Solutions To Selected Problems Chapter D Chapter F

Chapter E

Exercises E.1

1.

Answers will vary.
2.

“an”
3.

Answers will vary.
4.

opposite; opposite
5.

Answers will vary.
6.

velocity
7.

velocity
8.

$F(x)+G(x)$
9.

$3x^{4}/4+C$
10.

$x^{9}/9+C$
11.

$10x^{3}/3-2x+C$
12.

$t+C$
13.

$-1/(3t)+C$
14.

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

$2\sqrt{x}+C$
16.

$\tan\theta+C$
17.

$-\cos\theta+C$
18.

$\sec x-\csc x+C$
19.

$5e^{\theta}+C$
20.

$\frac{e^{t}}{2}+C$
21.

$4/3t^{3}+6t^{2}+9t+C$
22.

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

$x^{6}/6+C$
24.

$e^{\pi}x+C$
25.

$-x^{-3}+C$
26.

$\dfrac{4}{3}x^{3}+\frac{7}{2}x^{-2}+C$
27.

$\dfrac{2}{9}x^{9/2}+C$
28.

$\dfrac{2}{7}x^{7/2}-\frac{14}{3}x^{3/2}+C$
29.

$5x-\frac{2}{9}x^{3}+\frac{3}{16}x^{4}+C$
30.

$\frac{1}{7}u^{7}-\frac{1}{3}u^{6}-\frac{1}{4}u^{4}+\frac{2}{7}u+C$
31.

$\frac{2}{3}u^{3}+\frac{9}{2}u^{2}+4u+C$
32.

$\frac{2}{7}t^{7/2}+\frac{6}{5}t^{5/2}+\frac{4}{3}t^{3/2}+C$
33.

$2\sqrt{x}+x+\frac{2}{3}x\sqrt{x}+C$
34.

$x+C$
35.

$\theta+\tan\theta+C$
36.

$\tan t+\sec t+C$
37.

$-\cot t-t+C$
38.

$2\sin x+C$
39.

$8\sqrt{u}+4u\sqrt{u}+C$
40.

$-\cos t+C$
41.

$6t^{1/3}+\frac{3}{4}t^{4/3}+C$
42.

$\frac{4}{9}x^{9/4}+\frac{4}{9}x^{9/5}+C$
43.

$-\cos x+3$
44.

$5e^{x}+5$
45.

$x^{4}-x^{3}+7$
46.

$\tan x+4$
47.

$\frac{7x^{3}}{6}-\frac{9x}{2}+\frac{40}{3}$
48.

$5e^{x}-2x$
49.

$\theta-\sin(\theta)-\pi+4$
50.

$3x-2$
51.

$x^{-2}+1$
52.

$2\sqrt{x}-4$
53.
- $x>0$
  
  $1/x$
  
  $x<0$
  
  $1/x$
  
  $\ln\left\lvert x\right\rvert+C$ . Explanations will vary.
54.

$s(t)=2t^{3/2}$ .
55.

$s(t)=241.67-16t^{2}$ ft, so $s(t)=0$ at $t=3.89$ sec.
56.

Other antiderivatives are vertical shifts of this one.
57.

Other antiderivatives are vertical shifts of this one.
58.

Use technology to verify
59.

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

Exercises E.2

1.

Answers will vary.
2.

Answers will vary.
3.

0
4.

$\int 0^{2}(2x+3)\operatorname{d}\!x$
5.
- 3
  
  4
  
  3
  
  0
  
  $-4$
  
  9
6.
- $-4$
  
  $-5$
  
  $-3$
  
  1
  
  $-2$
  
  10
7.
- $4$
  
  $2$
  
  $4$
  
  2
  
  $1$
  
  2
8.
- $-1/2$
  
  $0$
  
  $3/2$
  
  $3/2$
  
  $9/2$
  
  $15/2$
9.
- $\pi$
  
  $\pi$
  
  $2\pi$
  
  $10\pi$
10.
- $15$
  
  $12$
  
  $0$
  
  $3(b-a)$
11.
- $-59$
  
  $-48$
  
  $-27$
  
  $-33$
  
  $70$
  
  $91$
12.
- $4/\pi$
  
  $-4/\pi$
  
  $0$
  
  $2/\pi$
  
  $4/\pi$
  
  $8/\pi$
13.
- $4$
  
  $4$
  
  $-4$
  
  $-2$
  
  $6$
  
  $2$
14.
- $40/3$
  
  $26/3$
  
  $8/3$
  
  $38/3$
15.
- 2ft/s
  
  2ft
  
  1.5ft
16.
- 3ft/s
  
  9.5ft
  
  9.5ft
17.
- 64ft/s
  
  64ft
  
  $t=2$
  
  $t=2+\sqrt{7}\approx 4.65$ seconds
18.
- 96ft/s
  
  6 seconds
  
  6 seconds
  
  Never; the maximum height is 208ft.
19.

2
20.

5
21.

16
22.

Answers can vary; one solution is $a=-2$ , $b=7$
23.

22
24.

$-7$
25.

$0$
26.

Answers can vary; one solution is $a=-11$ , $b=18$
27.

This is a triangle with base $b$ and height $m b$ .
28.
29.

$1/4x^{4}-2/3x^{3}+7/2x^{2}-9x+C$
30.

$-\cos x-\sin x+\tan x+C$
31.

$3/4t^{4/3}-1/t+2^{t}/\ln 2+C$
32.

$\ln\left\lvert x\right\rvert+\csc x+C$

Exercises E.3

1.

limits
2.

14
3.

Rectangles.
4.

T
5.

$2^{2}+3^{2}+4^{2}=29$
6.

$-6-2+2+6+10=10$
7.

$0-1+0+1+0=0$
8.

$5+5+5+5+5+5+5+5+5+5=50$
9.

$1+1/2+1/3+1/4+1/5=137/60$
10.

$-1+2-3+4-5+6=3$
11.

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

$1+1+1+1+1+1=6$
13.

Answers may vary; $\sum_{i=1}^{5}3i$
14.

Answers may vary; $\sum_{i=0}^{8}(i^{2}-1)$
15.

Answers may vary; $\sum_{i=1}^{4}\frac{i}{i+1}$
16.

Answers may vary; $\sum_{i=0}^{4}(-1)^{i}e^{i}$
17.

50
18.

325
19.

1045
20.

28,650
21.

$-8525$
22.

$2050$
23.

$5050$
24.

$2870$
25.

$155$
26.

$91,225$
27.

$24$
28.

$11,700$
29.

$\displaystyle\int_{0}^{\pi}\frac{\sin x}{1+x}\operatorname{d}\!x$
30.

$\displaystyle\int_{2}^{5}x\sqrt{1+x^{3}}\operatorname{d}\!x$
31.

$\displaystyle\int_{2}^{7}5x^{3}-4x+7\operatorname{d}\!x$
32.

$\displaystyle\int_{1}^{3}\frac{x}{x^{2}+4}\operatorname{d}\!x$
33.

$\displaystyle\lim_{n\to\infty}\left[\frac{3}{n}\sum_{i=1}^{n}4-2\biggl(2+\frac% {3i}{n}\biggr)\right]$
34.

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

$\displaystyle\lim_{n\to\infty}\frac{\pi}{n}\sum_{i=1}^{n}\frac{\sin^{3}(-\pi/2% +\pi i/n)}{2+\cos(-\pi/2+\pi i/n)}$
36.

$\displaystyle\lim_{n\to\infty}\frac{2}{n}\sum_{i=1}^{n}e^{2i/n}$
37.

$19$
38.

$59/8$
39.

$\pi/3+\pi/(2\sqrt{3})\approx 1.954$
40.

$8.16986$
41.

$0.388584$
42.

$496/315\approx 1.5746$
43.
- Exact expressions will vary; $\frac{(1+n)^{2}}{4n^{2}}$ .
  
  $121/400$ , $10201/40000$ , $1002001/4000000$
  
  $1/4$
44.
- Exact expressions will vary; $2+4/n^{2}$ .
  
  $51/25$ , $5001/2500$ , $500001/250000$
  
  $2$
45.
- 8.
  
  8, 8, 8
  
  $8$
46.
- Exact expressions will vary; $20/3-96/(3n)+64/(3n^{2})$ .
  
  $92/25$ , $3968/625$ , $103667/15625$
  
  $20/3$
47.
- Exact expressions will vary; $100-200/n$ .
  
  $80$ , $98$ , $499/5$
  
  $100$
48.
- Exact expressions will vary; $-(1-1/n^{2})/12$ .
  
  $-33/400$ , $-3333/40000$ , $-333333/4000000$
  
  $-1/12$
49.
- Exact expressions will vary; $80.5$ .
  
  $72.25$
  
  $62.5$
50.
- $(5\text{ s})((0+6+14+23+30+36)\text{ mph})=545\frac{\text{mi s}}{\text{hr}}% \times\frac{1\text{ hr}}{3600\text{ s}}\times\frac{5280\text{ ft}}{1\text{ mi}% }=799\text{ ft}$
  
  $(5\text{ s})((6+14+23+30+36+40)\text{ mph})=585\frac{\text{mi s}}{\text{hr}}% \times\frac{1\text{ hr}}{3600\text{ s}}\times\frac{5280\text{ ft}}{1\text{ mi}% }=858\text{ ft}$

51.

$\displaystyle\int_{a}^{b}k\cdot f(x)\operatorname{d}\!x$	$\displaystyle=\lim_{n\to\infty}\sum_{i=1}^{n}k\cdot f(c_{i})\Delta x$	T5.3.2.2
	$\displaystyle=\lim_{n\to\infty}k\cdot\sum_{i=1}^{n}k\cdot f(c_{i})\Delta x$	T5.3.1.3
	$\displaystyle=k\cdot\lim_{n\to\infty}\sum_{i=1}^{n}k\cdot f(c_{i})\Delta x$	T1.3.1.4
	$\displaystyle=k\int_{a}^{b}f(x)\operatorname{d}\!x$	T5.3.2.2

52.

Let $f$ and $M$ be as given.

$\displaystyle\int_{a}^{b}f(x)\operatorname{d}\!x$ $\displaystyle=\lim_{n\to\infty}\sum_{i=1}^{n}f(c_{i})\Delta x$ T5.3.2.2

$\displaystyle\leq\lim_{n\to\infty}\sum_{i=1}^{n}M\Delta x$

$\displaystyle=\int_{a}^{b}M\operatorname{d}\!x$ T5.3.2.2

$\displaystyle=M(b-a)$
53.

$F(x)=5\tan x+4$
54.

$F(x)=7\ln\left\lvert x\right\rvert+14$
55.

$G(t)=4/6t^{6}-5/4t^{4}+8t+9$
56.

$G(t)=5\cdot e^{t}+900$
57.

$G(t)=\sin t-\cos t-78$
58.

$F(x)=2\sqrt{x}-\pi$

Exercises E.4

1.

Answers will vary.
2.

0
3.

T
4.

Answers will vary.
5.

20
6.

$28/3$
7.

$0$
8.

$1$
9.

$1$
10.

$1$
11.

$23/2$
12.

$-4$
13.

$e^{3}-e$
14.

$16/3$
15.

$4$
16.

$45/4$
17.

$\ln 2$
18.

$1/2$
19.

$1/4$
20.

$1/101$
21.

$15$
22.

$2-2/\sqrt{3}$
23.

$2$
24.

$\frac{7}{2}$
25.

$\frac{63}{2}$
26.

$\frac{88}{3}$
27.

$\frac{6\pi}{7}$
28.

$0$
29.

$\sqrt{2}$
30.

$1$
31.

$36$
32.

$\frac{1}{2}$
33.

$\frac{69}{4}$
34.

$\frac{8}{9}$
35.

Explanations will vary. A sketch will help.
36.

$\int_{a}^{a+2\pi}\sin t\operatorname{d}\!t=\cos(a+2\pi)-\cos(a)$ . Since cosine is periodic with period $2\pi$ , $\cos(a+2\pi)=\cos(a)$ , and hence the integral is 0.
37.

$c=2/\sqrt{3}$
38.

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

$c=\ln(e-1)\approx 0.54$
40.

$c=64/9\approx 7.1$
41.

$2/\pi$
42.

$2/pi$
43.

$2$
44.

$16/3$
45.

$16$
46.

$1/(e-1)$
47.

(a) $-300$ ft; (b) $312.5$ ft
48.

(a) $400$ ft; (b) $850$ ft
49.

(a) $-1$ ft; (b) 3ft
50.

(a) $128/5$ ft; (b) same
51.

$-64$ ft/s
52.

$50$ ft/s
53.

$2$ ft/s
54.

$0$ ft/s
55.

$F^{\prime}(x)=(3x^{2}+1)\frac{1}{x^{3}+x}$
56.

$F^{\prime}(x)=-3x^{11}$
57.

$F^{\prime}(x)=2x(x^{2}+2)-(x+2)$
58.

$F^{\prime}(x)=e^{x}\sin(e^{x})-\frac{1}{x}\sin(\ln x)$
59.

$F^{\prime}(x)=\frac{\ln x+4}{x^{2}+7}$
60.

$F^{\prime}(x)=[\cos^{3}(\sin x)+3\tan^{3}(\sin x)]\cos x$
61.

$F^{\prime}(x)=-\frac{15x^{2}\sqrt{\cos(5x^{3})+5}}{25x^{6}+e^{5x^{3}}}$
62.

$F^{\prime}(x)=-2\tan x\sec^{2}x[\ln(\tan^{2}x)+e^{\tan^{4}x-7}]$
63.
- x 0 1 2 3 4 5 6
  
  g(x) 0 .5 0 -.5 0 1.5 4
  
  $g(7)\approx 5.7$
  
  min at $x=3$ ; max at $x=7$
  
  Approximately
64.
- This is a consequence of Theorem 5.4.1.
  
  The derivative of the left is $g^{\prime}(xy)y=\frac{1}{xy}y$ . The derivative of the right is $g^{\prime}(x)=\frac{1}{x}$ . Theorem 5.1.1 tells us that the left and right therefore differ by a constant. Looking at $x=1$ and $g(1)=0$ tells us that this difference is 0.
  
  The derivative of the left is $g^{\prime}(x^{r})rx^{r-1}=\frac{1}{x^{r}}rx^{r-1}$ . The derivative of the right is $rg^{\prime}(x)=r\frac{1}{x}$ . Theorem 5.1.1 tells us that the left and right therefore differ by a constant. Looking at $x=1$ and $g(1)=0$ tells us that this difference is 0.
65.
- $b_{n}=4/n\pi$ for odd $n$ and $b_{n}=0$ for even $n$
  
  answers will vary

Exercises E.5

1.

Chain Rule.
2.

T
3.

$\frac{1}{8}(x^{3}-5)^{8}+C$
4.

$\frac{1}{4}(x^{2}-5x+7)^{4}+C$
5.

$\frac{1}{18}\left(x^{2}+1\right)^{9}+C$
6.

$\frac{1}{3}(3x^{2}+7x-1)^{6}+C$
7.

$\frac{1}{2}\ln\left\lvert 2x+7\right\rvert+C$
8.

$\sqrt{2x+3}+C$
9.

$\frac{2}{3}(x+3)^{3/2}-6(x+3)^{1/2}+C=\frac{2}{3}(x-6)\sqrt{x+3}+C$
10.

$\frac{2}{21}x^{3/2}\left(3x^{2}-7\right)+C$
11.

$2e^{\sqrt{x}}+C$
12.

$\frac{2\sqrt{x^{5}+1}}{5}+C$
13.

$-\frac{1}{2x^{2}}-\frac{1}{x}+C$
14.

$\frac{\ln^{2}(x)}{2}+C$
15.

$\frac{\sin^{3}(x)}{3}+C$
16.

$-\frac{\cos^{4}(x)}{4}+C$
17.

$-\frac{1}{6}\sin(3-6x)+C$
18.

$-\tan(4-x)+C$
19.

$\frac{1}{2}\ln\left\lvert\sec(2x)+\tan(2x)\right\rvert+C$
20.

$\frac{\sin\left(x^{2}\right)}{2}+C$
21.

$\tan(x)-x+C$
22.

The key is to rewrite $\cot x$ as $\cos x/\sin x$ , and let $u=\sin x$ .
23.

The key is to multiply $\csc x$ by 1 in the form $(\csc x+\cot x)/(\csc x+\cot x)$ .
24.

$\frac{1}{3}e^{3x-1}+C$
25.

$\frac{e^{x^{3}}}{3}+C$
26.

$\frac{1}{2}e^{(x-1)^{2}}+C$
27.

$x-e^{-x}+C$
28.

$\ln(e^{x}+1)+C$
29.

$\frac{e^{-3x}}{3}-e^{-x}+C$
30.

$\frac{1}{2}\ln^{2}(x)+C$
31.

$\frac{\bigl(\ln x\bigr)^{3}}{3}+C$
32.

$\frac{3}{2}\ln^{2}(x)+C$
33.

$\frac{1}{2}\ln\left\lvert\ln\left(x^{2}\right)\right\rvert+C$
34.

$\frac{x^{2}}{2}+3x+\ln\left\lvert x\right\rvert+C$
35.

$\frac{x^{3}}{3}+\frac{x^{2}}{2}+x+\ln\left\lvert x\right\rvert+C$
36.

$-\frac{1}{3\left(x^{3}+3\right)}+C$
37.

$\frac{1}{45}(5x^{3}+5x^{2}+2)^{9}+C$
38.

$-\sqrt{1-x^{2}}+C$
39.

$-\frac{1}{3}\cot\left(x^{3}+1\right)+C$
40.

$-\frac{2}{3}\cos^{\frac{3}{2}}(x)+C$
41.

$-\frac{1}{5}\cos(5x+1)+C$
42.

$\ln\left\lvert x-5\right\rvert+C$
43.

$\frac{7}{3}\ln\left\lvert 3x+2\right\rvert+C$
44.

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

$3\ln\left\lvert 3x^{2}+9x+7\right\rvert+C$
46.

$3\sqrt{x^{2}-2x-6}+C$
47.

$\sqrt{x^{2}-6x+8}+C$
48.

$2\sin\sqrt{x}+C$
49.

$\frac{1}{2}\sec^{2}\theta+C$ or $\frac{1}{2}\tan^{2}\theta+C$
50.

$\frac{1}{10}(2x+3)^{5/2}-\frac{1}{2}(2x+3)^{3/2}+C$
51.

$-\frac{1}{2(x^{2}+1)}+\frac{1}{4(x^{2}+1)^{2}}+C$
52.

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

$\frac{1}{11}(x^{3}+2)^{11}-\frac{2}{5}(x^{3}+2)^{10}+\frac{4}{9}(x^{3}+2)^{9}+C$
54.

$-3\cos(\frac{x}{3})+C$
55.

$\frac{2}{3}\sin^{6}(\frac{x}{4})+C$
56.

$\frac{2}{3}\sin(x^{3/2}+1)+C$
57.

$-\ln 2$
58.

$352/15$
59.

$2/3$
60.

$1/5$
61.

$(1-e)/2$
62.

$e-1$
63.

$0$
64.

$\ln\left(\frac{4}{1+e}\right)$
65.

$\frac{2}{3}$
66.

$\frac{1}{12}$

$\displaystyle\int_{a}^{b}f(x)\operatorname{d}\!x$	$\displaystyle=\lim_{n\to\infty}\sum_{i=1}^{n}f(c_{i})\Delta x$	T5.3.2.2
	$\displaystyle\leq\lim_{n\to\infty}\sum_{i=1}^{n}M\Delta x$
	$\displaystyle=\int_{a}^{b}M\operatorname{d}\!x$	T5.3.2.2
	$\displaystyle=M(b-a)$

x	0	1	2	3	4	5	6
g(x)	0	.5	0	-.5	0	1.5	4