use linear interpolation to estimate the desired quantity.
A sensor measures the position $f(t)$ of a particle $t$ microseconds after a collision as given in the table.
$$\begin{array}{|l|l|l|l|}\hline t & 5 & 10 & 15 \\\hline f(t) & 8 & 14 & 18 \\\hline\end{array}$$
Estimate the position of the particle at times (a) $t=8$ and (b) $t=12$.

Grace Muhihu
Grace Muhihu
Numerade Educator
02:30

Problem 13

In exercises 13-16, use Newton's method with the given $x_{0}$ to (a) compute $x_{1}$ and $x_{2}$ by hand and (b) use a computer or calculator to find the root to at least five decimal places of accuracy.
$$
x^{3}+3 x^{2}-1=0, x_{0}=1
$$

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
04:36

Problem 14

Use Newton's method with the given $x_{0}$ to (a) compute $x_{1}$ and $x_{2}$ by hand and (b) use a computer or calculator to find the root to at least five decimal places of accuracy.
$$
x^{3}+4 x^{2}-x-1=0, x_{0}=-1
$$

Grace Muhihu
Grace Muhihu
Numerade Educator
04:27

Problem 15

Use Newton's method with the given $x_{0}$ to (a) compute $x_{1}$ and $x_{2}$ by hand and (b) use a computer or calculator to find the root to at least five decimal places of accuracy.
$$
x^{4}-3 x^{2}+1=0, x_{0}=1
$$

Grace Muhihu
Grace Muhihu
Numerade Educator
04:32

Problem 16

Use Newton's method with the given $x_{0}$ to (a) compute $x_{1}$ and $x_{2}$ by hand and (b) use a computer or calculator to find the root to at least five decimal places of accuracy.
$$
x^{4}-3 x^{2}+1=0, x_{0}=-1
$$

Grace Muhihu
Grace Muhihu
Numerade Educator
02:54

Problem 17

In exercises $17-24,$ use Newton's method to find an approximate root (accurate to six decimal places). Sketch the graph and explain how you determined your initial guess.
$$
x^{3}+4 x^{2}-3 x+1=0
$$

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
07:16

Problem 19

Use Newton's method to find an approximate root (accurate to six decimal places). Sketch the graph and explain how you determined your initial guess.
$$
x^{5}+3 x^{3}+x-1=0
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
07:44

Problem 20

Use Newton's method to find an approximate root (accurate to six decimal places). Sketch the graph and explain how you determined your initial guess.
$$
\cos x-x=0
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
08:11

Problem 21

Use Newton's method to find an approximate root (accurate to six decimal places). Sketch the graph and explain how you determined your initial guess.
$$
\sin x=x^{2}-1
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
08:26

Problem 22

Use Newton's method to find an approximate root (accurate to six decimal places). Sketch the graph and explain how you determined your initial guess.
$$
\cos x^{2}=x
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:57

Problem 23

Use Newton's method to find an approximate root (accurate to six decimal places). Sketch the graph and explain how you determined your initial guess.
$$
e^{x}=-x
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
08:24

Problem 24

Use Newton's method to find an approximate root (accurate to six decimal places). Sketch the graph and explain how you determined your initial guess.
$$
e^{-x}=\sqrt{x}
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
02:35

Problem 25

In exercises 25-30, use Newton's method [state the function $f(x)$ you use] to estimate the given number.
$$
\sqrt{11}
$$

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
07:52

Problem 26

Use Newton's method [state the function $f(x)$ you use] to estimate the given number.
$$
\sqrt{23}
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
05:42

Problem 27

Use Newton's method [state the function $f(x)$ you use] to estimate the given number.
$$
\sqrt[3]{11}
$$

Grace Muhihu
Grace Muhihu
Numerade Educator
07:52

Problem 28

Use Newton's method [state the function $f(x)$ you use] to estimate the given number.
$$
\sqrt[3]{23}
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
03:37

Problem 29

Use Newton's method [state the function $f(x)$ you use] to estimate the given number.
$$
\sqrt[4 \cdot 4]{24}
$$

Grace Muhihu
Grace Muhihu
Numerade Educator
03:37

Problem 30

Use Newton's method [state the function $f(x)$ you use] to estimate the given number.
$$
\sqrt[4.6]{24}
$$

Grace Muhihu
Grace Muhihu
Numerade Educator
03:56

Problem 31

In exercises 31-36, Newton's method fails. Explain why the method fails and, if possible, find a root by correcting the problem.
$$
4 x^{3}-7 x^{2}+1=0, x_{0}=0
$$

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
09:24

Problem 32

Newton's method fails. Explain why the method fails and, if possible, find a root by correcting the problem.
$$
4 x^{3}-7 x^{2}+1=0, x_{0}=1
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
05:08

Problem 33

Newton's method fails. Explain why the method fails and, if possible, find a root by correcting the problem.
$$
x^{2}+1=0, x_{0}=0
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
04:41

Problem 34

Newton's method fails. Explain why the method fails and, if possible, find a root by correcting the problem.
$$
x^{2}+1=0, x_{0}=1
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
07:31

Problem 35

Newton's method fails. Explain why the method fails and, if possible, find a root by correcting the problem.
$$
\frac{4 x^{2}-8 x+1}{4 x^{2}-3 x-7}=0, x_{0}=-1
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:17

Problem 36

Newton's method fails. Explain why the method fails and, if possible, find a root by correcting the problem.
$$
\left(\frac{x+1}{x-2}\right)^{1 / 3}=0, x_{0}=0.5
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
08:50

Problem 37

Use Newton's method with (a) $x_{0}=1.2$ and (b) $x_{0}=2.2$ to find a zero of $f(x)=x^{3}-5 x^{2}+8 x-4$. Discuss the difference in the rates of convergence in each case.

Ahmad Reda
Ahmad Reda
Numerade Educator
09:30

Problem 38

Use Newton's method with (a) $x_{0}=0.2$ and (b) $x_{0}=3.0$ to find a zero of $f(x)=x \sin x .$ Discuss the difference in the rates of convergence in each case.

Ahmad Reda
Ahmad Reda
Numerade Educator
09:26

Problem 39

Use Newton's method with (a) $x_{0}=-1.1$ and (b) $x_{0}=2.1$ to find a zero of $f(x)=x^{3}-3 x-2$. Discuss the difference in the rates of convergence in each case.

Ahmad Reda
Ahmad Reda
Numerade Educator
06:09

Problem 40

Factor the polynomials in exercises 37 and $39 .$ Find a relationship between the factored polynomial and the rate at which Newton's method converges to a zero. Explain how the function in exercise $38,$ which does not factor, fits into this relationship. (Note: The relationship will be explored further in exploratory exercise $1 .)$

Ahmad Reda
Ahmad Reda
Numerade Educator
04:30

Problem 41

In exercises $41-44,$ find the linear approximation at $x=0$ to show that the following commonly used approximations are valid for "small" $x$. Compare the approximate and exact values for $x=0.01, x=0.1$ and $x=1$.
$$
\tan x \approx x
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
02:27

Problem 42

Find the linear approximation at $x=0$ to show that the following commonly used approximations are valid for "small" $x$. Compare the approximate and exact values for $x=0.01, x=0.1$ and $x=1$.
$$
\sqrt{1+x} \approx 1+\frac{1}{2} x
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
02:44

Problem 43

Find the linear approximation at $x=0$ to show that the following commonly used approximations are valid for "small" $x$. Compare the approximate and exact values for $x=0.01, x=0.1$ and $x=1$.
$$
\sqrt{4+x} \approx 2+\frac{1}{4} x
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
02:15

Problem 44

Find the linear approximation at $x=0$ to show that the following commonly used approximations are valid for "small" $x$. Compare the approximate and exact values for $x=0.01, x=0.1$ and $x=1$.
$$
e^{x} \approx 1+x
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
05:57

Problem 45

(a) Find the linear approximation at $x=0$ to each of $f(x)=$ $(x+1)^{2}, g(x)=1+\sin (2 x)$ and $h(x)=e^{2 x} .$ Compare
your results.
(b) Graph each function in part (a) together with its linear approximation derived in part (a). Which function has the closest fit with its linear approximation?

Ahmad Reda
Ahmad Reda
Numerade Educator
06:12

Problem 46

(a) Find the linear approximation at $x=0$ to each of $f(x)=$ $\sin x, g(x)=\tan ^{-1} x \quad$ and $\quad h(x)=\sinh x=\frac{e^{x}-e^{-x}}{2}$ Compare your results.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:40

Problem 47

For exercise $7,$ compute the errors (the absolute value of the difference between the exact values and the linear approximations). Thinking of exercises $7 a-7 c$ as numbers of the form $\sqrt[4]{16+\Delta x},$ denote the errors as $e(\Delta x)$ (where $\Delta x=0.04$ $\Delta x=0.08$ and $\Delta x=0.16$ ). Based on these three computations, conjecture a constant $c$ such that $e(\Delta x) \approx c \cdot(\Delta x)^{2}$.

Grace Muhihu
Grace Muhihu
Numerade Educator
02:22

Problem 48

Use a computer algebra system (CAS) to determine the range of $x$ 's in exercise 41 for which the approximation is accurate to within $0.01 .$ That is, find $x$ such that $|\tan x-x|<0.01$.

Ahmad Reda
Ahmad Reda
Numerade Educator
02:04

Problem 49

Given the graph of $y=f(x)$, draw in the tangent lines used in Newton's method to determine $x_{1}$ and $x_{2}$ after starting at $x_{0}=2 .$ Which of the zeros will Newton's method converge to? Repeat with $x_{0}=-2$ and $x_{0}=0.4$.

Ahmad Reda
Ahmad Reda
Numerade Educator
02:34

Problem 50

What would happen to Newton's method in exercise 49 if you had a starting value of $x_{0}=0 ?$ Consider the use of Newton's method with $x_{0}=0.2$ and $x_{0}=10 .$ Obviously, $x_{0}=0.2$ is much closer to a zero of the function, but which initial guess would work better in Newton's method? Explain.

Ahmad Reda
Ahmad Reda
Numerade Educator
06:23

Problem 51

Show that Newton's method applied to $x^{2}-c=0$ (where $c>0$ is some constant) produces the iterative scheme $x_{n+1}=\frac{1}{2}\left(x_{n}+c / x_{n}\right)$ for approximating $\sqrt{c}$. This scheme has been known for over 2000 years. To understand why it works, suppose that your initial guess $\left(x_{0}\right)$ for $\sqrt{c}$ is a little too small. How would $c / x_{0}$ compare to $\sqrt{c}$ ? Explain why the average of $x_{0}$ and $c / x_{0}$ would give a better approximation to $\sqrt{c}$.

Ahmad Reda
Ahmad Reda
Numerade Educator
06:11

Problem 52

Show that Newton's method applied to $x^{n}-c=0$ (where $n$ and $c$ are positive constants) produces the iterative scheme $x_{n+1}=\frac{1}{n}\left[(n-1) x_{n}+c x_{n}^{1-n}\right]$ for approximating $\sqrt[n]{c}$.

Ahmad Reda
Ahmad Reda
Numerade Educator
02:52

Problem 53

Applying Newton's method to $x^{2}-x-1=0,$ show that (a) if $x_{0}=\frac{3}{2}, x_{1}=\frac{13}{8} ;$ (b) if $x_{0}=\frac{5}{3}, x_{1}=\frac{34}{21}$ (c) if $x_{0}=\frac{8}{5}$ $x_{1}=\frac{89}{55} ;$ (d) The Fibonacci sequence is defined by $F_{1}=1$ $F_{2}=1, F_{3}=2, F_{4}=3$ and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 3$ Write each number in parts $(\mathrm{a})-(\mathrm{c})$ as a ratio of Fibonacci numbers. Fill in the subscripts $m$ and $k$ in the following: If $x_{0}=\frac{F_{n+1}}{F_{n}},$ then $x_{1}=\frac{F_{m}}{F_{k}}$ (e) Assuming that Newton's method converges from $x_{0}=\frac{3}{2},$ determine $\lim _{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:52

Problem 54

Determine the behavior of Newton's method applied to (a) $f_{1}(x)=\frac{1}{5}(8 x-3)$ (b) $f_{2}(x)=\frac{1}{5}(16 x-3),$ (c) $f_{3}(x)=$ $\frac{1}{5}(32 x-3) ;(d) f(x) \quad$ when $\quad f(x)=f_{1}(x)$ if $\frac{1}{2} < x < 1$ $f(x)=f_{2}(x)$ if $\frac{1}{4} < x \leq \frac{1}{2}, f(x)=f_{3}(x)$ if $\frac{1}{8} < x \leq \frac{1}{4}$ and so on, with $x_{0}=\frac{3}{4} .$ Does Newton's method converge to a zero of $f ?$ (See Peter Horton's article in the December 2007 issue of Mathematics Magazine.)

Amrita Bhasin
Amrita Bhasin
Numerade Educator
05:51

Problem 55

A water wave of length $L$ meters in water of depth $d$ meters has velocity $v$ satisfying the equation
$$v^{2}=\frac{4.9 L}{\pi} \frac{e^{2 \pi d / L}-e^{-2 \pi d / L}}{e^{2 \pi d / L}+e^{-2 \pi d / L}}$$
Treating $L$ as a constant and thinking of $v^{2}$ as a function $f(d)$ use a linear approximation to show that $f(d) \approx 9.8 d$ for small values of $d$. That is, for small depths, the velocity of the wave is approximately $\sqrt{9.8 d}$ and is independent of the wavelength $L$.

Ahmad Reda
Ahmad Reda
Numerade Educator
03:21

Problem 56

Planck's law states that the energy density of blackbody radiation of wavelength $x$ is given by
$$f(x)=\frac{8 \pi h c x^{-5}}{e^{h c /(k T x)}-1}$$
Use the linear approximation in exercise 44 to show that $f(x) \approx 8 \pi k T / x^{4},$ which is known as the Rayleigh-Jeans law.

Ahmad Reda
Ahmad Reda
Numerade Educator
09:13

Problem 57

Newton's theory of gravitation states that the weight of a person at elevation $x$ feet above sea level is $W(x)=P R^{2} /(R+x)^{2}$, where $P$ is the person's weight at sea level and $R$ is the radius of the earth (approximately 20,900,000 feet). Find the linear approximation of $W(x)$ at $x=0$. Use the linear approximation to estimate the elevation required to reduce the weight of a 120 -pound person by $1 \%$.

Ahmad Reda
Ahmad Reda
Numerade Educator
04:53

Problem 58

One important aspect of Einstein's theory of relativity is that mass is not constant. For a person with mass $m_{0}$ at rest, the mass will equal $m=m_{0} / \sqrt{1-v^{2} / c^{2}}$ at velocity $v$ (where $c$ is the speed of light). Thinking of $m$ as a function of $v$, find the linear approximation of $m(v)$ at $v=0$. Use the linear approximation to show that mass is essentially constant for small velocities.

Ahmad Reda
Ahmad Reda
Numerade Educator
07:31

Problem 59

The spruce budworm is an enemy of the balsam fir tree. In one model of the interaction between these organisms, possible long-term populations of the budworm are solutions of the equation $r(1-x / k)=x /\left(1+x^{2}\right),$ for positive constants $r$ and $k$ (see Murray's Mathematical Biology). (a) Find all positive solutions of the equation with $r=0.5$ and $k=7$. (b) Repeat with $r=0.5$ and $k=7.5 .$ For a small change in the environmental constant $k$ (from 7 to 7.5 ), how did the solution change? The largest solution corresponds to an "infestation" of the spruce budworm.

Ahmad Reda
Ahmad Reda
Numerade Educator
12:06

Problem 60

Suppose that a species reproduces as follows: with probability $p_{0},$ an organism has no offspring; with probability $p_{1},$ an organism has one offspring; with probability $p_{2},$ an organism has two offspring and so on. The probability that the species goes extinct is given by the smallest nonnegative solution of the equation $p_{0}+p_{1} x+p_{2} x^{2}+\cdots=x$ (see Sigmund's Games of Life). Find the positive solutions of the equations $0.1+0.2 x+0.3 x^{2}+0.4 x^{3}=x$ and $0.4+0.3 x+0.2 x^{2}+0.1 x^{3}=x .$ Explain in terms of species going extinct why the first equation has a smaller solution than the second.

Ahmad Reda
Ahmad Reda
Numerade Educator
06:23

Problem 61

(a) In the diagram, a hockey player is $D$ feet from the net on the central axis of the rink. The goalie blocks off a segment of width $w$ and stands $d$ feet from the net. The shooting angle to one side of the goalie is given by $\phi=\tan ^{-1}\left[\frac{3(1-d / D)-w / 2}{D-d}\right] .$ Use a linear approximation of $\tan ^{-1} x$ at $x=0$ to show that if $d=0$, then $\phi \approx \frac{3-w / 2}{D} .$ Based on this, describe how $\phi$ changes if there is an increase in (i) $w$ or (ii) $D$.

Ahmad Reda
Ahmad Reda
Numerade Educator
03:28

Problem 62

In Einstein's theory of relativity, the length of an object depends on its velocity. If $L_{0}$ is the length of the object at rest, $v$ is the object's velocity and $c$ is the speed of light, the Lorentz contraction formula for the length of the object is $L=L_{0} \sqrt{1-v^{2} / c^{2}} .$ Treating $L$ as a function of $v,$ find the linear approximation of $L$ at $v=0$.

Ahmad Reda
Ahmad Reda
Numerade Educator