Solutions To Selected Problems Chapter 1 Chapter 3

Chapter 2

Exercises 2.0

1.

$80x^{12}y^{17}$
2.

$\dfrac{a}{16b^{7}}$
3.

$\dfrac{x^{3}}{16y^{22}z^{35}}$
4.

$x^{2}y^{4}z^{5}\sqrt[4]{z}=x^{2}y^{4}z^{21/4}$
5.

$3x(x^{2}+9x+3)$
6.

$\frac{5(x-1)}{3x^{\frac{1}{3}}}$
7.

$\frac{-5x+4}{2x^{\frac{1}{2}}(x+4)^{2}}$
8.

$6x(3x^{2}+2)^{3}(x^{2}-5)^{2}(7x^{2}-18)$
9.
(a) 8 (b) 44 (c) $x^{2}-6x+8$ (d) $x^{2}+2x-4$
10.
(a) $-\frac{1}{3}$ (b) undefined (c) $\dfrac{1}{\sqrt{x-2}-5}$ (d) $\sqrt{\dfrac{1}{x-5}-2}$
11.
(a) Possible solution: $f(x)=\frac{5}{x}$ and $g(x)=x+4$ (b) Possible solution: $f(x)=\left\lvert x\right\rvert$ and $g(x)=4-x^{2}$ (c) Possible solution: $f(x)=\sqrt{x-5}$ and $g(x)=(x+2)^{2}$
12.
(a) Possible solution: $f(x)=\sqrt[3]{x}$ , $g(x)=x^{2}$ , and $h(x)=2x+1$ (b) Possible solution: $f(x)=2x+1$ , $g(x)=\sqrt[3]{x}$ , and $h(x)=x^{2}$

Exercises 2.1

1.

T
3.

Answers will vary.
5.

(a) $f\,^{\prime}(x)=0$ , (b) $y=6$
7.

(a) $f\,^{\prime}(x)=-3$ , (b) $y=4-3x$
9.

(a) $h^{\prime}(x)=2-2x$ (b) $y=1$
11.

(a) $g^{\prime}(x)=\frac{1}{2\sqrt{x+3}}$ , (b) $y=\frac{x}{4}+\frac{7}{4}$
13.

(a) $h^{\prime}(x)=-\dfrac{3}{2x\sqrt{x}}$ , (b) $y=-\dfrac{3x}{16}+\dfrac{9}{4}$
15.

$f(x)=\sqrt{x}$ , $c=16$ .
17.

$f(x)=\dfrac{1}{x}$ , $c=2$
19.

$y=8.1(x-3)+16$
21.

$y=-0.099(x-9)+1$
23.

$y=.49(x-2)+\ln 2$
25.
(a) Approximations will vary; they should match (c) closely. (b) $f\,^{\prime}(x)=2x$ (c) At $(-1,0)$ , slope is $-2$ . At $(0,-1)$ , slope is 0. At $(2,3)$ , slope is 4.
27.
29.
31.
33.
(a) Approximately on $(-2,0)$ and $(2,\infty)$ . (b) Approximately on $(-\infty,-2)$ and $(0,2)$ . (c) Approximately at $x=0,\ \pm 2$ . (d) Approximately on $(-\infty,-1)$ and $(1,\infty)$ . (e) Approximately on $(-1,1)$ . (f) Approximately at $x=\pm 1$ .
35.
37.

Approximately $0.54$ .
39.
(a) 1 (b) 3 (c) Does not exist (d) $(-\infty,-3)\cup(3,\infty)$

Exercises 2.2

1.

Velocity
3.

Linear functions.
5.

$-17$
7.

$f(10.1)$ is likely most accurate, as accuracy is lost the farther from $x=10$ we go.
9.

$6$
11.

ft/s ${}^{2}$
13.
(a) thousands of dollars per car (b) It is likely that $P(0)<0$ . That is, negative profit for not producing any cars.
15.

$f(x)=g^{\prime}(x)$
17.

$g(x)=f\,^{\prime}(x)$
19.

$f(6)=1$ , $f\,^{\prime}(6)=-\frac{3}{4}$
21.
Answers vary. Possible solution
23.

$f\,^{\prime}(x)=10x$
25.

$f\,^{\prime}(\pi)\approx 0$ .

Exercises 2.3

1.

Power Rule.
3.

One answer is $f(x)=10e^{x}$ .
5.

$f(x)$ , $g(x)$ , $h(x)$ , and $m(x)$
7.

One possible answer is $f(x)=17x-205$ .
9.

$f\,^{\prime}(x)$ is a velocity function, and $f\,^{\prime\prime}(x)$ is acceleration.
11.

$f\,^{\prime}(x)=14x-5$
13.

$m^{\prime}(t)=45t^{4}-\frac{3}{8}t^{2}+3$
15.

$f\,^{\prime}(r)=6e^{r}$
17.

$f\,^{\prime}(x)=\frac{2}{x}-1$
19.

$h^{\prime}(t)=e^{t}-\cos t+\sin t$
21.

$f\,^{\prime}(t)=0$
23.

$g^{\prime}(x)=24x^{2}-120x+150$
25.

$f\,^{\prime}(x)=18x-12$
27.

$f\,^{\prime}(x)=\dfrac{3}{2}\sqrt{x}-\dfrac{1}{2x\sqrt{x}}$
29.
31.

$a$ is $f$ , $b$ is $f\,^{\prime}$ , $c$ is $f\,^{\prime\prime}$
33.

$f\,^{\prime}(x)=6x^{5}$ $f\,^{\prime\prime}(x)=30x^{4}$ $f\,^{\prime\prime\prime}(x)=120x^{3}$ $f^{(4)}(x)=360x^{2}$
35.

$h^{\prime}(t)=2t-e^{t}$ $h^{\prime\prime}(t)=2-e^{t}$ $h^{\prime\prime\prime}(t)=-e^{t}$ $h^{(4)}(t)=-e^{t}$
37.

$f\,^{\prime}(\theta)=\cos\theta+\sin\theta$ $f\,^{\prime\prime}(\theta)=-\sin\theta+\cos\theta$ $f\,^{\prime\prime\prime}(\theta)=-\cos\theta-\sin\theta$ $f^{(4)}(\theta)=\sin\theta-\cos\theta$
39.
(a) $v(t)=4t^{3}-8t$ , $a(t)=12t^{2}-8$ (b) $a(1.5)=19\ \text{ft/s}^{2}$ (c) $t=0$ sec and $t=\sqrt{2}$ sec
41.

Tangent line: $y=2(x-1)$
43.

Tangent line: $y=x-1$
45.

Tangent line: $y=\sqrt{2}(x-\frac{\pi}{4})-\sqrt{2}$
47.

$n=-3,2$
49.

The tangent line to $f(x)=x^{4}$ at $x=3$ is $y=108(x-3)+81$ ; thus $(3.01)^{4}\approx y(3.01)=108(.01)+81=82.08$ .

Exercises 2.4

1.

F
3.

T
5.

F
7.
$\displaystyle\frac{\operatorname{d}\!}{\operatorname{d}\!x}(\cot x)$ $\displaystyle=\frac{\operatorname{d}\!}{\operatorname{d}\!x}\left(\frac{\cos x% }{\sin x}\right)$ $\displaystyle=\frac{\sin x(-\sin x)-(\cos x)(\cos x)}{(\sin x)^{2}}$ $\displaystyle=\frac{-[(\sin x)^{2}+(\cos x)^{2}]}{(\sin x)^{2}}$ $\displaystyle=\frac{-1}{(\sin x)^{2}}=-\csc^{2}x$
9.
(a) $f\,^{\prime}(x)=(x^{2}+3x)+x(2x+3)$ (b) $f\,^{\prime}(x)=3x^{2}+6x$ (c) They are equal.
11.
(a) $h^{\prime}(s)=2(s+4)+(2s-1)(1)$ (b) $h^{\prime}(s)=4s+7$ (c) They are equal.
13.
(a) $f\,^{\prime}(x)=\frac{x(2x)-(x^{2}+3)1}{x^{2}}$ (b) $f\,^{\prime}(x)=1-\frac{3}{x^{2}}$ (c) They are equal.
15.
(a) $h^{\prime}(s)=\frac{4s^{3}(0)-3(12s^{2})}{16s^{6}}$ (b) $h^{\prime}(s)=-9/4s^{-4}$ (c) They are equal.
17.

$f\,^{\prime}(x)=\sin x+x\cos x$
19.

$H^{\prime}(y)=(y^{5}-2y^{3})(14y+1)+(5y^{4}-6y^{2})(7y^{2}+y-8)$
21.

$g^{\prime}(x)=\frac{-12}{(x-5)^{2}}$
23.

$g^{\prime}(x)=\dfrac{\sqrt{x}+8}{2(\sqrt{x}+4)^{2}}$
25.

$h^{\prime}(x)=-\csc^{2}x-e^{x}$
27.

$f\,^{\prime}(x)=\frac{(x+2)(4x^{3}+6x^{2})-(x^{4}+2x^{3})(1)}{(x+2)^{2}}$
29.

$y^{\prime}=-2x-5-\dfrac{10}{x^{2}}=-\dfrac{2x^{3}+5x^{2}+10}{x^{2}}$
31.

$p^{\prime}(x)=-\dfrac{1}{x^{2}}-\dfrac{2}{x^{3}}-\dfrac{3}{x^{4}}=-\dfrac{x^{2% }+2x+3}{x^{4}}$
33.

$f\,^{\prime}(t)=5t^{4}(\sec t+e^{t})+t^{5}(\sec t\tan t+e^{t})$
35.

$g^{\prime}(x)=0$
37.

$f\,^{\prime}(y)=y(2y^{3}-5y-1)(12y)+y(6y^{2}-5)(6y^{2}+7)+1(2y^{3}-5y-1)(6y^{2% }+7)=72y^{5}-64y^{3}-18y^{2}-70y-7$
39.

$h^{\prime}(x)=\frac{(t^{2}\cos t+2)(2t\sin t+t^{2}\cos t)-(t^{2}\sin t+3)(2t% \cos t-t^{2}\sin t)}{(t^{2}\cos t+2)^{2}}$
41.

$g^{\prime}(x)=2\sin x\sec x+2x\cos x\sec x+2x\sin x\sec x\tan x=2\tan x+2x+2x% \tan^{2}x=2\tan x+2x\sec^{2}x$
43.

$y=2x+2$
45.

$y=4$
47.

$x=3/2$
49.

$f\,^{\prime}(x)$ is never 0.
51.

$f\,^{\prime\prime}(x)=2\cos x-x\sin x$
53.

$f\,^{\prime\prime}(x)=\cot^{2}x\csc x+\csc^{3}x$
55.

$1$
57.

$-4$
59.

$-\dfrac{1}{25}$
61.

(a) $-\frac{7}{2}$ (b) $\frac{1}{18}$ (c) $-\frac{9}{2}$ (d) $\frac{15}{2}$

Exercises 2.5

1.

T
3.

F
5.

T
7.

$f\,^{\prime}(x)=10(4x^{3}-x)^{9}\cdot(12x^{2}-1)=(120x^{2}-10)(4x^{3}-x)^{9}$
9.

$g^{\prime}(\theta)=3(\sin\theta+\cos\theta)^{2}(\cos\theta-\sin\theta)$
11.

$f\,^{\prime}(x)=4\big{(}x+\frac{1}{x}\big{)}^{3}\big{(}1-\frac{1}{x^{2}}\big{)}$
13.

$f\,^{\prime}(x)=-3\sin(3x)$
15.

$h^{\prime}(x)=(2\theta+4)\sec^{2}(\theta^{2}+4\theta)$
17.

$h^{\prime}(t)=8\sin^{3}(2t)\cos(2t)$
19.

$g^{\prime}(x)=2(\tan x\sec^{2}x-x\sec^{2}(x^{2}))$
21.

$f\,^{\prime}(x)=-\tan x$
23.

$f\,^{\prime}(x)=2/x$
25.

$r^{\prime}(x)=\dfrac{-6(x-1)}{x^{3}\sqrt{4x-3}}$
27.

$h^{\prime}(x)=200(2x+1)^{9}[(2x+1)^{10}+1]^{9}$
29.

$F\mkern 1.35mu ^{\prime}(x)=2(2x+1)(2x+3)^{2}(24x^{2}+26x+3)$
31.

$f\,^{\prime}(x)=5(x^{2}+x)^{4}(2x+1)(3x^{4}+2x)^{3}+(x^{2}+x)^{5}3(3x^{4}+2x)^% {2}(12x^{3}+2)$
33.

$g^{\prime}(t)=10t\cos(\frac{1}{t})e^{5t^{2}}+\frac{1}{t^{2}}\sin(\frac{1}{t})e% ^{5t^{2}}$
35.

$f\,^{\prime}(x)=\dfrac{\tan(5x)8(4x+1)-(4x+1)^{2}5\sec^{2}(5x)}{\tan^{2}(5x)}$
37.

$y^{\prime}=\dfrac{-\cos x\sin x\cos(\cos^{2}x)}{\sqrt{\sin(\cos^{2}x)}}$
39.

$f\,^{\prime}(x)=\frac{1}{2}x^{-1/2}-\frac{1}{2}x^{-3/2}=\frac{1}{2\sqrt{x}}-% \frac{1}{2\sqrt{x^{3}}}$
41.

$f\,^{\prime}(t)=\frac{-t}{\sqrt{1-t^{2}}}$
43.

$h^{\prime}(x)=1.5x^{0.5}=1.5\sqrt{x}$
45.

$g^{\prime}(x)=\frac{\sqrt{x}(1)-(x+7)(1/2x^{-1/2})}{x}=\frac{1}{2\sqrt{x}}-% \frac{7}{2\sqrt{x^{3}}}$
47.

$15$
49.

(a) $6$ (b) $1$ (c) $-3$ (d) $1.5$
51.

$y=0$
53.

$y=-3(\theta-\pi/2)+1$
55.

In both cases the derivative is the same: $1/x$ .
57.
Let $g(x)=-x$ . Then (a) $f\circ g=f$ , so $f\,^{\prime}(-x)=f\,^{\prime}\circ g(x)=-f\,^{\prime}\circ g(x)g^{\prime}(x)=-% (f\circ g)^{\prime}(x)=-f\,^{\prime}(x)$ (b) $f\circ g=-f$ , so $f\,^{\prime}(-x)=f\,^{\prime}\circ g(x)=-f\,^{\prime}\circ g(x)g^{\prime}(x)=-% (f\circ g)^{\prime}(x)=f\,^{\prime}(x)$
59.

$[f(g(x))]^{\prime\prime}=[f\,^{\prime}(g(x))g^{\prime}(x)]^{\prime}=[f\,^{% \prime}(g(x))]^{\prime}g^{\prime}(x)+f\,^{\prime}(g(x))g^{\prime\prime}(x)=f\,% ^{\prime\prime}(g(x))g^{\prime}(x)g^{\prime}(x)+f\,^{\prime}(g(x))g^{\prime% \prime}(x)=f\,^{\prime\prime}(g(x))[g^{\prime}(x)]^{2}+f\,^{\prime}(g(x))g^{% \prime\prime}(x)$
61.

$2xe^{x}\cot x+x^{2}e^{x}\cot x-x^{2}e^{x}\csc^{2}x$

Exercises 2.6

1.

Answers will vary.
3.

T
5.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{-4x^{3}}{2y+1}$
7.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\sin x\sec y$
9.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{y}{x}$
11.

$-\frac{2\sin(y)\cos(y)}{x}$
13.

$\frac{1}{2y+2}$
15.

$\frac{-\cos(x)(x+\cos(y))+\sin(x)+y}{\sin(y)(\sin(x)+y)+x+\cos(y)}$
17.

$-\frac{2x+y}{2y+x}$
19.

$\dfrac{3x^{2}y\cos(x^{3})-\sin(y^{3})}{3xy^{2}\cos(y^{3})-\sin(x^{3})}$
21.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{y(y-2x)}{x(x-2y)}$
23.
(a) $y=0$ (b) $y=-1.859(x-0.1)+0.281$
25.
(a) $y=4$ (b) $y=0.93(x-2)-\sqrt[4]{108}$
27.
(a) $y=-\frac{1}{\sqrt{3}}(x-\frac{7}{2})+\frac{6+3\sqrt{3}}{2}$ (b) $y=\sqrt{3}(x-\frac{4+3\sqrt{3}}{2})+\frac{3}{2}$
29.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{(2y+1)(-12x^{2})% +4x^{3}\left(2\frac{-4x^{3}}{2y+1}\right)}{(2y+1)^{2}}$
31.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{\cos x\cos y+% \sin^{2}x\tan y}{\cos^{2}y}$
33.

In each, $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=-\frac{y}{x}$ .