Solutions To Selected Problems Chapter I Chapter K

Chapter J

Exercises J.0

1.

$y=\frac{1}{2}(x-3)^{2}+\frac{3}{2}$
2.

$y=\frac{-1}{12}(x+1)^{2}-1$
3.

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

$x=y^{2}$
5.

$y=-\frac{1}{4}(x-1)^{2}+2$
6.

$x=-\frac{1}{12}y^{2}$
7.

$y=4x^{2}$
8.

$x=-\frac{1}{8}(y-3)^{2}+2$
9.
10.
11.

$\frac{(x+1)^{2}}{9}+\frac{(y-2)^{2}}{4}=1$
12.

$\frac{(x-1)^{2}}{1/4}+\frac{y^{2}}{9}=1$
13.

$\frac{(x-1)^{2}}{2}+(y-2)^{2}=1$
14.

$\frac{x^{2}}{3}+\frac{y^{2}}{5}=1$
15.

$\frac{x^{2}}{4}+\frac{(y-3)^{2}}{6}=1$
16.

$\frac{(x-2)^{2}}{4}+\frac{(y-2)^{2}}{4}=1$
17.

$x^{2}-\frac{y^{2}}{3}=1$
18.

$y^{2}-\frac{x^{2}}{24}=1$
19.

$\frac{(y-3)^{2}}{4}-\frac{(x-1)^{2}}{9}=1$
20.

$\frac{(x-1)^{2}}{9}-\frac{(y-3)^{2}}{4}=1$
21.
22.
23.

$\frac{x^{2}}{4}-\frac{y^{2}}{3}=1$
24.

$\frac{x^{2}}{3}-\frac{(y-1)^{2}}{9}=1$
25.

$(y-2)^{2}-\frac{x^{2}}{10}=1$
26.

$4y^{2}-\frac{x^{2}}{4}=1$

Exercises J.1

1.

T
2.

F
3.

$\sqrt{2}$
4.

$6$
5.

$4/3$
6.

$6$
7.

$109/2$
8.

$3/2$
9.

$12/5$
10.

$79953333/400000\approx 199.883$
11.

$-\ln(2-\sqrt{3})\approx 1.31696$
12.

$\sinh^{-1}1$
13.

$\int_{0}^{1}\sqrt{1+4x^{2}}\operatorname{d}\!x$
14.

$\int_{0}^{1}\sqrt{1+100x^{18}}\operatorname{d}\!x$
15.

$\int_{0}^{1}\sqrt{1+\frac{1}{4x}}\operatorname{d}\!x$
16.

$\int_{1}^{e}\sqrt{1+\frac{1}{x^{2}}}\operatorname{d}\!x$
17.

$\int_{-1}^{1}\sqrt{1+\frac{x^{2}}{1-x^{2}}}\operatorname{d}\!x$
18.

$\int_{-3}^{3}\sqrt{1+\frac{x^{2}}{81-9x^{2}}}\operatorname{d}\!x$
19.

$\int_{1}^{2}\sqrt{1+\frac{1}{x^{4}}}\operatorname{d}\!x$
20.

$\int_{-\pi/4}^{\pi/4}\sqrt{1+\sec^{2}x\tan^{2}x}\operatorname{d}\!x$
21.

$1.4790$
22.

$1.8377$
23.

Simpson’s Rule fails, as it requires one to divide by 0. However, recognize the answer should be the same as for $y=x^{2}$ ; why?
24.

$2.1300$
25.

Simpson’s Rule fails.
26.

Simpson’s Rule fails.
27.

$1.4058$
28.

$1.7625$
29.

$2\pi\int_{0}^{1}2x\sqrt{5}\operatorname{d}\!x=2\pi\sqrt{5}$
30.

$2\pi\int_{0}^{1}x^{3}\sqrt{1+9x^{4}}\operatorname{d}\!x=\pi/27(10\sqrt{10}-1)$
31.

$2\pi\int_{0}^{1}\sqrt{x}\sqrt{1+1/(4x)}\operatorname{d}\!x=\pi/6(5\sqrt{5}-1)$
32.

$2\pi\int_{0}^{1}\sqrt{1-x^{2}}\sqrt{1+x/(1-x^{2})}\operatorname{d}\!x=4\pi$

Exercises J.2

1.

T
2.

orientation
3.

rectangular
4.

Answers will vary.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
- Traces the parabola $y=x^{2}$ , moves from left to right.
  
  Traces the parabola $y=x^{2}$ , but only from $-1\leq x\leq 1$ ; traces this portion back and forth infinitely.
  
  Traces the parabola $y=x^{2}$ , but only for $0<x$ . Moves left to right.
  
  Traces the parabola $y=x^{2}$ , moves from right to left.
20.
- Traces a circle of radius $1$ counterclockwise once.
  
  Traces a circle of radius $1$ counterclockwise over $6$ times.
  
  Traces a circle of radius $1$ clockwise infinite times.
  
  Traces an arc of a circle of radius $1$ , from an angle of $-1$ radians to $1$ radian, twice.
21.

Possible Answer: $x=t$ , $y=9-4t$
22.

Possible Answer: $x=\frac{5+t^{2}}{4}$ , $y=t$
23.

Possible Answer: $x=-9+7\cos t$ , $y=4+7\sin t$
24.

Possible Answer: $x=2+5\sec t$ , $y=-3+5\tan t$
25.

Possible Answer: $x=\frac{5}{4}t+\frac{11}{4}$ , $y=t$ , $[-3,1]$
26.

Possible Answer: $x=-1+4t$ , $y=3-5t$ , $[0,1]$
27.

Possible Answer: $x=t$ , $y=t^{2}+2t$ , $(-\infty,-1]$
28.

Possible Answer: $x=2t-t^{2}$ , $y=1-t$ , $[1,\infty)$
29.

$x=(t+11)/6$ , $y=(t^{2}-97)/12$ . At $t=1$ , $x=2$ , $y=-8$ .

$y^{\prime}=6x-11$ ; when $x=2$ , $y^{\prime}=1$ .
30.

$x=\ln t$ , $y=t$ . At $t=1$ , $x=0$ , $y=1$ .

$y^{\prime}=e^{x}$ ; when $x=0$ , $y^{\prime}=1$ .
31.

$x=\cos^{-1}t$ , $y=\sqrt{1-t^{2}}$ . At $t=1$ , $x=0$ , $y=0$ .

$y^{\prime}=\cos x$ ; when $x=0$ , $y^{\prime}=1$ .
32.

$x=1/(4t^{2})$ , $y=1/(2t)$ . At $t=1$ , $x=1/4$ , $y=1/2$ .

$y^{\prime}=1/(2\sqrt{x})$ ; when $x=1/4$ , $y^{\prime}=1$ .
33.
Possible answers:
- $x=\sin t$ , $y=\cos t$ , $[\pi/2,5\pi/2]$
  
  $x=\cos t$ , $y=\sin t$ , $[0,2\pi]$
  
  $x=\sin t$ , $y=\cos t$ , $[\pi/2,9\pi/2]$
  
  $x=\cos t$ , $y=\sin t$ , $[0,4\pi]$
34.
Possible Answers:
- $x=a\sin t,y=b\cos t,[\pi/2,5\pi/2]$
  
  $x=a\cos t,y=b\sin t,[0,2\pi]$
  
  $x=a\sin t,y=b\cos t,[\pi/2,9\pi/2]$
  
  $x=a\cos t,y=b\sin t,[0,4\pi]$
35.

$x=4t$ , $y=-16t^{2}+64t$
36.

$x=50t$ , $y=-16t^{2}+64t$
37.

$x=10t$ , $y=-16t^{2}+320t$
38.

$x=2\cos t$ , $y=-2\sin t$ ; other answers possible
39.

$x=3\cos(2\pi t)+1$ , $y=3\sin(2\pi t)+1$ ; other answers possible
40.

$x=\cos t+1$ , $y=3\sin t+3$ ; other answers possible
41.

$x=5\cos t$ , $y=\sqrt{24}\sin t$ ; other answers possible
42.

$x=\pm\sec t+2$ , $y=\sqrt{8}\tan t-3$ ; other answers possible
43.

$x=2\tan t$ , $y=\pm 6\sec t$ ; other answers possible
44.

$x=10t-2\sin t$ , $y=10-2\cos t$ ; other answers possible
45.

$y=-1.5x+8.5$
46.

$x^{2}-y^{2}=1$
47.

$\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{9}=1$
48.

$y=x^{3/2}$
49.

$y=2x+3$
50.

$y=x^{3}-3$
51.

$y=e^{2x}-1$
52.

$y^{2}-x^{2}=1$
53.

$x^{2}-y^{2}=1$
54.

$x=1-2y^{2}$
55.

$y=\frac{b}{a}(x-x_{0})+y_{0}$ ; line through $(x_{0},y_{0})$ with slope $b/a$ .
56.

$x^{2}+y^{2}=r^{2}$ ; circle centered at $(0,0)$ with radius $r$ .
57.

$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$ ; ellipse centered at $(h,k)$ with horizontal axis of length $2a$ and vertical axis of length $2b$ .
58.

$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$ ; hyperbola centered at $(h,k)$ with horizontal transverse axis and asymptotes with slope $b/a$ . The parametric equations only give half of the hyperbola. When $a>0$ , the right half; when $a<0$ , the left half.
59.

$t=\pm 1$
60.

$t=-1,\ 2$
61.

$t=\pi/2,3\pi/2$
62.

$t=\pi/6,\pi/2,5\pi/6$
63.

$t=-1$
64.

$t=2$
65.

$t=k\pi$ for integer values of $k$
66.

$t=\dotsc 0,\ 2\pi,\ 4\pi,\ \dotsc$

Exercises J.3

1.

F
2.

$t$
3.

F
4.

T
5.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=2t$
  
  Tangent line: $y=2(x-1)+1$ ; normal line: $y=-1/2(x-1)+1$
6.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=10\sqrt{t}$
  
  Tangent line: $y=20(x-2)+22$ ; normal line: $y=-1/20(x-2)+22$
7.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{2t+1}{2t-1}$
  
  Tangent line: $y=3x+2$ ; normal line: $y=-1/3x+2$
8.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{3t^{2}-1}{2t}$
  
  $t=0$ : Tangent line: $x=-1$ ; normal line: $y=0$ $t=1$ : Tangent line: $y=x$ ; normal line: $y=-x$
9.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\csc t$
  
  $t=\pi/4$ : Tangent line: $y=\sqrt{2}(x-\sqrt{2})+1$ ; normal line: $y=-1/\sqrt{2}(x-\sqrt{2})+1$
10.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=-2\cos(2t)\csc t$
  
  $t=\pi/4$ : Tangent line: $y=1$ ; normal line: $x=\sqrt{2}/2$
11.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{\cos t\sin(2t)+2\sin t% \cos(2t)}{-\sin t\sin(2t)+2\cos t\cos(2t)}$
  
  Tangent line: $y=x-\sqrt{2}$ ; normal line: $y=-x$
12.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{\sin(t)+10\cos(t)}{\cos(% t)-10\sin(t)}$
  
  Tangent line: $y=-x/10+e^{\pi/20}$ ; normal line: $y=10x+e^{\pi/20}$
13.

horizontal: $t=0$ ; vertical: none
14.

horizontal: $t=0$ ; vertical: none (though this uses a one-sided limit, as $x(t)$ is not defined for $t<0$ .
15.

horizontal: $t=-1/2$ ; vertical: $t=1/2$
16.

horizontal: $t=\pm 1/\sqrt{3}$ ; vertical: $t=\pm 1$
17.

horizontal: none; vertical: $t=0$
18.

horizontal: $t=\pi/4,3\pi/4,5\pi/4,7\pi/4$ ; vertical: $t=0,\pi,2\pi$
19.

The solution is non-trivial; use identities $\sin(2t)=2\sin t\cos t$ and $\cos(2t)=\cos^{2}t-\sin^{2}t=1-2\sin^{2}t$ to rewrite $\operatorname{d}\!y/\operatorname{d}\!t=2\sin t(2\cos^{2}t-\sin^{2}t)$ and $\operatorname{d}\!x/\operatorname{d}\!t=2\cos t(1-3\sin^{2}t)$ . Horizontal: $\sin t=0$ when $t=0,\pi,2\pi$ , and $2\cos^{2}t-\sin^{2}t=0$ when $t=\tan^{-1}(\sqrt{2}),\ \pi\pm\tan^{-1}(\sqrt{2}),\ 2\pi-\tan^{-1}(\sqrt{2})$ . Vertical: $\cos t=0$ when $t=\pi/2,3\pi/2$ , and $1-3\sin^{2}t=0$ when $t=\sin^{-1}(1/\sqrt{3}),\pi-\sin^{-1}(1/\sqrt{3})$ .
20.

horizontal: $t=\tan^{-1}(-10)$ , $\tan^{-1}(-10)+\pi$ ; vertical: $t=\tan^{-1}(1/10)-\pi$ , $\tan^{-1}(1/10)$
21.

$t_{0}=0$ ; $\lim_{t\to 0}\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=0$ .
22.

$t_{0}=2$ ; $\lim_{t\to 2}\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=1$ .
23.

$t_{0}=1$ ; $\lim_{t\to 1}\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\infty$ .
24.

$t_{0}=\dotsc,-\pi/2,0,\pi/2,\pi,\dotsc$ ; $\lim_{t\to 0}\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=1$ .
25.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=2$ ; always concave up
26.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=10$ ; always concave up
27.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=-\frac{4}{(2t-1)^{3}}$ ; concave up on $(-\infty,1/2)$ ; concave down on $(1/2,\infty)$ .
28.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{3t^{2}+1}{4t^{3}}$ ; concave down on $(-\infty,0)$ ; concave up on $(0,\infty)$ .
29.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=-\cot^{3}t$ ; concave up on $(-\pi/2,0)$ ; concave down on $(0,\pi/2)$ .
30.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{\cos t\sin(2t)+2% \sin t\cos(2t)}{(-\sin t\sin(2t)+2\cos t\cos(2t))^{2}}$ ; concavity switches at

$t=\tan^{-1}(\frac{1}{\sqrt{2}}),\ \frac{\pi}{2},\ \pi-\tan^{-1}(\frac{1}{\sqrt% {2}}),\pi+\tan^{-1}(\frac{1}{\sqrt{2}}),\ \frac{3\pi}{2},\ 2\pi-\tan^{-1}(% \frac{1}{\sqrt{2}})$
31.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{4(13+3\cos(4t))}% {(\cos t+3\cos(3t))^{3}}$ , obtained with a computer algebra system; concave up on $\big(-\tan^{-1}(\frac{1}{\sqrt{2}}),\tan^{-1}(\frac{1}{\sqrt{2}})\big)$ , concave down on $\big(-\frac{\pi}{2},-\tan^{-1}(\frac{1}{\sqrt{2}})\big)$ ; $\big(\tan^{-1}(\frac{1}{\sqrt{2}}),\frac{\pi}{2}\big)$
32.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{1010}{e^{t/10}% \big(\cos t-10\sin t\big)^{3}}$ ; concavity switches at

$t=\tan^{-1}(1/10)+n\pi$ , where $n$ is an integer.
33.

$L=6\pi$
34.

On $[0,2\pi]$ , arc length is $L=\sqrt{101}(e^{\pi/5}-1)$ ; on $[2\pi,4\pi]$ , $L=\sqrt{101}(e^{2\pi/5}-1)$ .
35.

$L=2\sqrt{34}$
36.

$L=4\sqrt{2}-2$
37.

$2\pi$
38.

$4\sqrt{2}-2$
39.

$-\dfrac{\sqrt{10}}{3}+\ln(3+\sqrt{10})+\sqrt{2}-\ln(1+\sqrt{2})$
40.

$e^{3}+11-e^{-8}$
41.

$L\approx 2.4416$ (actual value: $L=2.42211$ )
42.

$L\approx 9.73004$ (actual value: $L=9.42943$ )
43.

$L\approx 4.19216$ (actual value: $L=4.18308$ )
44.

Formula: $C\approx 25.9062$ ; Simpson’s Rule: $C\approx 25.4786$ (actual value: $C=25.527$ )
45.

The answer is $16\pi$ for both (of course), but the integrals are different.
46.

$8\pi^{2}$ .
47.

$\frac{6\pi a^{2}}{5}$
48.

$\frac{24\pi(949\sqrt{26}+1)}{5}$
49.

$SA\approx 8.50101$ (actual value $SA=8.02851$
50.

$SA\approx 1.36751$ (actual value $SA=1.36707$
51.

$\frac{1}{2}\sinh\theta\cosh\theta-\frac{1}{2}\theta$

Exercises J.4

1.

Answers will vary.
2.

F
3.

T
4.

F
5.
6.
7.

$A(2.5,\pi/4)$ and $A(-2.5,5\pi/4)$ ;

$B(-1,5\pi/6)$ and $B(1,11\pi/6)$ ;

$C(3,4\pi/3)$ and $C(-3,\pi/3)$ ;

$D(1.5,2\pi/3)$ and $D(-1.5,5\pi/3)$
8.

$A(2,\pi/6)$ and $A(-2,-5\pi/6)$ ;

$B(1,-\pi/3)$ and $B(-1,2\pi/3)$ ;

$C(2,3\pi/4)$ and $C(-2,-\pi/4)$ ;

$D(2.5,\pi)$ and $D(2.5,-\pi)$
9.

$A=(\sqrt{2},\sqrt{2})$ ;

$B=(\sqrt{2},-\sqrt{2})$ ;

$C=(\sqrt{5},-0.46)$ ;

$D=(\sqrt{5},2.68)$
10.

$A=(-3,0)$ ;

$B=(-1/2,\sqrt{3}/2)$ ;

$C=(4,\pi/2)$ ;

$D=(2,-\pi/3)$
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.

$(x-1)^{2}+y^{2}=1$
34.

$x^{2}+(y+2)^{2}=4$
35.

$x^{2}+(y-\frac{3}{2})^{2}=\frac{9}{4}$
36.

$(x+\frac{3}{4})^{2}+y^{2}=\frac{9}{16}$
37.

$(x-1/2)^{2}+(y-1/2)^{2}=1/2$
38.

$y=2/5x+7/5$
39.

$x=3$
40.

$y=4$
41.

$x^{4}+x^{2}y^{2}-y^{2}=0$
42.

$y^{4}+x^{2}y^{2}-x^{2}=0$
43.

$x^{2}+y^{2}=4$
44.

$y=x/\sqrt{3}$
45.

$\theta=\pi/4$
46.

$r=7/(\sin\theta-4\cos\theta)$
47.

$r=5\sec\theta$
48.

$r=5\csc\theta$
49.

$r=\cos\theta/\sin^{2}\theta$
50.

$r=1/\sqrt[3]{\cos^{2}\theta\sin\theta}$
51.

$r=\sqrt{7}$
52.

$r=-2\cos\theta$
53.

$P(\sqrt{3}/2,\pi/6)$ , $P(0,\pi/2)$ , $P(-\sqrt{3}/2,5\pi/6)$
54.

$P(1,0)$ , $P(0,\pi/2)=P(0,\pi/4)$ , $P(-1/2,2\pi/3)$
55.

$P(0,0)=P(0,\pi/2)$ , $P(\sqrt{2},\pi/4)$
56.

$P(\sqrt{3}/2,\pi/3)=P(-\sqrt{3}/2,4\pi/3)$ , $P(\sqrt{3}/2,2\pi/3)=P(-\sqrt{3}/2,5\pi/3)$ , $P(0,\pi/2)$
57.

$P(\sqrt{2}/2,\pi/12)$ , $P(-\sqrt{2}/2,5\pi/12)$ , $P(\sqrt{2}/2,3\pi/4)$ , and the origin.
58.

$P(3/2,\pi/3)$ , $P(3/2,-\pi/3)$
59.

For all points, $r=1$ ;

$\theta=\frac{\pi}{12},\ \frac{5\pi}{12},\ \frac{7\pi}{12},\ \frac{11\pi}{12},% \ \frac{13\pi}{12},\ \frac{17\pi}{12},\ \frac{19\pi}{12},\ \frac{23\pi}{12}$ .
60.

$P(0,0)=P(0,3\pi/2)$ , $P(1+\sqrt{2}/2,3\pi/4)$ , $P(1-\sqrt{2}/2,7\pi/4)$
61.

Answers will vary. If $m$ and $n$ do not have any common factors, then an interval of $2n\pi$ is needed to sketch the entire graph.
62.

Answers will vary.

Exercises J.5

1.

Using $x=r\cos\theta$ and $y=r\sin\theta$ , we can write $x=f(\theta)\cos\theta$ , $y=f(\theta)\sin\theta$ .
2.

rectangles; sectors of circles
3.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=-\cot\theta$
  
  tangent line: $y=-(x-\sqrt{2}/2)+\sqrt{2}/2$ ; normal line: $y=x$
4.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=1/2(\tan\theta-\cot\theta)$
  
  tangent line: $y=1/2$ ; normal line: $x=1/2$
5.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{\cos\theta(1+2\sin\theta% )}{\cos^{2}\theta-\sin\theta(1+\sin\theta)}$
  
  tangent line: $x=3\sqrt{3}/4$ ; normal line: $y=3/4$
6.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{3\sin^{2}(t)+(1-3\cos(t)% )\cos(t)}{3\sin(t)\cos(t)-\sin(t)(1-3\cos(t))}$
  
  tangent line: $y=\frac{1}{1+3\sqrt{2}}(x+(1/\sqrt{2}+3/2))+1/\sqrt{2}+3/2\approx y=0.19(x+2.2% 1)+2.21$ ; normal line: $y=-(1+3\sqrt{2})(x+(1/\sqrt{2}+3/2))+1/\sqrt{2}+3/2$
7.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{\theta\cos\theta+\sin% \theta}{\cos\theta-\theta\sin\theta}$
  
  tangent line: $y=-(2/\pi)x+\pi/2$ ; normal line: $y=(\pi/2)x+\pi/2$
8.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{\cos\theta\cos(3\theta)-% 3\sin\theta\sin(3\theta)}{-\cos(3\theta)\sin\theta-3\cos\theta\sin(3\theta)}$
  
  tangent line: $y=x/\sqrt{3}$ ; normal line: $y=-\sqrt{3}x$
9.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{4\sin(\theta)\cos(4% \theta)+\sin(4\theta)\cos(\theta)}{4\cos(\theta)\cos(4\theta)-\sin(\theta)\sin% (4\theta)}$
  
  tangent line: $y=5\sqrt{3}(x+\sqrt{3}/4)-3/4$ ; normal line: $y=-1/5\sqrt{3}(x+\sqrt{3}/4)-3/4$
10.
- $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=1$
  
  tangent line: $y=x+1$ ; normal line: $y=-x-1$
11.

horizontal: $\theta=\pi/2,3\pi/2$ ;

vertical: $\theta=0,\pi,2\pi$
12.

horizontal: $\theta=0,\pi/2,\pi$ ;

vertical: $\theta=\pi/4,3\pi/4$
13.

horizontal: $\theta=\tan^{-1}(1/\sqrt{5}),\ \pi/2,$

$\pi-\tan^{-1}(1/\sqrt{5}),\ \pi+\tan^{-1}(1/\sqrt{5}),\ 3\pi/2,\ 2\pi-\tan^{-1% }(1/\sqrt{5})$ ;

vertical: $\theta=0,\ \tan^{-1}(\sqrt{5}),\ \pi-\tan^{-1}(\sqrt{5}),\ \pi,\ \pi+\tan^{-1}% (\sqrt{5}),\ 2\pi-\tan^{-1}(\sqrt{5})$
14.

horizontal: $\theta=\pi/3,\ 5\pi/3$ ;

vertical: $\theta=0,\ 2\pi/3,\ 4\pi/3,\ 2\pi$

At $\theta=\pi$ , $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=0/0$ ; apply L’Hôpital’s Rule to find that $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}\rightarrow 0$ as $\theta\rightarrow\pi$ .
15.

In polar: $\theta=0\ \cong\ \theta=\pi$

In rectangular: $y=0$
16.

In polar: $\theta=\frac{\pi}{6}$ , $\theta=\frac{\pi}{2}$ , and $\theta=-\frac{\pi}{6}$

In rectangular: $y=\sqrt{3}x$ , $x=0$ , and $y=-\sqrt{3}x$ .
17.

In polar: $\theta=\frac{\pi}{4}$ and $\theta=-\frac{\pi}{4}$

In rectangular: $y=x$ and $y=-x$ .
18.

In polar: $\theta=0\ \cong\ \theta=\pi$ and $\theta=\frac{\pi}{2}$

In rectangular: $y=0$ and $x=0$
19.

area = $\frac{4\pi}{3}+2\sqrt{3}$
20.

area = $25\pi$
21.

area = $\pi/12$
22.

area = $\pi/(4n)$
23.

area = $3\pi/2$
24.

area = $\pi-3\sqrt{3}/2$
25.

area = $2\pi+3\sqrt{3}/2$
26.

area = $\pi+3\sqrt{3}$
27.

area = $1$
28.

area = $\displaystyle\int_{\pi/12}^{\pi/3}\frac{1}{2}\sin^{2}(3\theta)\operatorname{d}% \!\theta-\int_{\pi/12}^{\pi/6}\frac{1}{2}\cos^{2}(3\theta)\operatorname{d}\!% \theta=\frac{1}{12}+\frac{\pi}{24}$
29.

area = $\frac{1}{32}(4\pi-3\sqrt{3})$
30.

area = $\displaystyle\int_{0}^{\pi/3}\frac{1}{2}(1-\cos\theta)^{2}\operatorname{d}\!% \theta+\int_{\pi/3}^{\pi/2}\frac{1}{2}(\cos\theta)^{2}\operatorname{d}\!\theta% =\frac{7\pi}{24}-\frac{\sqrt{3}}{2}\approx 0.0503$
31.

$x\,^{\prime}(\theta)=f\,^{\prime}(\theta)\cos\theta-f(\theta)\sin\theta$ , $y\,^{\prime}(\theta)=f\,^{\prime}(\theta)\sin\theta+f(\theta)\cos\theta$ . Square each and add; applying the Pythagorean Theorem twice achieves the result.
32.

$4\pi$
33.

$4\pi$
34.

area = $\pi\sqrt{2}$
35.

$L\approx 2.2592$ ; (actual value $L=2.22748$ )
36.

$L\approx 7.62933$ ; (actual value $L=8$ )
37.

$SA=16\pi$
38.

$SA=4\pi$
39.

$SA=32\pi/5$
40.

$SA=4\pi^{2}$
41.

$SA=36\pi$
42.

$SA=9\pi$