Solutions To Selected Problems Chapter K Chapter M

Chapter L

Exercises L.1

1.

parametric equations
2.

vectors
3.

displacement
4.

displacement
5.
6.
7.
8.
9.
10.
11.
12.
13.

(fullscreen)
14.

(fullscreen)
15.

(fullscreen)
16.

(fullscreen)
17.

$\left\lVert\vec{r}(t)\right\rVert=\sqrt{t^{2}+t^{4}}=\left\lvert t\right\rvert% \sqrt{t^{2}+1}$ .
18.

$\left\lVert\vec{r}(t)\right\rVert=\sqrt{25\cos^{2}t+9\sin^{2}t}$ .
19.

$\left\lVert\vec{r}(t)\right\rVert=\sqrt{4\cos^{2}t+4\sin^{2}t+t^{2}}=\sqrt{t^{% 2}+4}$ .
20.

$\left\lVert\vec{r}(t)\right\rVert=\sqrt{\cos^{2}t+t^{2}+t^{4}}$ .
21.

Answers may vary, though most direct solution is

$\vec{r}(t)=\left\langle 2\cos t+1,2\sin t+2\right\rangle$ .
22.

Answers may vary; three solutions are

$\vec{r}(t)=\left\langle 3\sin t+5,3\cos t+5\right\rangle$ ,

$\vec{r}(t)=\left\langle-3\cos t+5,3\sin t+5\right\rangle$ and

$\vec{r}(t)=\left\langle 3\cos t+5,-3\sin t+5\right\rangle$ .
23.

Answers may vary, though most direct solution is

$\vec{r}(t)=\left\langle 1.5\cos t,5\sin t\right\rangle$ .
24.

Answers may vary, though most direct solutions are

$\vec{r}(t)=\left\langle-3\cos t+3,2\sin t-2\right\rangle$ ,

$\vec{r}(t)=\left\langle 3\cos t+3,-2\sin t-2\right\rangle$ and

$\vec{r}(t)=\left\langle 3\sin t+3,2\cos t-2\right\rangle$ .
25.

Answers may vary, though most direct solutions are

$\vec{r}(t)=\left\langle t,5(t-2)+3\right\rangle$ and

$\vec{r}(t)=\left\langle t+2,5t+3\right\rangle$ .
26.

Answers may vary, though most direct solutions are

$\vec{r}(t)=\left\langle t,-1/2(t-1)+5\right\rangle$ ,

$\vec{r}(t)=\left\langle t+1,-1/2t+5\right\rangle$ ,

$\vec{r}(t)=\left\langle-2t+1,t+5\right\rangle$ and

$\vec{r}(t)=\left\langle 2t+1,-t+5\right\rangle$ .
27.

Answers may vary, though most direct solution is

$\vec{r}(t)=\left\langle 2\cos t,2\sin t,2t\right\rangle$ .
28.

Answers may vary, though most direct solution is

$\vec{r}(t)=\left\langle 3\cos(4\pi t),3\sin(4\pi t),3t\right\rangle$ .
29.

Answers may vary, though most direct solution is $\vec{r}(t)=\left\langle R\cos(2\pi Nt),R\sin(2\pi Nt),ht\right\rangle$ .
30.

Answers may vary, though most direct solution is $\vec{r}(t)=\left\langle\sqrt{5}\cos(2\pi t),2,\sqrt{5}\sin(2\pi t)\right\rangle$ .
31.

$\left\langle 1,0\right\rangle$
32.

$\left\langle 1,1\right\rangle$
33.

$\left\langle 0,0,1\right\rangle$
34.

$\left\langle 1,2,7\right\rangle$

Exercises L.2

1.

component
2.

displacement
3.

It is difficult to identify the points on the graphs of $\vec{r}(t)$ and $\vec{r}\thinspace^{\prime}(t)$ that correspond to each other.
4.

A scalar-vector product, a dot product and a cross product.
5.

$\left\langle 11,74,\sin 5\right\rangle$
6.

$\left\langle e^{3},0\right\rangle$
7.

$\left\langle 1,e\right\rangle$
8.

$\left\langle 2t,1,0\right\rangle$
9.

$(-\infty,0)\bigcup(0,\infty)$
10.

$(0,\infty)$
11.

$\vec{r}\thinspace^{\prime}(t)=\left\langle-\sin t,e^{t},1/t\right\rangle$
12.

$\vec{r}\thinspace^{\prime}(t)=\left\langle-1/t^{2},5/(3t+1)^{2},\sec^{2}t\right\rangle$
13.

$\vec{r}\thinspace^{\prime}(t)=(2t)\left\langle\sin t,2t+5\right\rangle+(t^{2})% \left\langle\cos t,2\right\rangle=\left\langle 2t\sin t+t^{2}\cos t,6t^{2}+10t\right\rangle$
14.

$r\mkern 1.35mu^{\prime}(t)=\left\langle 2t,1\right\rangle\cdot\left\langle\sin t% ,2t+5\right\rangle+\left\langle t^{2}+1,t-1\right\rangle\cdot\left\langle\cos t% ,2\right\rangle=$

$(t^{2}+1)\cos t+2t\sin t+4t+3$
15.

$\vec{r}\thinspace^{\prime}(t)=\left\langle 2t,1,0\right\rangle\times\left% \langle\sin t,2t+5,1\right\rangle+\left\langle t^{2}+1,t-1,1\right\rangle% \times\left\langle\cos t,2,0\right\rangle=$

$\left\langle-1,\cos t-2t,6t^{2}+10t+2+\cos t-\sin t-t\cos t\right\rangle$
16.

$\vec{r}\thinspace^{\prime}(t)=\left\langle\sinh t,\cosh t\right\rangle$
17.

$\vec{r}\thinspace^{\prime}(t)=\left\langle 2t+1,2t-1\right\rangle$
18.

$\vec{r}\thinspace^{\prime}(t)=\left\langle 2t-2,3t^{2}-6t+2\right\rangle$
19.

$\vec{r}\thinspace^{\prime}(t)=\left\langle 2t,3t^{2}-1\right\rangle$
20.

$\vec{r}\thinspace^{\prime}(t)=\left\langle 2t-4,3t^{2}-12t+11\right\rangle$
21.

$\ell(t)=\left\langle 2,0\right\rangle+t\left\langle 3,1\right\rangle$
22.

$\ell(t)=\left\langle 3\sqrt{2}/2,\sqrt{2}/2\right\rangle+t\left\langle-3\sqrt{% 2}/2,\sqrt{2}/2\right\rangle$
23.

$\ell(t)=\left\langle-3,0,\pi\right\rangle+t\left\langle 0,-3,1\right\rangle$
24.

$\ell(t)=\left\langle 1,0,0\right\rangle+t\left\langle 1,1,1\right\rangle$
25.

$t=2n\pi$ , where $n$ is an integer;

so $t=\dotsc-4\pi,-2\pi,0,2\pi,4\pi,\dotsc$
26.

$t=1$
27.

$\vec{r}(t)$ is not smooth at $t=3\pi/4+n\pi$ , where $n$ is an integer
28.

$t=\pm 1$
29.

Both derivatives return $\left\langle 5t^{4},4t^{3}-3t^{2},3t^{2}\right\rangle$ .
30.

Both derivatives return $2\sin t+t^{2}\cos t+te^{t}+1$ .
31.

Both derivatives return

$\langle 2t-e^{t}-1,\cos t-3t^{2},$

$(t^{2}+2t)e^{t}-(t-1)\cos t-\sin t\rangle$ .
32.

Both derivatives return $\left\langle 6t^{5},3t^{2},0\right\rangle$
33.

$\left\langle\frac{1}{4}t^{4},\sin t,te^{t}-e^{t}\right\rangle+\vec{C}$
34.

$\left\langle\tan^{-1}t,\tan t\right\rangle+\vec{C}$
35.

$\left\langle-2,0\right\rangle$
36.

$\left\langle 4,-4\right\rangle$
37.

$\vec{r}(t)=\left\langle\frac{1}{2}t^{2}+2,-\cos t+3\right\rangle$
38.

$\vec{r}(t)=\left\langle\ln\left\lvert t+1\right\rvert+1,-\ln\left\lvert\cos t% \right\rvert+2\right\rangle$
39.

$\vec{r}(t)=\left\langle t^{4}/12+t+4,\ t^{3}/6+2t+5,\ t^{2}/2+3t+6\right\rangle$
40.

$\vec{r}(t)=\left\langle-\cos t+1,t-\sin t,e^{t}-t-1\right\rangle$
41.

$2\sqrt{13}\pi$
42.

$10\pi$
43.

$\frac{1}{54}\left((22)^{3/2}-8\right)$
44.

$\sqrt{2}(1-e^{-1})$
45.

12
46.

$2\pi\sqrt{2+16\pi^{2}}+\ln(2\sqrt{2}\pi+\sqrt{1+8\pi^{2}})$

47.

As $\vec{r}(t)$ has constant length, $\vec{r}(t)\cdot\vec{r}(t)=c^{2}$ for some constant $c$ . Thus

	$\displaystyle\vec{r}(t)\cdot\vec{r}(t)$	$\displaystyle=c^{2}$
	$\displaystyle\frac{\operatorname{d}\!}{\operatorname{d}\!t}\big(\vec{r}(t)% \cdot\vec{r}(t)\big)$	$\displaystyle=\frac{\operatorname{d}\!}{\operatorname{d}\!t}\big(c^{2}\big)$
	$\displaystyle\vec{r}\thinspace^{\prime}(t)\cdot\vec{r}(t)+\vec{r}(t)\cdot\vec{% r}\thinspace^{\prime}(t)$	$\displaystyle=0$
	$\displaystyle 2\vec{r}(t)\cdot\vec{r}\thinspace^{\prime}(t)$	$\displaystyle=0$
	$\displaystyle\vec{r}(t)\cdot\vec{r}\thinspace^{\prime}(t)$	$\displaystyle=0.$

48.

$\sqrt{h^{2}+4\pi^{2}R^{2}N^{2}}$

Exercises L.3

1.

Velocity is a vector, indicating an objects direction of travel and its rate of distance change (i.e., its speed). Speed is a scalar.
2.

Displacement is a vector, indicating the difference between the starting and ending positions of an object. Distance traveled is a scalar, indicating the arc length of the path followed.
3.

The average velocity is found by dividing the displacement by the time traveled — it is a vector. The average speed is found by dividing the distance traveled by the time traveled — it is a scalar.
4.

arc length
5.

One example is traveling at a constant speed $s$ in a circle, ending at the starting position. Since the displacement is $\vec{0}$ , the average velocity is $\vec{0}$ , hence $\left\lVert\vec{0}\right\rVert=0$ . But traveling at constant speed $s$ means the average speed is also $s>0$ .
6.

Distance traveled is always greater than or equal to the magnitude of displacement, therefore average speed will always be at least as large as the magnitude of the average velocity.
7.

$\vec{v}(t)=\left\langle 2,5,0\right\rangle$ , $\vec{a}(t)=\left\langle 0,0,0\right\rangle$
8.

$\vec{v}(t)=\left\langle 6t-2,-2t+1\right\rangle$ , $\vec{a}(t)=\left\langle 6,-2\right\rangle$
9.

$\vec{v}(t)=\left\langle-\sin t,\cos t\right\rangle$ , $\vec{a}(t)=\left\langle-\cos t,-\sin t\right\rangle$
10.

$\vec{v}(t)=\left\langle 1/10,\sin t,\cos t\right\rangle$ , $\vec{a}(t)=\left\langle 0,\cos t,-\sin t\right\rangle$
11.

$\vec{v}(t)=\left\langle 1,\cos t\right\rangle$ , $\vec{a}(t)=\left\langle 0,-\sin t\right\rangle$
12.

$\vec{v}(t)=\left\langle 2t,2t\cos\left(t^{2}\right)\right\rangle$ ,

$\vec{a}(t)=\left\langle 2,2\left(\cos\left(t^{2}\right)-2t^{2}\sin\left(t^{2}% \right)\right)\right\rangle$
13.

$\vec{v}(t)=\left\langle 2t+1,-2t+2\right\rangle$ , $\vec{a}(t)=\left\langle 2,-2\right\rangle$
14.

$\vec{v}(t)=\left\langle-\frac{2\left(t^{2}+3t-1\right)}{\left(t^{2}+1\right)^{% 2}},2t\right\rangle$ , $\vec{a}(t)=\left\langle\frac{2\left(2t^{3}+9t^{2}-6t-3\right)}{\left(t^{2}+1% \right)^{3}},2\right\rangle$
15.

$\left\lVert\vec{v}(t)\right\rVert=\sqrt{4t^{2}+1}$ .

Min at $t=0$ ; Max at $t=\pm 1$ .
16.

$\left\lVert\vec{v}(t)\right\rVert=\left\lvert t\right\rvert\sqrt{9t^{2}-12t+8}$ .

min: $t=0$ ; max: $t=-1$
17.

$\left\lVert\vec{v}(t)\right\rVert=5$ .

Speed is constant, so there is no difference between min/max
18.

$\left\lVert\vec{v}(t)\right\rVert=\sqrt{4\sin^{2}t+25\cos^{2}t}$ .

min: $t=\pi/2,\ 3\pi/2$ ; max: $t=0,\ 2\pi$
19.

$\left\lVert\vec{v}(t)\right\rVert=\left\lvert\sec t\right\rvert\sqrt{\tan^{2}t% +\sec^{2}t}$ .

min: $t=0$ ; max: $t=\pi/4$
20.

$\left\lVert\vec{v}(t)\right\rVert=\sqrt{2-2\sin t}$ .

min: $t=\pi/2$ ; max: $t=3\pi/2$
21.

$\left\lVert\vec{v}(t)\right\rVert=13$ .

speed is constant, so there is no difference between min/max
22.

$\left\lVert\vec{v}(t)\right\rVert=\sqrt{8t^{2}+3}$ .

min: $t=0$ ; max: $t=1$
23.

$\left\lVert\vec{v}(t)\right\rVert=\sqrt{4t^{2}+1+t^{2}/(1-t^{2})}$ .

min: $t=0$ ; max: there is no max; speed approaches $\infty$ as $t\to\pm 1$
24.

$\left\lVert\vec{v}(t)\right\rVert=\sqrt{g^{2}t^{2}-(2gv_{0}\sin\theta)t+v_{0}^% {2}}$ .

min: $t=(v_{0}\sin\theta)/g$ ; max: $t=0$ , $t=(2v_{0}\sin\theta)/g$
25.
- $\vec{r}_{1}(1)=\left\langle 1,1\right\rangle$ ; $\vec{r}_{2}(1)=\left\langle 1,1\right\rangle$
  
  $\vec{v}_{1}(1)=\left\langle 1,2\right\rangle$ ; $\left\lVert\vec{v}_{1}(1)\right\rVert=\sqrt{5}$ ; $\vec{a}_{1}(1)=\left\langle 0,2\right\rangle$
  $\vec{v}_{2}(1)=\left\langle 2,4\right\rangle$ ; $\left\lVert\vec{v}_{2}(1)\right\rVert=2\sqrt{5}$ ; $\vec{a}_{2}(1)=\left\langle 2,12\right\rangle$
26.
- $\vec{r}_{1}(\pi/2)=\left\langle 0,3\right\rangle$ ; $\vec{r}_{2}(\pi/8)=\left\langle 0,3\right\rangle$
  
  $\vec{v}_{1}(\pi/2)=\left\langle-3,0\right\rangle$ ; $\left\lVert\vec{v}_{1}(\pi/2)\right\rVert=3$ ; $\vec{a}_{1}(\pi/2)=\left\langle 0,-3\right\rangle$
  $\vec{v}_{2}(\pi/8)=\left\langle-12,0\right\rangle$ ; $\left\lVert\vec{v}_{2}(\pi/8)\right\rVert=12$ ; $\vec{a}_{2}(\pi/8)=\left\langle 0,-48\right\rangle$
27.
- $\vec{r}_{1}(2)=\left\langle 6,4\right\rangle$ ; $\vec{r}_{2}(2)=\left\langle 6,4\right\rangle$
  
  $\vec{v}_{1}(2)=\left\langle 3,2\right\rangle$ ; $\left\lVert\vec{v}_{1}(2)\right\rVert=\sqrt{13}$ ; $\vec{a}_{1}(2)=\left\langle 0,0\right\rangle$
  $\vec{v}_{2}(2)=\left\langle 6,4\right\rangle$ ; $\left\lVert\vec{v}_{2}(2)\right\rVert=2\sqrt{13}$ ; $\vec{a}_{2}(2)=\left\langle 0,0\right\rangle$
28.
- $\vec{r}_{1}(1)=\left\langle 1,1\right\rangle$ ; $\vec{r}_{2}(\pi/2)=\left\langle 1,1\right\rangle$
  
  $\vec{v}_{1}(1)=\left\langle 1,1/2\right\rangle$ ; $\left\lVert\vec{v}_{1}(1)\right\rVert=\sqrt{5}/2$ ; $\vec{a}_{1}(1)=\left\langle 0,-1/4\right\rangle$
  $\vec{v}_{2}(\pi/2)=\left\langle 0,0\right\rangle$ ; $\left\lVert\vec{v}_{2}(\pi/2)\right\rVert=0$ ; $\vec{a}_{2}(\pi/2)=\left\langle-1,-1/2\right\rangle$
29.

$\vec{v}(t)=\left\langle 2t+1,3t+2\right\rangle$ ,

$\vec{r}(t)=\left\langle t^{2}+t+5,3t^{2}/2+2t-2\right\rangle$
30.

$\vec{v}(t)=\left\langle 2t-1,3t-1\right\rangle$ ,

$\vec{r}(t)=\left\langle t^{2}-t+5,3t^{2}/2-t-5/2\right\rangle$
31.

$\vec{v}(t)=\left\langle\sin t,\cos t\right\rangle$ , $\vec{r}(t)=\left\langle 1-\cos t,\sin t\right\rangle$
32.

$\vec{v}(t)=\left\langle 10,-32t+50\right\rangle$ , $\vec{r}(t)=\left\langle 10t,-16t^{2}+50t\right\rangle$
33.

Displacement: $\left\langle 0,0,6\pi\right\rangle$ ; distance traveled: $2\sqrt{13}\pi\approx 22.65$ ft; average velocity: $\left\langle 0,0,3\right\rangle$ ; average speed: $\sqrt{13}\approx 3.61$ ft/s
34.

Displacement: $\left\langle-10,0\right\rangle$ ; distance traveled: $5\pi\approx 15.71$ ft; average velocity: $\left\langle-10/\pi,0\right\rangle\approx\left\langle-3.18,0\right\rangle$ ; average speed: $5$ ft/s
35.

Displacement: $\left\langle 0,0\right\rangle$ ; distance traveled: $2\pi\approx 6.28$ ft; average velocity: $\left\langle 0,0\right\rangle$ ; average speed: $1$ ft/s
36.

Displacement: $\left\langle 10,20,-20\right\rangle$ ; distance traveled: $30$ ft; average velocity: $\left\langle 1,2,-2\right\rangle$ ; average speed: $3$ ft/s
37.

At $t$ -values of $\sin^{-1}(9/30)/(4\pi)+n/2\approx 0.024+n/2$ seconds, where $n$ is an integer.
38.
The stone, while whirling, can be modeled by $\vec{r}(t)=\left\langle 3\cos(8\pi t),3\sin(8\pi t)\right\rangle$ .
- For $t$ -values $t=\sin^{-1}(3/20)/(8\pi)+n/4\approx 0.006+n/4$ , where $n$ is an integer.
  
  $\left\lVert\vec{r}\thinspace^{\prime}(t)\right\rVert=24\pi\approx 51.4$ ft/s
  
  At $t=0.006$ , the stone is approximately $19.77$ ft from Goliath. Using the formula for projectile motion, we want the angle of elevation that lets a projectile starting at $\left\langle 0,6\right\rangle$ with a initial velocity of $51.4$ ft/s arrive at $\left\langle 19.77,9\right\rangle$ . The desired angle is $0.27$ radians, or $15.69^{\circ}$ .
39.
- Holding the crossbow at an angle of $0.013$ radians, $\approx 0.745^{\circ}$ will hit the target $0.4$ s later. (Another solution exists, with an angle of $89^{\circ}$ , landing $18.75$ s later, but this is impractical.)
  
  In the .4 seconds the arrow travels, a deer, traveling at 20mph or 29.33ft/s, can travel 11.7ft. So she needs to lead the deer by 11.7ft.
40.
The position function of the ball is

$\vec{r}(t)=\left\langle(146.67\cos\theta)t,-16t^{2}+(146.67\sin\theta)t+3\right\rangle$ ,

where $\theta$ is the angle of elevation.
- With $\theta=20^{\circ}$ , the ball reaches 310ft from home plate in 2.25 seconds; at this time, the height of the ball is 34.9ft, not enough to clear the Green Monster.
  
  With $\theta=21^{\circ}$ , the ball reaches 310ft from home plate in 2.26s, with a height of 40ft, clearing the wall.
41.

The position function is $\vec{r}(t)=\left\langle 220t,-16t^{2}+1000\right\rangle$ . The $y$ -component is 0 when $t=7.9$ ; $\vec{r}(7.9)=\left\langle 1739.25,0\right\rangle$ , meaning the box will travel about 1740ft horizontally before it lands.
42.
The position function of the ball is

$\vec{r}(t)=\left\langle(v_{0}\cos\theta)t,-16t^{2}+(v_{0}\sin\theta)t+6\right\rangle$ ,

where $\theta$ is the angle of elevation and $v_{0}$ is the initial ball speed.
- With $v_{0}=73.33$ ft/s, there are two angles of elevation possible. An angle of $\theta=9.47^{\circ}$ delivers the ball in 0.83s, while an angle of $79.57^{\circ}$ delivers the ball in $4.5$ s.
  
  With $\theta=8^{\circ}$ , the initial speed must be 53.8mph $\approx 78.9$ ft/s.

Exercises L.4

1.

1
2.

0
3.

$\vec{T}(t)$ and $\vec{N}(t)$ .
4.

the speed
5.

$\vec{T}(t)=\left\langle\frac{4t}{\sqrt{20t^{2}-4t+1}},\frac{2t-1}{\sqrt{20t^{2% }-4t+1}}\right\rangle$ ;

$\vec{T}(1)=\left\langle 4/\sqrt{17},1/\sqrt{17}\right\rangle$
6.

$\vec{T}(t)=\left\langle\frac{1}{\sqrt{1+\sin^{2}t}},-\frac{\sin t}{\sqrt{1+% \sin^{2}t}}\right\rangle$ ;

$\vec{T}(\pi/4)=\left\langle\sqrt{2/3},-1/\sqrt{3}\right\rangle$
7.

$\vec{T}(t)=\frac{\cos t\sin t}{\sqrt{\cos^{2}t\sin^{2}t}}\left\langle-\cos t,% \sin t\right\rangle$ . (Be careful; this cannot be simplified as $\left\langle-\cos t,\sin t\right\rangle$ because $\sqrt{\cos^{2}t\sin^{2}t}\neq\cos t\sin t$ , but rather $\left\lvert\cos t\sin t\right\rvert$ .) $\vec{T}(\pi/4)=\left\langle-\sqrt{2}/2,\sqrt{2}/2\right\rangle$
8.

$\vec{T}(t)=\left\langle-\sin t,\cos t\right\rangle$ ; $\vec{T}(\pi)=\left\langle 0,-1\right\rangle$
9.

$\ell(t)=\left\langle 2,0\right\rangle+t\left\langle 4/\sqrt{17},1/\sqrt{17}\right\rangle$ ; in parametric form,

$\ell(t)=\begin{cases}x=2+4t/\sqrt{17}\\ y=t/\sqrt{17}\end{cases}$
10.

$\ell(t)=\left\langle\pi/4,\sqrt{2}/{2}\right\rangle+t\left\langle\sqrt{2/3},-1% /\sqrt{3}\right\rangle$ ; in parametric form,

$\ell(t)=\begin{cases}x=\pi/4+\sqrt{2/3}t\\ y=\sqrt{2}/2-t/\sqrt{3}\end{cases}$
11.

$\ell(t)=\left\langle\sqrt{2}/4,\sqrt{2}/4\right\rangle+t\left\langle-\sqrt{2}/% 2,\sqrt{2}/2\right\rangle$ ; in parametric form,

$\ell(t)=\begin{cases}x=\sqrt{2}/4-\sqrt{2}t/2\\ y=\sqrt{2}/4+\sqrt{2}t/2\end{cases}$
12.

$\ell(t)=\left\langle-1,0\right\rangle+t\left\langle 0,-1\right\rangle$ ; in parametric form,

$\ell(t)=\begin{cases}x=-1\\ y=-t\end{cases}$
13.

$\vec{T}(t)=\left\langle-\sin t,\cos t\right\rangle$ ; $\vec{N}(t)=\left\langle-\cos t,-\sin t\right\rangle$
14.

$\vec{T}(t)=\left\langle\frac{1}{\sqrt{1+4t^{2}}},\frac{2t}{\sqrt{1+4t^{2}}}\right\rangle$ ; $\vec{N}(t)=\left\langle-\frac{2t}{\sqrt{1+4t^{2}}},\frac{1}{\sqrt{1+4t^{2}}}\right\rangle$
15.

$\vec{T}(t)=\left\langle-\frac{\sin t}{\sqrt{4\cos^{2}t+\sin^{2}t}},\frac{2\cos t% }{\sqrt{4\cos^{2}t+\sin^{2}t}}\right\rangle$ ;

$\vec{N}(t)=\left\langle-\frac{2\cos t}{\sqrt{4\cos^{2}t+\sin^{2}t}},-\frac{% \sin t}{\sqrt{4\cos^{2}t+\sin^{2}t}}\right\rangle$
16.

$\vec{T}(t)=\left\langle\frac{e^{t}}{\sqrt{e^{2t}+e^{-2t}}},-\frac{e^{-t}}{% \sqrt{e^{2t}+e^{-2t}}}\right\rangle$ ;

$\vec{N}(t)=\left\langle\frac{e^{-t}}{\sqrt{e^{2t}+e^{-2t}}},\frac{e^{t}}{\sqrt% {e^{2t}+e^{-2t}}}\right\rangle$
17.
- Be sure to show work
  
  $\vec{N}(\pi/4)=\left\langle-5/\sqrt{34},-3/\sqrt{34}\right\rangle$
18.
- Be sure to show work
  
  $\vec{N}(1)=\left\langle\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}}\right\rangle$
19.
- Be sure to show work
  
  $\vec{N}(0)=\left\langle-\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}}\right\rangle$
20.
- Be sure to show work
  
  $\vec{N}(\pi/4)=\left\langle\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right\rangle$
21.

$\vec{T}(t)=\frac{1}{\sqrt{5}}\left\langle 2,\cos t,-\sin t\right\rangle$ ; $\vec{N}(t)=\left\langle 0,-\sin t,-\cos t\right\rangle$
22.

$\vec{T}(t)=\left\langle-\sin t,3/5\cos t,4/5\cos t\right\rangle$ ;

$\vec{N}(t)=\left\langle-\cos t,-3/5\sin t,-4/5\sin t\right\rangle$
23.

$\vec{T}(t)=\frac{1}{\sqrt{a^{2}+b^{2}}}\left\langle-a\sin t,a\cos t,b\right\rangle$ ;

$\vec{N}(t)=\left\langle-\cos t,-\sin t,0\right\rangle$
24.

$\vec{T}(t)=\frac{1}{\sqrt{a^{2}+1}}\left\langle-a\sin(at),a\cos(at),1\right\rangle$ ;

$\vec{N}(t)=\left\langle-\cos t,-\sin t,0\right\rangle$
25.

$a_{\text{T}}=\frac{4t}{\sqrt{1+4t^{2}}}$ and $a_{\text{N}}=\sqrt{4-\frac{16t^{2}}{1+4t^{2}}}$

At $t=0$ , $a_{\text{T}}=0$ and $a_{\text{N}}=2$ ;

At $t=1$ , $a_{\text{T}}=4/\sqrt{5}$ and $a_{\text{N}}=2/\sqrt{5}$ .

At $t=0$ , all acceleration comes in the form of changing the direction of velocity and not the speed; at $t=1$ , more acceleration comes in changing the speed than in changing direction.
26.

$a_{\text{T}}=\frac{-2/t^{5}}{\sqrt{1+1/t^{4}}}$ and $a_{\text{N}}=\sqrt{\frac{4}{t^{6}}-\frac{4/t^{10}}{1+1/t^{4}}}$

At $t=1$ , $a_{\text{T}}=\sqrt{2}$ and $a_{\text{N}}=-\sqrt{2}$ ;

At $t=2$ , $a_{\text{T}}=-\frac{1}{4\sqrt{17}}$ and $a_{\text{N}}=\frac{1}{\sqrt{17}}$ .

At $t=1$ , acceleration comes from changing speed and changing direction in “equal measure;” at $t=2$ , acceleration is nearly $\vec{0}$ as it is; the low value of $a_{\text{T}}$ shows that the speed is nearly constant and the low value of $a_{\text{N}}$ shows the direction is not changing quickly.
27.

$a_{\text{T}}=0$ and $a_{\text{N}}=2$

At $t=0$ , $a_{\text{T}}=0$ and $a_{\text{N}}=2$ ;

At $t=\pi/2$ , $a_{\text{T}}=0$ and $a_{\text{N}}=2$ .

The object moves at constant speed, so all acceleration comes from changing direction, hence $a_{\text{T}}=0$ . $\vec{a}(t)$ is always parallel to $\vec{N}(t)$ , but twice as long, hence $a_{\text{N}}=2$ .
28.

$a_{\text{T}}=2$ and $a_{\text{N}}=4t^{2}$

At $t=\sqrt{\pi/2}$ , $a_{\text{T}}=2$ and $a_{\text{N}}=2\pi$ ;

At $t=\sqrt{\pi}$ , $a_{\text{T}}=2$ and $a_{\text{N}}=4\pi$ .

The object moves at increasing speed (increasing at a constant rate of acceleration), hence $a_{\text{T}}=2$ . Since the object is increasing speed yet always traveling in a circle of radius 1, the direction must change more quickly; the amount of acceleration that changes direction increases over time.
29.

$a_{\text{T}}=0$ and $a_{\text{N}}=a$

At $t=0$ , $a_{\text{T}}=0$ and $a_{\text{N}}=a$ ;

At $t=\pi/2$ , $a_{\text{T}}=0$ and $a_{\text{N}}=a$ .

The object moves at constant speed, meaning that $a_{\text{T}}$ is always 0. The object “rises” along the $z$ -axis at a constant rate, so all acceleration comes in the form of changing direction circling the $z$ -axis. The greater the radius of this circle the greater the acceleration, hence $a_{\text{N}}=a$ .
30.

$a_{\text{T}}=0$ and $a_{\text{N}}=5$

At $t=0$ , $a_{\text{T}}=0$ and $a_{\text{N}}=5$ ;

At $t=\pi/2$ , $a_{\text{T}}=0$ and $a_{\text{N}}=5$ .

The object moves at constant speed, meaning that $a_{\text{T}}$ is always 0. Acceleration is thus always perpendicular to the direction of travel; in this particular case, it is always 5 times the unit vector pointing orthogonal to the direction of travel.

Exercises L.5

1.

time and/or distance
2.

curvature
3.

Answers may include lines, circles, helixes
4.

Answers will vary; they should mention the circle is tangent to the curve and has the same curvature as the curve at that point.
5.

$\kappa$
6.

$a_{\text{T}}$ is not affected by curvature; the greater the curvature, the larger $a_{\text{N}}$ becomes.
7.

$s=3t$ , so $\vec{r}(s)=\left\langle 2s/3,s/3,-2s/3\right\rangle$
8.

$s=7t$ , so $\vec{r}(s)=\left\langle 7\cos(s/7),7\sin(s/7)\right\rangle$
9.

$s=\sqrt{13}t$ , so

$\vec{r}(s)=\left\langle 3\cos(s/\sqrt{13}),3\sin(s/\sqrt{13}),2s/\sqrt{13}\right\rangle$
10.

$s=13t$ , so

$\vec{r}(s)=\left\langle 5\cos(s/13),13\sin(s/13),12\cos(s/13)\right\rangle$
11.

$\kappa=\frac{\left\lvert 6x\right\rvert}{\left(1+(3x^{2}-1)^{2}\right)^{3/2}}$ ;

$\kappa(0)=0$ , $\kappa(1/2)=\frac{192}{17\sqrt{17}}\approx 2.74$ .
12.

$\kappa=\frac{\left\lvert\frac{6x^{2}-2}{(x^{2}+1)^{3}}\right\rvert}{\left(1+% \frac{4x^{2}}{(x^{2}+1)^{4}}\right)^{3/2}}$ ;

$\kappa(0)=2$ , $\kappa(2)=\frac{2750}{641\sqrt{641}}\approx 0.169$ .
13.

$\kappa=\frac{\left\lvert\cos x\right\rvert}{\left(1+\sin^{2}x\right)^{3/2}}$ ;

$\kappa(0)=1$ , $\kappa(\pi/2)=0$
14.

$\kappa=1$ ;

$\kappa(0)=1$ , $\kappa(1/2)=1$
15.

$\kappa=\frac{\left\lvert 2\cos t\cos(2t)+4\sin t\sin(2t)\right\rvert}{\left(4% \cos^{2}(2t)+\sin^{2}t\right)^{3/2}}$ ;

$\kappa(0)=1/4$ , $\kappa(\pi/4)=8$
16.

$\kappa=2$ ;

$\kappa(0)=2$ , $\kappa(\pi/3)=2$
17.

$\kappa=\frac{\left\lvert 6t^{2}+2\right\rvert}{\left(4t^{2}+(3t^{2}-1)^{2}% \right)^{3/2}}$ ;

$\kappa(0)=2$ , $\kappa(5)=\frac{19}{1394\sqrt{1394}}\approx 0.0004$
18.

$\kappa=\frac{\left\lvert\sec^{3}t\right\rvert}{\left(\sec^{4}t+\sec^{2}t\tan^{% 2}t\right)^{3/2}}$ ;

$\kappa(0)=1$ , $\kappa(\pi/6)=\frac{3\sqrt{3}}{5\sqrt{5}}\approx 0.465$
19.

$\kappa=0$ ;

$\kappa(0)=0$ , $\kappa(1)=0$
20.

$\kappa=\frac{2\sqrt{18t^{4}+15t^{2}+1}}{\left(18t^{4}-2t^{2}+1\right)^{3/2}}$ ;

$\kappa(0)=2$ , $\kappa(1)=2\sqrt{2}/17\approx 0.166378$
21.

$\kappa=\frac{3}{13}$ ;

$\kappa(0)=3/13$ , $\kappa(\pi/2)=3/13$
22.

$\kappa=\frac{1}{13}$ ;

$\kappa(0)=1/13$ , $\kappa(\pi/2)=1/13$
23.

maximized at $x=\pm\frac{\sqrt{2}}{\sqrt[4]{5}}$
24.

maximized at $x=\dotsc-3\pi/2,-\pi/2,\pi/2,\dotsc$
25.

maximized at $t=1/4$
26.

maximized at $t=\pm\sqrt{5}$
27.

radius of curvature is $5\sqrt{5}/4$ .
28.

radius of curvature is $5\sqrt{10}$ .
29.

radius of curvature is $9$ .
30.

radius of curvature is $1/45$ .
31.

$x^{2}+(y-1/2)^{2}=1/4$ , or

$\vec{c}(t)=\left\langle 1/2\cos t,1/2\sin t+1/2\right\rangle$
32.

$(x-8/3)^{2}+y^{2}=1/9$ , or $\vec{c}(t)=\left\langle\frac{1}{3}\cos t+\frac{8}{3},\frac{1}{3}\sin t\right\rangle$
33.

$x^{2}+(y+8)^{2}=81$ , or $\vec{c}(t)=\left\langle 9\cos t,9\sin t-8\right\rangle$
34.

$(x-1/2)^{2}+(y-1/2)^{2}=1/2$ , or

$\vec{c}(t)=\left\langle\frac{\sqrt{2}}{2}\cos t+\frac{1}{2},\frac{\sqrt{2}}{2}% \sin t+\frac{1}{2}\right\rangle$
35.

Let $\vec{r}(t)=\left\langle x(t),y(t),0\right\rangle$ and apply the second formula of part 3.
36.

Let $\vec{r}(t)=\left\langle t,f(t),0\right\rangle$ and apply part 2 or the second formula of part 3.
37.

Using Theorem 12.5.2 part 2, we have $\kappa(t)=\frac{\left\lvert f(t)^{2}+2f\,^{\prime}(t)^{2}-f\,^{\prime\prime}(t% )f(t)\right\rvert}{(f(t)^{2}+f\,^{\prime}(t)^{2})^{3/2}}$