4.E: Chapter 4 Review Exercises

12.1: Vector-Valued Functions

Terms and Concepts

1. Vector-valued functions are closely related to _______ ________ of graphs.

2. When sketching vector-valued functions, technically one isn't graphing points, but rather _______.

3. It can be useful to think of _______ as a vector that points from a starting position to an ending position.

Problems

In Exercises 4-11, sketch the vector-valued function on the given interval.

4. $\vec{r}(t) = \langle t^2,t^2-1 \rangle, \text{ for }-2 \le t \le 2$.

5. $\vec{r}(t) = \langle t^2,t^3 \rangle, \text{ for }-2 \le t \le 2$.

6. $\vec{r}(t) = \langle 1/t,t/t^2 \rangle, \text{ for }-2 \le t \le 2$.

7. $\vec{r}(t) = \langle \frac{1}{10}t^2,\sin t \rangle, \text{ for }-2\pi \le t \le 2\pi$.

8. $\vec{r}(t) = \langle \frac{1}{10}t^2,\sin t \rangle, \text{ for }-2\pi \le t \le 2\pi$.

9. $\vec{r}(t) = \langle 3\sin (\pi t),2\cos (\pi t) \rangle, \text{ on }[0,2]$.

10. $\vec{r}(t) = \langle 3\cos t,2\sin (2t) \rangle, \text{ on }[0,2\pi]$.

11. $\vec{r}(t) = \langle 2\sec t,\tan t \rangle, \text{ on }[-\pi,\pi]$.

In Exercises 12-15, sketch the vector-valued function on the given interval in $\mathbb{R}^3$. Technology may be useful in creating the sketch.

12. $\vec{r}(t) =\langle 2\cos t, t,2\sin t \rangle , \text{ on }[0,2\pi]$.

13. $\vec{r}(t) =\langle 3\cos t, \sin t,t/\pi \rangle , \text{ on }[0,2\pi]$.

14. $\vec{r}(t) =\langle \cos t,\sin t,\sin t \rangle , \text{ on }[0,2\pi]$.

15. $\vec{r}(t) =\langle \cos t,\sin t,\sin (2t) \rangle , \text{ on }[0,2\pi]$.

In Exercises 16-19, find $\lVert \vec{r}(t) \rVert$.

16. $\vec{r}(t) =\langle t,t^2 \rangle$.

17. $\vec{r}(t) =\langle 5\cos t, 3\sin t \rangle$.

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

19. $\vec{r}(t) =\langle \cos t, t, t^2 \rangle$.

In Exercises 20-27, create a vector-valued function whose graph matches the given description.

20. A circle of radius 2, centered $(1,2)$, traced counter-clockwise once on $[0,2\pi ]$.

21. A circle of radius 3, centered $(5,5)$, traced clockwise once on $[0,2\pi ]$.

22. An ellipse, centered at $(0,0)$ with vertical major axis length 10 and minor axis of length 3, traced once counter-clockwise on $[0,2\pi]$.

23. An ellipse, centered at $(3,-2)$ with horizontal major axis length 6 and minor axis of length 4, traced once clockwise on $[0,2\pi]$.

24. A line through $(2,3)$ with a slope of 5.

25. A line through $(1,5)$ with a slope of -1/2.

26. A vertically oriented helix with radius of 2 that starts at $(2,0,0)$ and ends at $(2,0,4\pi)$ after 1 revolution on $[0,2\pi]$.

27. A vertically oriented helix with radius of 3 that starts at $(3,0,0)$ and ends at $(3,0,3)$ after 2 revolutions on $[0,1]$.

In Exercises 28-31, find the average rate of change of $\vec{r}(t)$ on the given interval.

28. $\vec{r}(t) = \langle t,t^2 \rangle \text{ on }[-2,2].$

29. $\vec{r}(t) = \langle t,t+\sin t \rangle \text{ on }[0,2\pi].$

30. $\vec{r}(t) = \langle 3\cos t,2\sin t, t \rangle \text{ on }[0,2\pi].$

31. $\vec{r}(t) = \langle t,t^2,t^3 \rangle \text{ on }[-1,3].$

12.2: Calculus and Vector-Valued Functions

Terms and Concepts

1. Limits, derivatives and integrals of vector-valued functions are all evaluated _______ -wise.

2. The definite integral of a rate of change function gives ________.

3. Why is it generally not useful to graph both $\vec{r}(t)\text{ and }\vec{r}'(t)$ on the same axes?

Problems

In Exercises 4-7, evaluate the given limit.

4. $\lim\limits_{t\to 5}\langle 2t+1, 3t^2-1,\sin t \rangle$

5. $\lim\limits_{t\to 3}\langle e^t, \frac{t^2-9}{t+3} \rangle$

6. $\lim\limits_{t\to 0}\langle \frac{t}{\sin t},(1+t)^{\frac{1}{t}} \rangle$

7. $\lim\limits_{h\to 0}\frac{\vec{r}(t+h)-\vec{r}(t)}{h},\text{ where }\vec{r}(t) = \langle t^2,t,1 \rangle$

In Exercises 8-9, identify the interval(s) on which $\vec{r}(t)$ is continuous.

8. $\vec{r}(t) =\langle t^2, 1/t \rangle$

9. $\vec{r}(t) =\langle \cos t, e^t, \ln t \rangle$

In Exercises 10-14, find the derivative of the given function.

10. $\vec{r}(t) =\langle \cos t,e^t,\ln t \rangle$

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

12. $\vec{r}(t) =(t^2) \langle \sin t, 2t+5\rangle$

13. $\vec{r}(t) =\langle t^2+1,t-1 \rangle \cdot \langle \sin t,2t+5 \rangle$

14. $\vec{r}(t) =\langle t^2+1,t-1,1 \rangle \times \langle \sin t, 2t+5,1\rangle$

In Exercises 15-18, find $\vec{r}'(t)$. Sketch $\vec{r}(t)\text{ and }\vec{r}'(1)$, with the initial point of $\vec{r}'(1)\text{ at }\vec{r}(1)$.

15. $\vec{r}(t) =\langle t^2+t,t^2-t\rangle$

16. $\vec{r}(t) =\langle t^2-2t+2,t^3-3t^2+2t\rangle$

17. $\vec{r}(t) =\langle t^2+1,t^3-t\rangle$

18. $\vec{r}(t) =\langle t^2-4t+5, t^3-6t^2+11t-6\rangle$

In Exercises 19-22, give the equation of the line tangent to the graph of $\vec{r}(t)$ at the given t value.

19. $\vec{r}(t) =\langle t^2+t,t^2-t\rangle\text{ at }t=1.$

20. $\vec{r}(t) =\langle3\cos t, \sin t\rangle\text{ at }t=\pi/4.$

21. $\vec{r}(t) =\langle 3\cos t, 3\sin t, t\rangle\text{ at }t=\pi.$

22. $\vec{r}(t) =\langle e^t,\tan t, t\rangle\text{ at }t=0.$

In Exercises 23-26, find the value(s) of t for which $\vec{r}(t)$ is not smooth.

23. $\vec{r}(t)=\langle \cos t, \sin t -t\rangle$

24. $\vec{r}(t)=\langle t^2-2t+1,t^3+t^2-5t+3\rangle$

25. $\vec{r}(t)=\langle \cos t-\sin t, -\cos t, \cos (4t)\rangle$

26. $\vec{r}(t)=\langle t^3-3t+2, -\cos (\pi t),\sin^2 (\pi t)\rangle$

Exercises 27-29 ask you to verify parts of Theorem 92. In each let $f(t)=t^3,\vec{r}(t) = \langle t^2, t-1, 1 \rangle$ and $\vec{s}(t)=\langle \sin t, e^t, t \rangle$. Compute the various derivatives as indicated.

27. Simplify $f(t)\vec{r}(t)$, then find its derivative; show this is the same as $f'(t)\vec{r}'(t) +f(t)\vec{r}'(t)$.

28. Simplify $\vec{r}(t)\cdot\vec{s}(t)$, then find its derivative; show this is the same as $\vec{r}'(t)\cdot \vec{s}'(t) +\vec{r}(t)\cdot \vec{s}'(t)$.

29. Simplify $\vec{r}(t)\times \vec{s}(t)$, then find its derivative; show this is the same as $\vec{r}'(t)\times \vec{s}'(t) +\vec{r}(t)\times \vec{s}'(t)$.

In Exercises 30-33, evaluate the given definite or indefinite integral.

30. $\int \langle t^3,\cos t, te^t \rangle \,dt$

31. $\int \left \langle \frac{1}{1+t^2},\sec^2 t \right \rangle \,dt$

32. $\int_0^{\pi} \langle -\sin t, \cos t \rangle \,dt$

33. $\int_{-2}^{2} \langle 2t+1,2t-1 \rangle \,dt$

In Exercises 34-37, solve the given initial value problems.

34. Find $\vec{r}(t)$, given that $\vec{r}'(t)=\langle t,\sin t \rangle$ and $\vec{r}(0) =\langle 2,2 \rangle$.

35. Find $\vec{r}(t)$, given that $\vec{r}'(t)=\langle 1,(t+1),\tan t \rangle$ and $\vec{r}(0) =\langle 1,2 \rangle$.

36. Find $\vec{r}(t)$, given that $\vec{r}''(t)=\langle t^2,t,1 \rangle$, $\vec{r}'(0) =\langle 1,2,3 \rangle \text{ and }\vec{r}(0)=\langle 4,5,6 \rangle$.

37. Find $\vec{r}(t)$, given that $\vec{r}''(t)=\langle \cos t, \sin t, e^t \rangle$, $\vec{r}'(0) =\langle 0,0,0 \rangle \text{ and }\vec{r}(0)=\langle 0,0,0 \rangle$.

In Exercises 38-41, find the arc length of $\vec{r}(t)$ on the indicated interval.

38. $\vec{r}(t)=\langle 2\cos t,2\sin t,3t \rangle \text{ on }[0,2\pi]$

39. $\vec{r}(t)=\langle 5\cos t, 3\sin t, 4\sin t \rangle \text{ on }[0,2\pi]$.

40. $\vec{r}(t)=\langle t^3,t^2, t^3\rangle \text{ on }[0,1]$.

41. $\vec{r}(t)=\langle e^{-t}\cos t, e^{-t}\sin t\rangle \text{ on }[0,1]$.

42. Prove Theorem 93; that is, show if $\vec{r}(t)$ has constant length and is differentiable, then $\vec{r}(t)\cdot \vec{r}'(t)=0$. (Hint: use the Product Rule to compute $\frac{d}{dt} (\vec{r}(t)\cdot \vec{r}(t))$.)

12.3: The Calculus of Motion

Terms and Concepts

1. How is velocity different from speed?

2. What is the difference between displacement and distance traveled?

3. What is the difference between average velocity and average speed?

4. Distance traveled is the same as ______ _______, just viewed in a different context.

5. Describe a scenario where an object's average speed is a large number, but the magnitude of the average velocity is not a large number.

6. Explain why it is not possible to have an average velocity with a large magnitude but a small average speed.

Problems

In Exercises 7-10, a position function $\vec{r}(t)$ is given. Find $\vec{v}(t)$ and $\vec{a}(t)$.

7. $\vec{r}(t)=\langle 2t+1,5t-2, 7 \rangle$

8. $\vec{r}(t)=\langle 3t^2-2t+1, -t^2+t+14 \rangle$

9. $\vec{r}(t)=\langle \cos t, \sin t \rangle$

10. $\vec{r}(t)=\langle t/10, -\cos t, \sin t \rangle$

In Exercises 11-14, a position function $\vec{r}(t)$ is given. Sketch $\vec{r}(t)$ and $\vec{a}(t)$, then add $\vec{r}(t_0)$ and $\vec{a}(t_0)$ to your sketch, with their initial points at $\vec{r}(t_0)$, for the given value of $t_0$.

11. $\vec{r}(t)=\langle t,\sin t \rangle \text{ on }[0,\pi /2 ];\, t_0=\pi/4$

12. $\vec{r}(t)=\langle t^2,\sin t^2 \rangle \text{ on }[0,\pi /2 ];\, t_0=\sqrt{\pi/4}$

13. $\vec{r}(t)=\langle t^2+t,-t^2+2t \rangle \text{ on }[-2,2 ];\, t_0=1$

14. $\vec{r}(t)=\langle \frac{2t+3}{t^2+1},t^2 \rangle \text{ on }[-1,1 ];\, t_0=0$

In Exercises 15-24, a position function $\vec{r}(t)$ of an object is given. Find the speed of the object in terms of $t$, and find where the speed is minimized/maximized on the indicated interval.

15. $\vec{r}(t) = \langle t^2,t \rangle \text{ on }[-1,1]$

16. $\vec{r}(t) = \langle t^2,t^2-t^3 \rangle \text{ on }[-1,1]$

17. $\vec{r}(t) = \langle 5\cos t, 5\sin t \rangle \text{ on }[0,2\pi]$

18. $\vec{r}(t) = \langle 2\cos t, 5\sin t \rangle \text{ on }[0,2\pi]$

19. $\vec{r}(t) = \langle \sec t, \tan t \rangle \text{ on }[0,\pi/4]$

20. $\vec{r}(t) = \langle t+\cos t, 1-\sin t\rangle \text{ on }[0,2\pi]$

21. $\vec{r}(t) = \langle 12t, 5\cos t, 5\sin t \rangle \text{ on }[0,4\pi]$

22. $\vec{r}(t) = \langle t^2-t,t^2+t,t \rangle \text{ on }[0,1]$

23. $\vec{r}(t) = \left \langle t,t^2,\sqrt{1-t^2} \right \rangle \text{ on }[-1,1]$

24. Projectile Motion: $\vec{r}(t) = \left \langle (v_0 \cos \theta )t, -\frac{1}{2}gt^2+(v_0 \sin \theta )t\right \rangle \text{ on } \left [ 0,\frac{2v_0 \sin \theta}{g}\right ]$.

In Exercises 25-28, position functions $\vec{r}_1 (t)$ and $\vec{r}_2 (s)$ for two objects are given that follow the same path on the respective intervals.
(a) Show that the positions are the same at the indicated $t_0$ and $s_0$ values; ie., show $\vec{r}_1 (t_0)=\vec{r}_2 (s_0)$.
(b) Find the velocity, speed and acceleration of the two objects at $t_0$ and $s_0$, respectively.

25.
$\vec{r}_1 (t) =\langle t, t^2 \rangle \text{ on }[0,1]; t_0 =1$
$\vec{r}_2 (t) =\langle s^2, s^4 \rangle \text{ on }[0,1]; s_0 =1$

26.
$\vec{r}_1 (t) =\langle 3\cos t,3\sin t \rangle \text{ on }[0,2\pi]; t_0 =\pi/2$
$\vec{r}_2 (t) =\langle 3\cos (4s),3\sin (4s) \rangle \text{ on }[0,\pi/2]; s_0 =\pi/8$

27.
$\vec{r}_1 (t) =\langle 3t,2t \rangle \text{ on }[0,2]; t_0 =2$
$\vec{r}_2 (t) =\langle 6t-6,4t-4 \rangle \text{ on }[1,2]; s_0 =2$

28.
$\vec{r}_1 (t) =\langle t, \sqrt{t} \rangle \text{ on }[0,1]; t_0 =1$
$\vec{r}_2 (t) =\langle \sin t,\sqrt{\sin t} \rangle \text{ on }[0,\pi/2]; s_0 =\pi/2$

In Exercises 29-32, find the position function of an object given its acceleration and initial velocity and position.

29. $\vec{a}(t) =\langle 2,3\rangle; \quad \vec{v}(0)=\langle 1,2\rangle,\quad \vec{r}(0) = \langle 5,-2 \rangle$

30. $\vec{a}(t) =\langle 2,3\rangle; \quad \vec{v}(1)=\langle 1,2\rangle,\quad \vec{r}(1) = \langle 5,-2 \rangle$

31. $\vec{a}(t) =\langle \cos t,-\sin t \rangle; \quad \vec{v}(0)=\langle 0,1 \rangle,\quad \vec{r}(0) = \langle 0,0 \rangle$

32. $\vec{a}(t) =\langle 0,-32\rangle; \quad \vec{v}(0)=\langle 10,50 \rangle,\quad \vec{r}(0) = \langle 0,0 \rangle$

In Exercises 33-36, find the displacement, distance traveled, average velocity, and average speed of the described object on the given interval.

33. An object with position function $\vec{r}(t) =\langle 2\cos t,2\sin t,3t\rangle$, where distances are measured in feet and time is in seconds, on $[0,2\pi ]$.

34. An object with position function $\vec{r}(t) =\langle 5\cos t, -5\sin t\rangle$, where distances are measured in feet and time is in seconds, on $[0,\pi ]$.

35. An object with velocity function $\vec{v}(t) =\langle \cos t, \sin t \rangle$, where distances are measured in feet and time is in seconds, on $[0,2\pi ]$.

36. An object with velocity function $\vec{r}(t) =\langle 1,2,-1\rangle$, where distances are measured in feet and time is in seconds, on $[0,10 ]$.

Exercises 37-42 ask you to solve a variety of problems based on the principles of projectile motion.

37. A boy whirls a ball, attached to a 3 ft string, above his head in a counter–clockwise circle. The ball makes 2 revolutions per second.
At what t-values should the boy release the string so that the ball heads directly for a tree standing 10 ft in front of him?

38. David faces Goliath with only a stone in a 3 ft sling, which he whirls above his head at 4 revolutions per second. They stand 20 ft apart.
(a) At what t-values must David release the stone in his sling in order to hit Goliath?
(b) What is the speed at which the stone is traveling when released?
(c) Assume David releases the stone from a height of 6 ft and Goliath's forehead is 9 ft above the ground. What angle of elevation must David apply to the stone to hit Goliath's head?

39. A hunter aims at a deer which is 40 yards away. Her cross-bow is at a height of 5 ft, and she aims for a spot on the deer 4 ft above the ground. The crossbow fires her arrows at 300 ft/s.
(a) At what angle of elevation should she hold the crossbow to hit her target?
(b) If the deer is moving perpendicularly to her line of sight at a rate of 20 mph, by approximately how much should she lead the deer in order to hit in the desired location?

40. A baseball player hits a ball at 100 mph, with an initial height of 3 ft and an angle of elevation of 20$^\circ$, at Boston's Fenway Park. The ball flies towards the famed "Green Monster," a wall 37 ft high located 310 ft from home plate.
(a) Show that as hit, the ball hits the wall.
(b) Show that if the angle of elevation is 21$^\circ$, the ball clears the Green Monster.

41. A Cessna flies at 1000 ft at 150 mph and drops a box of supplies to the professor (and his wife) on an island. Ignoring wind resistance, how far horizontally will the supplies travel before they land?

42. A football quarterback throws a pass from a height of 6 ft, intending to hit his receiver 20 yds away at a height of 5 ft.
(a) If the ball is thrown at a rate of 50 mph, what angle of elevation is needed to hit his intended target?
(b) If the ball is thrown at with an angle of elevation of 8$^\circ$, what initial ball speed is needed to hit his target?

12.4: Unit Tangent and Normal Vectors

Terms and Concepts

1. If $\vec{T}(t)$ is a unit tangent vector, what is $\lVert \vec{T}(t)\rVert$?

2. If $\vec{N}(t)$ is a unit normal vector, what is $\vec{N}(t)\cdot \vec{r}'(t)$?

3. The acceleration vector $\vec{a}(t)$ lies in the plane defined by what two vectors?

4. $a_T$ measures how much the acceleration is affecting the _______ of an object.

Problems

In Exercises 5-8, given $\vec{r}(t)$, find $\vec{T}(t)$ and evaluate it at the indicated value of $t$.

5. $\vec{r}(t) = \langle 2t^2,t^2-1 \rangle ,\quad t=1$

6. $\vec{r}(t) = \langle t,\cos t \rangle ,\quad t=\pi/4$

7. $\vec{r}(t) = \langle \cos^3 t,\sin^3 t \rangle ,\quad t=\pi/4$

8. $\vec{r}(t) = \langle \cos t, \sin t \rangle ,\quad t=\pi$

In Exercises 9-12, find the equation of the line tangent to the curve at the indicated t-value using the unit tangent vector. Note: these are the same problems as in Exercises 5-8.

9. $\vec{r}(t) = \langle 2t^2,t^2-1 \rangle ,\quad t=1$

10. $\vec{r}(t) = \langle t,\cos t \rangle ,\quad t=\pi/4$

11. $\vec{r}(t) = \langle \cos^3 t,\sin^3 t \rangle ,\quad t=\pi/4$

12. $\vec{r}(t) = \langle \cos t, \sin t \rangle ,\quad t=\pi$

In Exercises 13-16, find $\vec{N}(t)$ using Definition 75. Confirm the result using the Theorem 97.

13. $\vec{r}(t) = \langle 3\cos t,3\sin t \rangle$

14. $\vec{r}(t) = \langle t,t^2 \rangle$

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

16. $\vec{r}(t) = \langle e^t, e^{-t} \rangle$

In Exercises 17-20, a position function $\vec{r}(t)$ is given along with its unit tangent vector $\vec{T}(t)$ evaluated at $t=a$, for some value of $a$.
(a) Confirm that $\vec{T}(a)$ is as stated.
(b) Using a graph of $\vec{r}(t)$ and Theorem 97, find $\vec{N}(a)$.

17. $\vec{r}(t) = \langle 3\cos t, 5\sin t \rangle;\quad \vec{T}(\pi/4)=\left \langle -\frac{3}{\sqrt{34}},\frac{5}{\sqrt{34}}\right \rangle$

18. $\vec{r}(t) = \left \langle t,\frac{1}{t^2+1} \right \rangle;\quad \vec{T}(1)=\left \langle \frac{2}{\sqrt{5}},-\frac{1}{\sqrt{5}}\right \rangle$

19. $\vec{r}(t) = (1+2\sin t)(\cos t, \sin t);\quad \vec{T}(0)=\left \langle \frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}}\right \rangle$

20. $\vec{r}(t) = \left \langle \cos^3 t,\sin^3 t \right \rangle;\quad \vec{T}(\pi/4)=\left \langle -\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right \rangle$

In Exercises 21-24, find $\vec{N}(t)$.

21. $\vec{r}(t)=\langle 4t, 2\sin t, 2\cos t \rangle$

22. $\vec{r}(t)=\langle 5\cos t ,3\sin t,4\sin t \rangle$

23. $\vec{r}(t)=\langle a\cos t, a\sin t,bt \rangle$

24. $\vec{r}(t)=\langle \cos (at),\sin (at),t \rangle$

In Exercises 25-30, find $a_T$ and $a_N$ given $\vec{r}(t)$. Sketch $\vec{r}(t)$ on the indicated interval, and comment on the relative sizes of $a_T$ and $a_N$ at the indicated t values.

25. $\vec{r}(t) = \langle t,t^2 \rangle \text{ on }[-1,1];\text{ consider }t=0\text{ and }t=1$.

26. $\vec{r}(t) = \langle t,1/t \rangle \text{ on }(0,4;\text{ consider }t=1\text{ and }t=2$.

27. $\vec{r}(t) = \langle 2\cos t, 2\sin t\rangle \text{ on }[0,2\pi];\text{ consider }t=1\text{ and }t=\pi/2$.

28. $\vec{r}(t) = \langle \cos (t^2),\sin (t^2)\rangle \text{ on }(0,2\pi];\text{ consider }t=\sqrt{\pi/2}\text{ and }t=\sqrt{\pi}$.

29. $\vec{r}(t) = \langle a\cos t, a\sin t ,bt\rangle \text{ on }[0,2\pi];\text{ where }a,b>0;\text{ consider }t=0 \text{ and }t=\pi/2$.

30. $\vec{r}(t) = \langle 5\cos t, 4\sin t, 3\sin t \rangle \text{ on }[0,2\pi];\text{ consider }t=0\text{ and }t=\pi/2$.

12.5: The Arc Length Parameter and Curvature

Terms and Concepts

1. It is common to describe position in terms of both ______ and/or _______.

2. A measure of the "curviness" of a curve is ________.

3. Give two shapes with constant curvature.

4. Describe in your own words what an "osculating circle" is.

5. Complete the identity: $\vec{T}'(s)=$_________$\vec{N}(s)$.

6. Given a position function $\vec{r}(t)$, how are $a_T$ and $a_N$ affected by the curvature?

Problems

In Exercises 7-10, a position function $\vec{r}(t)$ is given, where $t=0$ corresponds to the initial position. Find the arc length parameter $s$, and rewrite $\vec{r}(t)$ in terms of $s$; that is, find $\vec{r}(s)$.

7. $\vec{r}(t) = \langle 2t, t, -2t \rangle$

8. $\vec{r}(t) = \langle 7\cos t, 7\sin t \rangle$

9. $\vec{r}(t) = \langle 3\cos t, 3\sin t, 2t \rangle$

10. $\vec{r}(t) = \langle 5\cos t, 13\sin t,12\cos t \rangle$

In Exercises 11-22, a curve $C$ is described along with 2 points on $C$.
(a) Using a sketch, determine at which of these points the curvature is greater.
(b) Find the curvature $\kappa$ of $C$, and evaluate $\kappa$ at each of the 2 given points.

11. $C$ is defined by $y=x^3-x$; points given at $x=0$ and $x=1/2$.

12. $C$ is defined by $y=\frac{1}{x^2+1}$; points given at $x=0$ and $x=2$.

13. $C$ is defined by $y=\cos x$; points given at $x=0$ and $x=\pi /2$.

14. $C$ is defined by $y=\sqrt{1-x^2}$ on $(-1,1)$; points given at $x=0$ and $x=1/2$.

15. $C$ is defined by $\vec{r}(t)=\langle \cos t, \sin (2t) \rangle$; points given at $t=0$ and $t=\pi/4$.

16. $C$ is defined by $\vec{r}(t)=\langle \cos^2 (t),\sin t\cos t \rangle$; points given at $t=0$ and $t=\pi/3$.

17. $C$ is defined by $\vec{r}(t)=\langle t^2-1,t^3-t \rangle$; points given at $t=0$ and $t=\pi/6$.

18. $C$ is defined by $\vec{r}(t)=\langle \tan t,\sec t \rangle$; points given at $t=0$ and $t=\pi/6$.

19. $C$ is defined by $\vec{r}(t)=\langle 4t+2,3t-1,2t+5\rangle$; points given at $t=0$ and $t=1$.

20. $C$ is defined by $\vec{r}(t)=\langle t^3-t,t^3-4,t^2-1 \rangle$; points given at $t=0$ and $t=1$.

21. $C$ is defined by $\vec{r}(t)=\langle 3\cos t, 3\sin t, 2t \rangle$; points given at $t=0$ and $t=\pi/2$.

22. $C$ is defined by $\vec{r}(t)=\langle 5\cos t, 13\sin t, 12\cos t \rangle$; points given at $t=0$ and $t=\pi/2$.

In Exercises 23-26, find the value of $x$ or $t$ where curvature is maximized.

23. $y=\frac{1}{6}x^3$

24. $y=\sin x$

25. $\vec{r}(t) =\langle t^2+2t, 3t-t^2 \rangle$

26. $\vec{r}(t) = \langle t, 4/t, 3/t \rangle$

In Exercises 27-30, find the radius of curvature at the indicated value.

27. $y=\tan x , \text{ at }x=\pi/4$

28. $y=x^2+x-3 , \text{ at }x=\pi/4$

29. $\vec{r}(t) = \langle \cos t ,\sin (3t) \rangle , \text{ at }t=0$

30. $\vec{r}(t) = \langle 5\cos (3t),t \rangle , \text{ at }t=0$

In Exercises 31-34, find the equation of the osculating circle to the curve at the indicated $t$-value.

31. $\vec{r}(t) = \langle t,t^2 \rangle ,\text{ at }t=0.$

32. $\vec{r}(t) = \langle 3\cos t, \sin t \rangle ,\text{ at }t=0.$

33. $\vec{r}(t) = \langle 3\cos t,\sin t \rangle ,\text{ at }t=\pi/2.$

34. $\vec{r}(t) = \langle t^2-t,t^2+t \rangle ,\text{ at }t=0.$