# Calculus for AP

## Educators

### Problem 1

Find the coordinates at times $t=0,2,4$ of a particle following the path $x=1+t^{3}, y=9-3 t^{2}$

Eric M.

### Problem 2

Find the coordinates at $t=0, \frac{\pi}{4}, \pi$ of a particle moving along the path $c(t)=\left(\cos 2 t, \sin ^{2} t\right)$

Eric M.

### Problem 3

Show that the path traced by the bullet in Example 3 is a parabola by eliminating the parameter.

Eric M.

### Problem 4

Use the table of values to sketch the parametric curve $(x(t), y(t)),$ indicating the direction of motion. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline t & {-3} & {-2} & {-1} & {0} & {1} & {2} & {3} \\ \hline x & {-15} & {0} & {3} & {0} & {-3} & {0} & {15} \\ \hline y & {5} & {0} & {-3} & {-4} & {-3} & {0} & {5} \\ \hline\end{array}$$

Eric M.

### Problem 5

Graph the parametric curves. Include arrows indicating the direction of motion.
$$(\mathbf{a}) \quad(t, t) , -\infty < t < -\infty$$
$$(b) (\sin t, \sin t) , 0 \leq t \leq 2 \pi$$
$$(\mathrm{c})\left(e^{t}, e^{t}\right) , -\infty < t < -\infty$$
$$(\mathrm{d})\left(t^{3}, t^{3}\right) , -1 \leq t \leq 1$$

Eric M.

### Problem 6

6. Give two different parametrizations of the line through $(4,1)$ with slope $2 .$

Eric M.

### Problem 7

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.
$$x=t+3, \quad y=4 t$$

Eric M.

### Problem 8

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.
$$x=t^{-1}, \quad y=t^{-2}$$

Eric M.

### Problem 9

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.
$$x=t, \quad y=\tan ^{-1}\left(t^{3}+e^{t}\right)$$

Eric M.

### Problem 10

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.
$$x=t^{2}, \quad y=t^{3}+1$$

Eric M.

### Problem 11

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.
$$x=e^{-2 t}, \quad y=6 e^{4 t}$$

Eric M.

### Problem 12

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.
$$x=1+t^{-1}, \quad y=t^{2}$$

Eric M.

### Problem 13

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.
$$x=\ln t, \quad y=2-t$$

Eric M.

### Problem 14

In Exercises $7-14,$ express in the form $y=f(x)$ by eliminating the parameter.$$x=\cos t, \quad y=\tan t$$

Eric M.

### Problem 15

In Exercises $15-18,$ graph the curve and draw an arrow specifying the direction corresponding to motion. $$x=\frac{1}{2} t, \quad y=2 t^{2}$$

Eric M.

### Problem 16

In Exercises $15-18,$ graph the curve and draw an arrow specifying the direction corresponding to motion. $$x=2+4 t, \quad y=3+2 t$$

Eric M.

### Problem 17

In Exercises $15-18,$ graph the curve and draw an arrow specifying the direction corresponding to motion. $$x=\pi t, \quad y=\sin t$$

Eric M.

### Problem 18

In Exercises $15-18,$ graph the curve and draw an arrow specifying the direction corresponding to motion. $$x=t^{2}, \quad y=t^{3}$$

Eric M.

### Problem 19

Match the parametrizations $(a)-(d)$ below with their plots in Figure $14,$ and draw an arrow indicating the direction of motion.
$$\begin{array}{ll}{\text { (a) } c(t)=(\sin t,-t)} & {\text { (b) } c(t)=\left(t^{2}-9,8 t-t^{3}\right)} \\ {\text { (c) } c(t)=\left(1-t, t^{2}-9\right)} & {\text { (d) } c(t)=(4 t+2,5-3 t)}\end{array}$$

Eric M.

### Problem 20

A particle follows the trajectory $$x(t)=\frac{1}{4} t^{3}+2 t, \quad y(t)=20 t-t^{2}$$
with $t$ in seconds and distance in centimeters.
(a) What is the particle's maximum height?
(b) When does the particle hit the ground and how far from the origin does it land?

Eric M.

### Problem 21

Find an interval of $t$ -values such that $c(t)=(\cos t, \sin t)$ traces the lower half of the unit circle.

Eric M.

### Problem 22

Find an interval of $t$ -values such that $c(t)=(2 t+1,4 t-5)$ parametrizes the segment from $(0,-7)$ to $(7,7) .$

Eric M.

### Problem 23

In Exercises $23-38,$ find parametric equations for the given curve.
$$y=9-4 x$$

Eric M.

### Problem 24

In Exercises $23-38,$ find parametric equations for the given curve.
$$y=8 x^{2}-3 x$$

Eric M.

### Problem 25

In Exercises $23-38,$ find parametric equations for the given curve.
$$4 x-y^{2}=5$$

Eric M.

### Problem 26

In Exercises $23-38,$ find parametric equations for the given curve.
$$x^{2}+y^{2}=49$$

Eric M.

### Problem 27

In Exercises $23-38,$ find parametric equations for the given curve.
$$(x+9)^{2}+(y-4)^{2}=49$$

Eric M.

### Problem 28

In Exercises $23-38,$ find parametric equations for the given curve.
$$\left(\frac{x}{5}\right)^{2}+\left(\frac{y}{12}\right)^{2}=1$$

Eric M.

### Problem 29

In Exercises $23-38,$ find parametric equations for the given curve.
Line of slope 8 through $(-4,9)$

Eric M.

### Problem 30

In Exercises $23-38,$ find parametric equations for the given curve.
Line through $(2,5)$ perpendicular to $y=3 x$

Eric M.

### Problem 31

In Exercises $23-38,$ find parametric equations for the given curve.
Line through $(3,1)$ and $(-5,4)$

Eric M.

### Problem 32

In Exercises $23-38,$ find parametric equations for the given curve.
Line through $\left(\frac{1}{3}, \frac{1}{6}\right)$ and $\left(-\frac{7}{6}, \frac{5}{3}\right)$

Eric M.

### Problem 33

In Exercises $23-38,$ find parametric equations for the given curve.
Segment joining $(1,1)$ and $(2,3)$

Eric M.

### Problem 34

In Exercises $23-38,$ find parametric equations for the given curve.
Segment joining $(-3,0)$ and $(0,4)$

Eric M.

### Problem 35

In Exercises $23-38,$ find parametric equations for the given curve.
Circle of radius 4 with center $(3,9)$

Eric M.

### Problem 36

In Exercises $23-38,$ find parametric equations for the given curve.
Ellipse of Exercise $28,$ with its center translated to $(7,4)$

Eric M.

### Problem 37

In Exercises $23-38,$ find parametric equations for the given curve.
$y=x^{2},$ translated so that the minimum occurs at $(-4,-8)$

Eric M.

### Problem 38

In Exercises $23-38,$ find parametric equations for the given curve.
$y=\cos x$ translated so that a maximum occurs at $(3,5)$

Eric M.

### Problem 39

In Exercises $39-42,$ find a parametrization $c(t)$ of the curve satisfying
the given condition.
$$y=3 x-4, \quad c(0)=(2,2)$$

Eric M.

### Problem 40

In Exercises $39-42,$ find a parametrization $c(t)$ of the curve satisfying
the given condition.
$$y=3 x-4, \quad c(3)=(2,2)$$

Eric M.

### Problem 41

In Exercises $39-42,$ find a parametrization $c(t)$ of the curve satisfying
the given condition.
$$y=x^{2}, \quad c(0)=(3,9)$$

Eric M.

### Problem 42

In Exercises $39-42,$ find a parametrization $c(t)$ of the curve satisfying the given condition.
$$x^{2}+y^{2}=4, \quad c(0)=\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$

Check back soon!

### Problem 43

Describe $c(t)=(\sec t, \tan t)$ for $0 \leq t<\frac{\pi}{2}$ in the form $y=f(x)$ .
Specify the domain of $x .$

Eric M.

### Problem 44

Find a parametrization of the right branch $(x>0)$ of the hyperbola $$\left(\frac{x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}=1$$
using the functions cosh $t$ and $\sinh t .$ How can you parametrize the branch $x<0 ?$

Eric M.

### Problem 45

The graphs of $x(t)$ and $y(t)$ as functions of $t$ are shown in Figure 15$($ A) . Which of $(1)-(1 I)$ is the plot of $c(t)=(x(t), y(t)) ?$ Explain.

Eric M.

### Problem 46

Which graph, (I) or (II), is the graph of $x(t)$ and which is the graph of $y(t)$ for the parametric curve in Figure 16$(\mathrm{A})$ ?

Eric M.

### Problem 47

Sketch $c(t)=\left(t^{3}-4 t, t^{2}\right)$ following the steps in Example 7

Eric M.

### Problem 48

Sketch $c(t)=\left(t^{2}-4 t, 9-t^{2}\right)$ for $-4 \leq t \leq 10$

Eric M.

### Problem 49

In Exercises $49-52,$ use $E q .(7)$ to find $d y / d x$ at the given point.
$$\left(t^{3}, t^{2}-1\right), \quad t=-4$$

Eric M.

### Problem 50

In Exercises $49-52,$ use $E q .(7)$ to find $d y / d x$ at the given point.
$$(2 t+9,7 t-9), \quad t=1$$

Eric M.

### Problem 51

In Exercises $49-52,$ use $E q .(7)$ to find $d y / d x$ at the given point.
$$\left(s^{-1}-3 s, s^{3}\right), \quad s=-1$$

Eric M.

### Problem 52

In Exercises $49-52,$ use $E q .(7)$ to find $d y / d x$ at the given point.
$$(\sin 2 \theta, \cos 3 \theta), \quad \theta=\frac{\pi}{6}$$

Eric M.

### Problem 53

In Exercises $53-56,$ find and an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using $E q .($ O) and by differentiating $f(x)$
$$c(t)=(2 t+1,1-9 t)$$

Eric M.

### Problem 54

In Exercises $53-56,$ find and an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using $E q .($ O) and by differentiating $f(x)$
$$c(t)=\left(\frac{1}{2} t, \frac{1}{4} t^{2}-t\right)$$

Eric M.

### Problem 55

In Exercises $53-56,$ find and an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using $E q .($ O) and by differentiating $f(x)$
$$x=s^{3}, \quad y=s^{6}+s^{-3}$$

Eric M.

### Problem 56

In Exercises $53-56,$ find and an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using $E q .($ O) and by differentiating $f(x)$
$$x=\cos \theta, \quad y=\cos \theta+\sin ^{2} \theta$$

Eric M.

### Problem 57

Find the points on the curve $c(t)=\left(3 t^{2}-2 t, t^{3}-6 t\right)$ where the tangent line has slope $3 .$

Eric M.

### Problem 58

Find the equation of the tangent line to the cycloid generated by a circle of radius 4 at $t=\frac{\pi}{2}$ .

Eric M.

### Problem 59

In Exercises $59-62,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)$ (see Figure 17$)$
Draw an arrow indicating the direction of motion, and determine
the interval of $t-$ values corresponding to the portion of the curve in each

Eric M.

### Problem 60

In Exercises $59-62,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)$ (see Figure 17$)$
Find the equation of the tangent line at $t=4$

Eric M.

### Problem 61

In Exercises $59-62,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)$ (see Figure 17$)$
Find the points where the tangent has slope $\frac{1}{2}$

Eric M.

### Problem 62

In Exercises $59-62,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)$ (see Figure 17$)$
Find the points where the tangent is horizontal or vertical.

Eric M.

### Problem 63

Let $A$ and $B$ be the points where the ray of angle $\theta$ intersects the two concentric circles of radii $r<R$ centered at the origin (Figure 18$) .$ Let $P$ be the point of intersection of the horizontal line through $A$ and the vertical line through $B$ . Express the coordinates of $P$ as a function of $\theta$ and describe the curve traced by $P$ for $0 \leq \theta \leq 2 \pi$

Eric M.

### Problem 64

$64 . \mathrm{A} 10-\mathrm{ft}$ ladder slides down a wall as its bottom $B$ is pulled away from the wall (Figure 19). Using the angle $\theta$ as parameter, find the
parametric equations for the path followed by (a) the top of the ladder A, (b) the bottom of the ladder $B$ , and (c) the point $P$ located 4 from the top of the ladder. Show that $P$ describes an ellipse.

Eric M.

### Problem 65

In Exercises $65-68,$ refer to the Bézier curve defined by $E q s .(8)$ and $(9) .$
Show that the Bézier curve with control points
$$P_{0}=(1,4), \quad P_{1}=(3,12), \quad P_{2}=(6,15), \quad P_{3}=(7,4)$$

Eric M.

### Problem 66

In Exercises $65-68,$ refer to the Bézier curve defined by $E q s .(8)$ and $(9) .$
$$\begin{array}{l}{\text { 66. Find an equation of the tangent line to the Bézier curve in Exercise }} \\ {65 \text { at } t=\frac{1}{3} .}\end{array}$$

Eric M.

### Problem 67

In Exercises $65-68,$ refer to the Bézier curve defined by $E q s .(8)$ and $(9) .$
67. $C A S$ Find and plot the Bézier curve $c(t)$ passing through the
control points $$P_{0}=(3,2), \quad P_{1}=(0,2), \quad P_{2}=(5,4), \quad P_{3}=(2,4)$$

Eric M.

### Problem 68

In Exercises $65-68,$ refer to the Bézier curve defined by $E q s .(8)$ and $(9) .$
Show that a cubic Bézier curve is tangent to the segment $\overline{P_{2} P_{3}}$
at $P_{3} .$

Eric M.

### Problem 69

A bullet fired from a gun follows the trajectory $$x=a t, \quad y=b t-16 t^{2} \quad(a, b>0)$$ Show that the bullet leaves the gun at an angle $\theta=\tan ^{-1}\left(\frac{b}{a}\right)$ and lands at a distance $a b / 16$ from the origin.

Eric M.

### Problem 70

70. $C R S$ Plot $c(t)=\left(t^{3}-4 t, t^{4}-12 t^{2}+48\right)$ for $-3 \leq t \leq 3$ .
Find the points where the tangent line is horizontal or vertical.

Eric M.

### Problem 71

71. $CCA S$ Plot the astroid $x=\cos ^{3} \theta, y=\sin ^{3} \theta$ and find the equa-
tion of the tangent line at $\theta=\frac{\pi}{3} .$

Eric M.

### Problem 72

Find the equation of the tangent line at $t=\frac{\pi}{4}$ to the cycloid generated by the unit circle with parametric equation $(5) .$

Eric M.

### Problem 73

Find the points with horizontal tangent line on the cycloid with parametric equation $(5) .$

Eric M.

### Problem 74

Property of the Cycloid Prove that the tangent line at a point $P$ on the cycloid always passes through the top point on the rolling circle as indicated in Figure $20 .$ Assume the generating circle of the cycloid has radius $1 .$

Eric M.

### Problem 75

A curtate cycloid (Figure 21$)$ is the curve traced by a point at a distance $h$ from the center of a circle of radius $R$ rolling along the $x$ -axis where $h<R .$ Show that this curve has parametric equations $x=R t-h \sin t, y=R-h \cos t$

Eric M.

### Problem 76

$CAS$ Use a computer algebra system to explore what happens
when $h>R$ in the parametric equations of Exercise $75 .$ Describe the
result.

Eric M.

### Problem 77

Show that the line of slope $t$ through $(-1,0)$ intersects the unit circle in the point with coordinates $$x=\frac{1-t^{2}}{t^{2}+1}, \quad y=\frac{2 t}{t^{2}+1}$$
Conclude that these equations parametrize the unit circle with the point $(-1,0)$ excluded (Figure 22$) .$ Show further that $t=y /(x+1)$ .

Eric M.

### Problem 78

The folium of Descartes is the curve with equation $x^{3}+y^{3}=$ $3 a x y,$ where $a \neq 0$ is a constant (Figure 23$) .$
(a) Show that the line $y=t x$ intersects the folium at the origin and at one other point $P$ for all $t \neq-1,0 .$ Express the coordinates of $P$ in terms of $t$ to obtain a parametrization of the folium. Indicate the direction of the parametrization on the graph.
(b) Describe the interval of $t$ -values parametrizing the parts of the
curve in quadrants I, II, and IV. Note that $t=-1$ is a point of discontinuity of the parametrization.
(c) Calculate $d y / d x$ as a function of $t$ and find the points with hori-
zontal or vertical tangent.

Eric M.

### Problem 79

79. Use the results of Exercise 78 to show that the asymptote of the folium is the line $x+y=-a . H i n t :$ Show that $\lim _{t \rightarrow-1}(x+y)=-a$

Eric M.

### Problem 80

Find a parametrization of $x^{2 n+1}+y^{2 n+1}=a x^{n} y^{n},$ where $a$ and $n$ are constants.

Eric M.

### Problem 81

Second Derivative for a Parametrized Curve Given a parametrized curve $c(t)=(x(t), y(t)),$ show that $$\frac{d}{d t}\left(\frac{d y}{d x}\right)=\frac{x^{\prime}(t) y^{\prime \prime}(t)-y^{\prime}(t) x^{\prime \prime}(t)}{x^{\prime}(t)^{2}}$$

Eric M.

### Problem 82

The second derivative of $y=x^{2}$ is $d y^{2} / d^{2} x=2 .$ Verify that Eq. (11) applied to $c(t)=\left(t, t^{2}\right)$ yields $d y^{2} / d^{2} x=2 .$ In fact, any parametrization may be used. Check that $c(t)=\left(t^{3}, t^{6}\right)$ and $c(t)=$ (tan $t, \tan ^{2} t )$ also yield $d y^{2} / d^{2} x=2$

Eric M.

### Problem 83

In Exercises $83-86, u s e E q .(11)$ to find $d^{2} y / d x^{2}$.
$$x=t^{3}+t^{2}, \quad y=7 t^{2}-4, \quad t=2$$

Eric M.

### Problem 84

In Exercises $83-86, u s e E q .(11)$ to find $d^{2} y / d x^{2}$.
$$x=s^{-1}+s, \quad y=4-s^{-2}, \quad s=1$$

Eric M.

### Problem 85

In Exercises $83-86, u s e E q .(11)$ to find $d^{2} y / d x^{2}$.
$$x=8 t+9, \quad y=1-4 t, \quad t=-3$$

Eric M.

### Problem 86

In Exercises $83-86, u s e E q .(11)$ to find $d^{2} y / d x^{2}$.
$$x=\cos \theta, \quad y=\sin \theta, \quad \theta=\frac{\pi}{4}$$

Eric M.

### Problem 87

Use Eq. $(11)$ to find the $t$ -intervals on which $c(t)=\left(t^{2}, t^{3}-4 t\right)$
is concave up.

Eric M.

### Problem 88

Use Eq. $(11)$ to find the $t$ -intervals on which $c(t)=\left(t^{2}, t^{4}-4 t\right)$
is concave up.

Eric M.

### Problem 89

Area Under a Parametrized Curve Let $c(t)=(x(t), y(t)),$ where $y(t)>0$ and $x^{\prime}(t)>0($ Figure 24$) .$ Show that the area $A$ under $c(t)$ for $t_{0} \leq t \leq t_{1}$ is
$$A=\int_{t_{0}}^{t_{1}} y(t) x^{\prime}(t) d t$$
Hint: Because it is increasing, the function $x(t)$ has an inverse $t=g(x)$
and $c(t)$ is the graph of $y=y(g(x)) .$ Apply the change-of-variables
formula to $A=\int_{x\left(t_{0}\right)}^{x\left(t_{1}\right)} y(g(x)) d x$

Eric M.

### Problem 90

Calculate the area under $y=x^{2}$ over $[0,1]$ using $\mathrm{Eq} .(12)$ with the
parametrizations $\left(t^{3}, t^{6}\right)$ and $\left(t^{2}, t^{4}\right) .$

Eric M.

### Problem 91

What does Eq. $(12)$ say if $c(t)=(t, f(t)) ?$

Eric M.

### Problem 92

Sketch the graph of $c(t)=(\ln t, 2-t)$ for $1 \leq t \leq 2$ and compute the area under the graph using $\mathrm{Eq} .(12) .$

Eric M.

### Problem 93

Galileo tried unsuccessfully to find the area under a cycloid. Around 1630 . Gilles de Roberval proved that the area under one arch of the cycloid $c(t)=(R t-R \sin t, R-R \cos t)$ generated by a circle of radius $R$ is equal to three times the area of the circle (Figure 25$) .$ Verify Roberval's result using Eq. $(12)$

Eric M.

### Problem 94

Prove the following generalization of Exercise $93 :$ For all $t>0$ the area of the cycloidal sector $O P C$ is equal to three times the area of the circular segment cut by the chord $P C$ in Figure $26 .$

Eric M.

### Problem 95

Derive the formula for the slope of the tangent line to a parametric curve $c(t)=(x(t), y(t))$ using a method different from that presented in the text. Assume that $x^{\prime}\left(t_{0}\right)$ and $y^{\prime}\left(t_{0}\right)$ exist and that
$x^{\prime}\left(t_{0}\right) \neq 0 .$ Show that
$$\lim _{h \rightarrow 0} \frac{y\left(t_{0}+h\right)-y\left(t_{0}\right)}{x\left(t_{0}+h\right)-x\left(t_{0}\right)}=\frac{y^{\prime}\left(t_{0}\right)}{x^{\prime}\left(t_{0}\right)}$$
Then explain why this limit is equal to the slope $d y / d x$ . Draw a diagram showing that the ratio in the limit is the slope of a secant line.

Eric M.

### Problem 96

Verify that the tractrix curve $(\ell>0)$ $$c(t)=\left(t-\ell \tanh \frac{t}{\ell}, \ell \operatorname{sech} \frac{t}{\ell}\right)$$ has the following property: For all $t,$ the segment from $c(t)$ to $(t, 0)$ is tangent to the curve and has length $\ell$ (Figure 27$) .$

Eric M.

### Problem 97

In Exercise 54 of Section $9.1,$ we described the tractrix by the differential equation $$\frac{d y}{d x}=-\frac{y}{\sqrt{\ell^{2}-y^{2}}}$$ Show that the curve $c(t)$ identified as the tractrix in Exercise 96 satisfies this differential equation. Note that the derivative on the left is taken with respect to $x,$ not $t$ .

Eric M.

### Problem 98

In Exercises 98 and 99 , refer to Figure 28
In the parametrization $c(t)=(a \cos t, b \sin t)$ of an ellipse, $t$ is not an angular parameter unless $a=b$ (in which case the ellipse is a circle). However, $t$ can be interpreted in terms of area: Show that if $c(t)=(x, y),$ then $t=(2 / a b) A,$ where $A$ is the area of the shaded region in Figure $28 .$ Hint: Use Eq. $(12)$ .

Eric M.
Show that the parametrization of the ellipse by the angle $\theta$ is
\begin{aligned} x &=\frac{a b \cos \theta}{\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}} \\ y &=\frac{a b \sin \theta}{\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}} \end{aligned}