Calculus: Early Transcendentals

Jon Rogawski, Colin Adams, Robert Franzosa

Chapter 11

Parametric Equations, Polar Coordinates, and Conic Sections - all with Video Answers

Educators

WM

Section 1

Parametric Equations

Problem 1

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

WM

William Mead

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 3

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

Linda Hand

Numerade Educator

Problem 4

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 5

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 6

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

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 9

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

Linda Hand

Numerade Educator

Problem 10

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

Linda Hand

Numerade Educator

Problem 11

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

Mukesh Devi

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 14

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

Linda Hand

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 19

Match the parametrizations (a)-(d) with their plots in Figure 15 , and draw an arrow indicating the direction of motion.
FIGURE 15
(a) $c(t)=(\sin t,-t)$
(b) $c(t)=\left(t^{2}-9,8 t-t^{3}\right)$
(c) $c(t)=\left(1-t, t^{2}-9\right)$
(d) $c(t)=(4 t+2,5-3 t)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 20

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

Sheryl Ezze

Numerade Educator

Problem 21

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 Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 23

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 24

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 25

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 26

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 27

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 29

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 30

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 31

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 33

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 34

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 35

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 37

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 38

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 39

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 40

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 41

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

Willis James

Numerade Educator

Problem 42

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

Linda Hand

Numerade Educator

Problem 43

Find a parametrization of the top half of the ellipse $4 x^{2}+5 y^{2}=100$, starting at (-5,0) and ending at (5,0) .

Linda Hand

Numerade Educator

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 $\cosh t$ and $\sinh t$. How can you parametrize the branch $x<0 ?$

JC

Jeremy Cooper

Numerade Educator

Problem 45

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 46

Show that $x=a+q t, y=b+p t,$ with $q \neq 0$, parametrizes a line with slope $m=p / q$. What are the $x$ - and $y$ -intercepts of the line?

Linda Hand

Numerade Educator

Problem 47

The graphs of $x(t)$ and $y(t)$ as functions of $t$ are shown in Figure $16(\mathrm{~A})$. Which of (I)-(III) is the plot of $c(t)=(x(t), y(t)) ?$ Explain.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 48

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 $17(\mathrm{~A}) ?$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 49

Figure 18 shows a parametric curve $c(t)=(q(t), p(t))$ that models the changing population sizes of a predator ( $p$ ) and its prey (q).
FIGURE 18
(a) $\square$ Discuss how you expect the predator and prey populations to change as time increases at points $C$ and $D$ on the parametric curve.
(b) As functions of $t,$ sketch graphs of $p(t)$ and $q(t)$ for three cycles around the parametric curve, beginning at point $C$.
(c) $\quad$ Both graphs in (b) should show oscillations between minimum and maximum values. Indicate which (predator or prey) has its peaks shortly after the other has its peaks, and explain why that makes sense in terms of an interaction between a predator and its prey.

Carson Merrill

Numerade Educator

Problem 50

For many years, the Hudson's Bay Company in Canada kept records of the number of snowshoe hare and lynx pelts traded each year. It is natural to expect that these values are roughly proportional to the sizes of the populations. Data for odd years between 1861 and 1891 appear in the table, where the number of pelts for lynx, $L,$ and snowshoe hares, $H,$ are shown (both in thousands). Plot the data on an $L H$ -coordinate system, connecting consecutive data points by a segment to create a parametric curve traced out by the data.
$$
\begin{aligned}
&\begin{array}{|l|c|c|c|c|c|c|c|c|}
\hline \text { Year } & \mathbf{1 8 6 1} & \mathbf{1 8 6 3} & \mathbf{1 8 6 5} & \mathbf{1 8 6 7} & \mathbf{1 8 6 9} & \mathbf{1 8 7 1} & \mathbf{1 8 7 3} & \mathbf{1 8 7 5} \\
\hline \boldsymbol{H} & 36 & 150 & 110 & 60 & 7 & 10 & 70 & 100 \\
\hline \boldsymbol{L} & 6 & 6 & 65 & 70 & 40 & 9 & 20 & 34 \\
\hline
\end{array}\\
&\begin{array}{|l|c|c|c|c|c|c|c|c|}
\hline \text { Year } & 1877 & 1879 & 1881 & 1883 & 1885 & 1887 & 1889 & 1891 \\
\hline H & 92 & 70 & 10 & 11 & 137 & 137 & 18 & 22 \\
\hline L & 45 & 40 & 15 & 15 & 60 & 80 & 26 & 18 \\
\hline
\end{array}
\end{aligned}
$$

Carson Merrill

Numerade Educator

Problem 51

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
\left(t^{3}, t^{2}-1\right), \quad t=-4
$$

Linda Hand

Numerade Educator

Problem 52

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
(2 t+9,7 t-9), \quad t=1
$$

Linda Hand

Numerade Educator

Problem 53

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
\left(s^{-1}-3 s, s^{3}\right), \quad s=-1
$$

Linda Hand

Numerade Educator

Problem 54

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
(\sin 2 \theta, \cos 3 \theta), \quad \theta=\frac{\pi}{6}
$$

Linda Hand

Numerade Educator

Problem 55

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
\left(\sin ^{3} \theta, \cos \theta\right), \quad \theta=\frac{\pi}{4}
$$

Linda Hand

Numerade Educator

Problem 56

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
(\sec \theta, \tan \theta), \quad t=\frac{\pi}{4}
$$

Linda Hand

Numerade Educator

Problem 57

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
\left(\ln t, \frac{1}{t}\right), \quad t=4
$$

Linda Hand

Numerade Educator

Problem 58

In Exercises $51-58$, use Eq. (6) to find $d y / d x$ at the given point.
$$
\left(e^{t}, t^{2}\right), \quad t=1
$$

Linda Hand

Numerade Educator

Problem 59

In Exercises $59-64$, find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in nwo ways: using $E q \cdot(6)$ and by differentiating $f(x)$.
$$
c(t)=(2 t+1,1-9 t)
$$

Linda Hand

Numerade Educator

Problem 60

In Exercises $59-64$, find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in nwo ways: using $E q \cdot(6)$ and by differentiating $f(x)$.
$$
c(t)=\left(\frac{1}{2} t, \frac{1}{4} t^{2}-t\right)
$$

Linda Hand

Numerade Educator

Problem 61

In Exercises $59-64$, find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in nwo ways: using $E q \cdot(6)$ and by differentiating $f(x)$.
$$
x=s^{3}, \quad y=s^{6}+s^{-3}
$$

Linda Hand

Numerade Educator

Problem 62

In Exercises $59-64$, find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in nwo ways: using $E q \cdot(6)$ and by differentiating $f(x)$.
$$
x=\cos \theta, \quad y=\cos \theta+\sin ^{2} \theta
$$

Linda Hand

Numerade Educator

Problem 63

In Exercises $59-64$, find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in nwo ways: using $E q \cdot(6)$ and by differentiating $f(x)$.
$$
x=1-e^{t}, \quad y=t-1
$$

Linda Hand

Numerade Educator

Problem 64

In Exercises $59-64$, find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in nwo ways: using $E q \cdot(6)$ and by differentiating $f(x)$.
$$
x=1+\ln t, \quad y=\frac{1}{t}
$$

Linda Hand

Numerade Educator

Problem 65

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

Linda Hand

Numerade Educator

Problem 66

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 67

In Exercises $67-70,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 19$) .$
Draw an arrow indicating the direction of motion, and determine the interval(s) of $t$ -values corresponding to the portion(s) of the curve in each of the four quadrants.

Linda Hand

Numerade Educator

Problem 68

In Exercises $67-70,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 19$) .$
Find the equation of the tangent line at $t=4$.

Linda Hand

Numerade Educator

Problem 69

In Exercises $67-70,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 19$) .$
Find the points where the tangent has slope $\frac{1}{2}$.

Linda Hand

Numerade Educator

Problem 70

In Exercises $67-70,$ let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 19$) .$
Find the points where the tangent is horizontal or vertical.

Linda Hand

Numerade Educator

Problem 71

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 20). Let $P$ De the point of intersection of the horizontal line through $A$ and the verical line through $B$. Express the coordinates of $P$ as a function of $\theta$ and Jescribe the curve traced by $P$ for $0 \leq \theta \leq 2 \pi$.

Carson Merrill

Numerade Educator

Problem 72

A 10-ft ladder slides down a wall as its bottom $B$ is pulled away from the wall (Figure 21 ). Using the angle $\theta$ as a 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 $P$ on the ladder, located $4 \mathrm{ft}$ from the top. Show that $P$ describes an ellipse.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 73

In Exercises $73-76,$ refer to the Bezier curve defined by Eqs. (7) and $(8) .$
Show that the Bézier curve with control points
$$
P_{0}=(1,4), \quad P_{1}=(3,12), \quad P_{2}=(6,15), \quad P_{3}=(7,4)
$$
has parametrization
$$
c(t)=\left(1+6 t+3 t^{2}-3 t^{3}, 4+24 t-15 t^{2}-9 t^{3}\right)
$$
Verify that the slope at $t=0$ is equal to the slope of the segment $\overline{P_{0} P_{1}}$

Carson Merrill

Numerade Educator

Problem 74

In Exercises $73-76,$ refer to the Bezier curve defined by Eqs. (7) and $(8) .$
Find an equation of the tangent line to the Bézier curve in Exercise 73 at $t=\frac{1}{3}$,

Linda Hand

Numerade Educator

Problem 75

In Exercises $73-76,$ refer to the Bezier curve defined by Eqs. (7) and $(8) .$
CAS Find and plot the Bézier curve $c(t)$ with control points
$$
P_{0}=(3,2), \quad P_{1}=(0,2), \quad P_{2}=(5,4), \quad P_{3}=(2,4)
$$

Carson Merrill

Numerade Educator

Problem 76

In Exercises $73-76,$ refer to the Bezier curve defined by Eqs. (7) and $(8) .$
Show that a cubic Bézier curve is tangent to the segment $\overline{P_{2} P_{3}}$ at $P_{3}$.

Carson Merrill

Numerade Educator

Problem 77

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

Carson Merrill

Numerade Educator

Problem 78

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.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 79

Plot the astroid $x=\cos ^{3} \theta, y=\sin ^{3} \theta$ and find the equation of the tangent line at $\theta=\frac{\pi}{3}$.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 80

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

Linda Hand

Numerade Educator

Problem 81

Find the points with a horizontal tangent line on the cycloid with para-

Linda Hand

Numerade Educator

Problem 82

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 22 . Assume the generating circle of the cycloid has

Carson Merrill

Numerade Educator

Problem 83

A curtate cycloid (Figure 23 ) 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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 84

Use a computer algebra system to explore what happens when $h \gg R$ in the parametric equations of Exercise $83 .$ Describe the result.

Carson Merrill

Numerade Educator

Problem 85

A 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 24). Show further that $t=y /(x+1)$,

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 86

The follum of Descartes is the curve with equation $x^{3}+y^{3}=3 a x y$, where $a \neq 0$ is a constant (Figure 25 ).
(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 $\mathbf{I}, \mathbf{I I},$ and $\mathrm{IV}$. 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 horizontal or vertical tangent.

Carson Merrill

Numerade Educator

Problem 87

Use the results of Exercise 86 to show that the asymptote of the folium is the line $x+y=-a$. Hint: Show that $\lim _{1}(x+y)=-a$.

Carson Merrill

Numerade Educator

Problem 88

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 89

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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 90

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 $c(t)=\left(\tan t, \tan ^{2} t\right)$ also yield $d y^{2} / d^{2} x=2$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 91

In Exercises $91-94$, use Eq. (11) to find $d^{2} y / d x^{2}$.
$$
x=t^{3}+t^{2}, \quad y=7 t^{2}-4, \quad t=2
$$

Linda Hand

Numerade Educator

Problem 92

In Exercises $91-94$, use Eq. (11) to find $d^{2} y / d x^{2}$.
$$
x=s^{-1}+s, \quad y=4-s^{-2}, \quad s=1
$$

Linda Hand

Numerade Educator

Problem 93

In Exercises $91-94$, use Eq. (11) to find $d^{2} y / d x^{2}$.
$$
x=8 t+9, \quad y=1-4 t, \quad t=-3
$$

Linda Hand

Numerade Educator

Problem 94

In Exercises $91-94$, use Eq. (11) to find $d^{2} y / d x^{2}$.
$$
x=\cos \theta, \quad y=\sin \theta, \quad \theta=\frac{\pi}{4}
$$

Linda Hand

Numerade Educator

Problem 95

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 96

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 97

Calculate the area under $y=x^{2}$ over [0,1] using Eq. (9) with the parametrizations $\left(t^{3}, t^{6}\right)$ and $\left(t^{2}, t^{4}\right)$.

Linda Hand

Numerade Educator

Problem 98

What does Eq. (9) say if $c(t)=(t, f(t)) ?$

Linda Hand

Numerade Educator

Problem 99

Consider the curve $c(t)=\left(t^{2}, t^{3}\right)$ for $0 \leq t \leq 1$.
(a) Find the area under the curve using Eq. (9).
(b) Find the area under the curve by expressing $y$ as a function of $x$ and finding the area using the standard method.

Linda Hand

Numerade Educator

Problem 100

Compute the area under the parametrized curve $c(t)=\left(e^{t}, t\right)$ for $0 \leq t \leq 1$ using Eq. (9) .

Linda Hand

Numerade Educator

Problem 101

Compute the area under the parametrized curve given by $c(t)=$ $\left(\sin t, \cos ^{2} t\right)$ for $0 \leq t \leq \pi / 2$ using Eq. (9)

Linda Hand

Numerade Educator

Problem 102

Sketch the graph of $c(t)=(\ln t, 2-t)$ for $1 \leq t \leq 2$ and compute the area under the graph using Eq. (9).

Linda Hand

Numerade Educator

Problem 103

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 $R$

Linda Hand

Numerade Educator

Problem 104

Prove the following generalization of Exercise 103: 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 27 .

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 105

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 $x^{\prime}\left(t_{0}\right) \neq 0 .$ 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 Mockensturm

Eric Mockensturm

Numerade Educator

Problem 106

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 28 ).

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 107

In Exercise 62 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 parametric curve $c(t)$ identified as the tractrix in Exercise 106 satisfies this differential equation. Note that the derivative on the left is taken with respect to $x,$ not $t$.

Linda Hand

Numerade Educator

Problem 108

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 29 . Hint: Use Eq. (9).

Carson Merrill

Numerade Educator

Problem 109

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator