Calculus for AP

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $x<0 ?$

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.

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$(\mathrm{A})$ ?

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

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

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

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

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

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

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 $f(x)$
$$c(t)=(2 t+1,1-9 t)$$

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 $f(x)$
$$c(t)=\left(\frac{1}{2} t, \frac{1}{4} t^{2}-t\right)$$

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 $f(x)$
$$x=s^{3}, \quad y=s^{6}+s^{-3}$$

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 $f(x)$
$$x=\cos \theta, \quad y=\cos \theta+\sin ^{2} \theta$$

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 $3 .$

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

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
of the four quadrants.

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 $t=4$

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 $\frac{1}{2}$

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.

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 $0 \leq \theta \leq 2 \pi$

$64 . \mathrm{A} 10-\mathrm{ft}$ 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.

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
$$P_{0}=(1,4), \quad P_{1}=(3,12), \quad P_{2}=(6,15), \quad P_{3}=(7,4)$$

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

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 $$
P_{0}=(3,2), \quad P_{1}=(0,2), \quad P_{2}=(5,4), \quad P_{3}=(2,4)
$$

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 $P_{3} .$

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.

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.

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

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

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

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 $1 .$

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 $x=R t-h \sin t, y=R-h \cos t$

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

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 $t=y /(x+1)$ .

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.

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 $\lim _{t \rightarrow-1}(x+y)=-a$

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

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

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 $d y^{2} / d^{2} x=2$

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

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

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

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

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

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

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 $A=\int_{x\left(t_{0}\right)}^{x\left(t_{1}\right)} y(g(x)) d x$

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

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

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

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. $(12)$

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 $26 .$

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.

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$) .$

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 $t$ .

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. $(12)$ .

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