Solutions To Selected Problems Chapter 4 Chapter 6

Chapter 5

Exercises 5.1

1.

Answers will vary.
3.

Answers will vary.
5.

Answers will vary.
7.

velocity
9.

$3x^{4}/4+C$
11.

$10x^{3}/3-2x+C$
13.

$-1/(3t)+C$
15.

$2\sqrt{x}+C$
17.

$-\cos\theta+C$
19.

$5e^{\theta}+C$
21.

$4/3t^{3}+6t^{2}+9t+C$
23.

$x^{6}/6+C$
25.

$-x^{-3}+C$
27.

$\dfrac{2}{9}x^{9/2}+C$
29.

$5x-\frac{2}{9}x^{3}+\frac{3}{16}x^{4}+C$
31.

$\frac{2}{3}u^{3}+\frac{9}{2}u^{2}+4u+C$
33.

$2\sqrt{x}+x+\frac{2}{3}x\sqrt{x}+C$
35.

$\theta+\tan\theta+C$
37.

$-\cot t-t+C$
39.

$8\sqrt{u}+4u\sqrt{u}+C$
41.

$6t^{1/3}+\frac{3}{4}t^{4/3}+C$
43.

$-\cos x+3$
45.

$x^{4}-x^{3}+7$
47.

$\frac{7x^{3}}{6}-\frac{9x}{2}+\frac{40}{3}$
49.

$\theta-\sin(\theta)-\pi+4$
51.

$x^{-2}+1$
53.
(a) $x>0$ (b) $1/x$ (c) $x<0$ (d) $1/x$ (e) $\ln\left\lvert x\right\rvert+C$ . Explanations will vary.
55.

$s(t)=241.67-16t^{2}$ ft, so $s(t)=0$ at $t=3.89$ sec.
57.
Other antiderivatives are vertical shifts of this one.
59.

$\operatorname{d}\!y=(2xe^{x}\cos x+x^{2}e^{x}\cos x-x^{2}e^{x}\sin x)% \operatorname{d}\!x$

Exercises 5.2

1.

Answers will vary.
3.

0
5.
(a) 3 (b) 4 (c) 3 (d) 0 (e) $-4$ (f) 9
7.
(a) $4$ (b) $2$ (c) $4$ (d) 2 (e) $1$ (f) 2
9.
(a) $\pi$ (b) $\pi$ (c) $2\pi$ (d) $10\pi$
11.
(a) $-59$ (b) $-48$ (c) $-27$ (d) $-33$ (e) $70$ (f) $91$
13.
(a) $4$ (b) $4$ (c) $-4$ (d) $-2$ (e) $6$ (f) $2$
15.
(a) 2ft/s (b) 2ft (c) 1.5ft
17.
(a) 64ft/s (b) 64ft (c) $t=2$ (d) $t=2+\sqrt{7}\approx 4.65$ seconds
19.

2
21.

16
23.

22
25.

$0$
27.

This is a triangle with base $b$ and height $m b$ .
29.

$1/4x^{4}-2/3x^{3}+7/2x^{2}-9x+C$
31.

$3/4t^{4/3}-1/t+2^{t}/\ln 2+C$

Exercises 5.3

1.

limits
3.

Rectangles.
5.

$2^{2}+3^{2}+4^{2}=29$
7.

$0-1+0+1+0=0$
9.

$1+1/2+1/3+1/4+1/5=137/60$
11.

$1/2+1/6+1/12+1/20=4/5$
13.

Answers may vary; $\sum_{i=1}^{5}3i$
15.

Answers may vary; $\sum_{i=1}^{4}\frac{i}{i+1}$
17.

50
19.

1045
21.

$-8525$
23.

$5050$
25.

$155$
27.

$24$
29.

$\displaystyle\int_{0}^{\pi}\frac{\sin x}{1+x}\operatorname{d}\!x$
31.

$\displaystyle\int_{2}^{7}5x^{3}-4x+7\operatorname{d}\!x$
33.

$\displaystyle\lim_{n\to\infty}\left[\frac{3}{n}\sum_{i=1}^{n}4-2\biggl{(}2+% \frac{3i}{n}\biggr{)}\right]$
35.

$\displaystyle\lim_{n\to\infty}\frac{\pi}{n}\sum_{i=1}^{n}\frac{\sin^{3}(-\pi/2% +\pi i/n)}{2+\cos(-\pi/2+\pi i/n)}$
37.

$19$
39.

$\pi/3+\pi/(2\sqrt{3})\approx 1.954$
41.

$0.388584$
43.
(a) Exact expressions will vary; $\frac{(1+n)^{2}}{4n^{2}}$ . (b) $121/400$ , $10201/40000$ , $1002001/4000000$ (c) $1/4$
45.
(a) 8. (b) 8, 8, 8 (c) $8$
47.
(a) Exact expressions will vary; $100-200/n$ . (b) $80$ , $98$ , $499/5$ (c) $100$
49.
(a) Exact expressions will vary; $80.5$ . (b) $72.25$ (c) $62.5$
51.
$\displaystyle\int_{a}^{b}k\cdot f(x)\operatorname{d}\!x$ $\displaystyle=\lim_{n\to\infty}\sum_{i=1}^{n}k\cdot f(c_{i})\Delta x\quad\text% {T\ref{thm:riemannSum}.2}$ $\displaystyle=\lim_{n\to\infty}k\cdot\sum_{i=1}^{n}k\cdot f(c_{i})\Delta x% \quad\text{T\ref{thm:summation}.3}$ $\displaystyle=k\cdot\lim_{n\to\infty}\sum_{i=1}^{n}k\cdot f(c_{i})\Delta x% \quad\text{T\ref{thm:limit_algebra}.4}$ $\displaystyle=k\int_{a}^{b}f(x)\operatorname{d}\!x\quad\text{T\ref{thm:% riemannSum}.2}$
53.

$F(x)=5\tan x+4$
55.

$G(t)=4/6t^{6}-5/4t^{4}+8t+9$
57.

$G(t)=\sin t-\cos t-78$

Exercises 5.4

1.

Answers will vary.
3.

T
5.

20
7.

$0$
9.

$1$
11.

$23/2$
13.

$e^{3}-e$
15.

$4$
17.

$\ln 2$
19.

$1/4$
21.

$15$
23.

$2$
25.

$\frac{63}{2}$
27.

$\frac{6\pi}{7}$
29.

$\sqrt{2}$
31.

$36$
33.

$\frac{69}{4}$
35.

Explanations will vary. A sketch will help.
37.

$c=2/\sqrt{3}$
39.

$c=\ln(e-1)\approx 0.54$
41.

$2/\pi$
43.

$2$
45.

$16$
47.

(a) $-300$ ft; (b) $312.5$ ft
49.

(a) $-1$ ft; (b) 3ft
51.

$-64$ ft/s
53.

$2$ ft/s
55.

$F^{\prime}(x)=(3x^{2}+1)\frac{1}{x^{3}+x}$
57.

$F^{\prime}(x)=2x(x^{2}+2)-(x+2)$
59.

$F^{\prime}(x)=\frac{\ln x+4}{x^{2}+7}$
61.

$F^{\prime}(x)=-\frac{15x^{2}\sqrt{\cos(5x^{3})+5}}{25x^{6}+e^{5x^{3}}}$
63.
(a) x 0 1 2 3 4 5 6 g(x) 0 .5 0 -.5 0 1.5 4 (b) $g(7)\approx 5.7$ (c) min at $x=3$ ; max at $x=7$ (d) Approximately
65.
(a) $b_{n}=4/n\pi$ for odd $n$ and $b_{n}=0$ for even $n$ (b) answers will vary

Exercises 5.5

1.

Chain Rule.
3.

$\frac{1}{8}(x^{3}-5)^{8}+C$
5.

$\frac{1}{18}\left(x^{2}+1\right)^{9}+C$
7.

$\frac{1}{2}\ln\left\lvert 2x+7\right\rvert+C$
9.

$\frac{2}{3}(x+3)^{3/2}-6(x+3)^{1/2}+C=\frac{2}{3}(x-6)\sqrt{x+3}+C$
11.

$2e^{\sqrt{x}}+C$
13.

$-\frac{1}{2x^{2}}-\frac{1}{x}+C$
15.

$\frac{\sin^{3}(x)}{3}+C$
17.

$-\frac{1}{6}\sin(3-6x)+C$
19.

$\frac{1}{2}\ln\left\lvert\sec(2x)+\tan(2x)\right\rvert+C$
21.

$\tan(x)-x+C$
23.

The key is to multiply $\csc x$ by 1 in the form $(\csc x+\cot x)/(\csc x+\cot x)$ .
25.

$\frac{e^{x^{3}}}{3}+C$
27.

$x-e^{-x}+C$
29.

$\frac{e^{-3x}}{3}-e^{-x}+C$
31.

$\frac{\bigl{(}\ln x\bigr{)}^{3}}{3}+C$
33.

$\frac{1}{2}\ln\left\lvert\ln\left(x^{2}\right)\right\rvert+C$
35.

$\frac{x^{3}}{3}+\frac{x^{2}}{2}+x+\ln\left\lvert x\right\rvert+C$
37.

$\frac{1}{45}(5x^{3}+5x^{2}+2)^{9}+C$
39.

$-\frac{1}{3}\cot\left(x^{3}+1\right)+C$
41.

$-\frac{1}{5}\cos(5x+1)+C$
43.

$\frac{7}{3}\ln\left\lvert 3x+2\right\rvert+C$
45.

$3\ln\left\lvert 3x^{2}+9x+7\right\rvert+C$
47.

$\sqrt{x^{2}-6x+8}+C$
49.

$\frac{1}{2}\sec^{2}\theta+C$ or $\frac{1}{2}\tan^{2}\theta+C$
51.

$-\frac{1}{2(x^{2}+1)}+\frac{1}{4(x^{2}+1)^{2}}+C$
53.

$\frac{1}{11}(x^{3}+2)^{11}-\frac{2}{5}(x^{3}+2)^{10}+\frac{4}{9}(x^{3}+2)^{9}+C$
55.

$\frac{2}{3}\sin^{6}(\frac{x}{4})+C$
57.

$-\ln 2$
59.

$2/3$
61.

$(1-e)/2$
63.

$0$
65.

$\frac{2}{3}$