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Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

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Problem 1

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

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Problem 2

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

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Problem 3

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

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Problem 4

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

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Problem 5

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

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

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

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Problem 10

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

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Problem 11

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

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Problem 12

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

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Problem 13

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

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Problem 14

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

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Problem 15

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

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Problem 16

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

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Problem 17

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

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Problem 18

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

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Problem 19

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

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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?

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Problem 21

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

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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 $(7,7) .$

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Problem 23

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

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Problem 24

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

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Problem 25

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

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Problem 26

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

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Problem 27

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

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Problem 28

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

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Problem 29

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

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Problem 30

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

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Problem 31

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

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Problem 32

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

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Problem 33

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

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Problem 34

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

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Problem 35

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

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Problem 36

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

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Problem 37

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

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Problem 38

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

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Problem 39

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

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Problem 40

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

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Problem 41

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

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Problem 42

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

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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 $x .$

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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 $x<0 ?$

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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.

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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$(\mathrm{A})$ ?

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Problem 47

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

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Problem 48

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

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Problem 49

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

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Problem 50

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

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Problem 51

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

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Problem 52

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

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

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

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

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

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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 $3 .$

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Problem 58

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

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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
of the four quadrants.

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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 $t=4$

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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 $\frac{1}{2}$

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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.

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

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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.

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

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Problem 66

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

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

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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 $P_{3} .$

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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.

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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.

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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 $\theta=\frac{\pi}{3} .$

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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 $(5) .$

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Problem 73

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

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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 $1 .$

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

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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.

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

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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.

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

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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.

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Problem 81

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

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

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Problem 83

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

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Problem 84

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

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Problem 85

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

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Problem 86

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

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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.

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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.

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

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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 $\left(t^{2}, t^{4}\right) .$

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Problem 91

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

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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 $\mathrm{Eq} .(12) .$

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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. $(12)$

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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 $26 .$

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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.

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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$) .$

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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 $t$ .

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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. $(12)$ .

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Problem 99

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

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