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Calculus for AP

Jon Rogawski & Colin Adams

Chapter 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS - all with Video Answers

Educators

+ 4 more educators

Section 1

Parametric Equations

01:00

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 Mockensturm
Eric Mockensturm
Numerade Educator
00:56

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
00:57

Problem 3

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

ML
Melony Liu
Numerade Educator
01:27

Problem 4

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:09

Problem 5

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:01

Problem 6

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

Bobby Barnes
Bobby Barnes
University of North Texas
00:46

Problem 7

Express in the form $y=f(x)$ by eliminating the parameter.
$x=t+3, \quad y=4 t$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:05

Problem 8

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:16

Problem 9

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:18

Problem 10

Express in the form $y=f(x)$ by eliminating the parameter.
$x=t^{2}, \quad y=t^{3}+1$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:35

Problem 11

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

Carson Merrill
Carson Merrill
Numerade Educator
01:02

Problem 12

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:10

Problem 13

Express in the form $y=f(x)$ by eliminating the parameter.
$x=\ln t, \quad y=2-t$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:20

Problem 14

Express in the form $y=f(x)$ by eliminating the parameter.
$x=\cos t, \quad y=\tan t$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
04:52

Problem 15

Graph the curve and draw an arrow specifying the direction corresponding to motion.
$x=\frac{1}{2} t, \quad y=2 t^{2}$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
04:57

Problem 16

Graph the curve and draw an arrow specifying the direction corresponding to motion.
$x=2+4 t, \quad y=3+2 t$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:19

Problem 17

Graph the curve and draw an arrow specifying the direction corresponding to motion.
$x=\pi t, \quad y=\sin t$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
04:16

Problem 18

Graph the curve and draw an arrow specifying the direction corresponding to motion.
$x=t^{2}, \quad y=t^{3}$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:22

Problem 19

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:30

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
Sheryl Ezze
Numerade Educator
02:19

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
01:13

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
01:13

Problem 23

Find parametric equations for the given curve.
$y=9-4 x$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:12

Problem 24

Find parametric equations for the given curve.
$y=8 x^{2}-3 x$

William Semus
William Semus
Numerade Educator
00:51

Problem 25

Find parametric equations for the given curve.
$4 x-y^{2}=5$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:49

Problem 26

Find parametric equations for the given curve.
$x^{2}+y^{2}=49$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:17

Problem 27

Find parametric equations for the given curve.
$(x+9)^{2}+(y-4)^{2}=49$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:55

Problem 28

Find parametric equations for the given curve.
$\left(\frac{x}{5}\right)^{2}+\left(\frac{y}{12}\right)^{2}=1$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:49

Problem 29

Find parametric equations for the given curve.
Line of slope 8 through $(-4,9)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:27

Problem 30

Find parametric equations for the given curve.
Line through $(2,5)$ perpendicular to $y=3 x$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:15

Problem 31

Find parametric equations for the given curve.
Line through $(3,1)$ and $(-5,4)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:13

Problem 32

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:47

Problem 33

Find parametric equations for the given curve.
Segment joining $(1,1)$ and $(2,3)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:08

Problem 34

Find parametric equations for the given curve.
Segment joining $(-3,0)$ and $(0,4)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:02

Problem 35

Find parametric equations for the given curve.
Circle of radius 4 with center $(3,9)$

Carson Merrill
Carson Merrill
Numerade Educator
02:00

Problem 36

Find parametric equations for the given curve.
Ellipse of Exercise 28, with its center translated to $(7,4)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:52

Problem 37

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:51

Problem 38

Find parametric equations for the given curve.
$y=\cos x,$ translated so that a maximum occurs at $(3,5)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:07

Problem 39

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:13

Problem 40

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:42

Problem 41

Find a parametrization $c(t)$ of the curve satisfying the given condition.
$y=x^{2}, \quad c(0)=(3,9)$

Carson Merrill
Carson Merrill
Numerade Educator
01:58

Problem 42

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:09

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 Mockensturm
Eric Mockensturm
Numerade Educator
07:49

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
03:13

Problem 45

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:07

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 17$(\mathrm{A}) ?$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:10

Problem 47

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:20

Problem 48

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:24

Problem 49

Use Eq. (8) to find $d y / d x$ at the given point.
$\left(t^{3}, t^{2}-1\right), \quad t=-4$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:15

Problem 50

Use Eq. (8) to find $d y / d x$ at the given point.
$(2 t+9,7 t-9), \quad t=1$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:10

Problem 51

Use Eq. (8) to find $d y / d x$ at the given point.
$\left(s^{-1}-3 s, s^{3}\right), \quad s=-1$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:23

Problem 52

Use Eq. (8) to find $d y / d x$ at the given point.
$(\sin 2 \theta, \cos 3 \theta), \quad \theta=\frac{\pi}{6}$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
05:26

Problem 53

Use Eq. (8) to find $d y / d x$ at the given point.
$\left(\sin ^{3} \theta, \cos \theta\right), \quad \theta=\frac{\pi}{4}$

Willis James
Willis James
Numerade Educator
01:45

Problem 54

Use Eq. (8) to find $d y / d x$ at the given point.
$\left(e^{t}, t^{2}\right), \quad t=1$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:57

Problem 55

Find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using Eq. (8) and by differentiating $f(x)$.
$c(t)=(2 t+1,1-9 t)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:28

Problem 56

Find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using Eq. (8) and by differentiating $f(x)$.
$c(t)=\left(\frac{1}{2} t, \frac{1}{4} t^{2}-t\right)$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:52

Problem 57

Find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using Eq. (8) and by differentiating $f(x)$.
$x=s^{3}, \quad y=s^{6}+s^{-3}$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:15

Problem 58

Find an equation $y=f(x)$ for the parametric curve and compute $d y / d x$ in two ways: using Eq. (8) and by differentiating $f(x)$.
$x=\cos \theta, \quad y=\cos \theta+\sin ^{2} \theta$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
00:57

Problem 59

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.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:28

Problem 60

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
09:35

Problem 61

Let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 18$)$
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.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:32

Problem 62

Let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 18$)$
Find the equation of the tangent line at $t=4$.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:54

Problem 63

Let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 18$)$
Find the points where the tangent has slope $\frac{1}{2}$.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:38

Problem 64

Let $c(t)=\left(t^{2}-9, t^{2}-8 t\right)($ see Figure 18$)$
Find the points where the tangent is horizontal or vertical.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:55

Problem 65

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 19). 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 Mockensturm
Eric Mockensturm
Numerade Educator
03:05

Problem 66

A 10-ft ladder slides down a wall as its bottom $B$ is pulled away from the wall (Figure 20). 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 $P$ located 4 ft from the top of the ladder. Show that $P$ describes an ellipse.

Carson Merrill
Carson Merrill
Numerade Educator
10:04

Problem 67

Refer to the Bezier curve defined by Eqs. (9) and (10).
Show that the Bezier 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 $\overline{P_{0} P_{1}}$.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
06:12

Problem 68

Refer to the Bezier curve defined by Eqs. (9) and (10).
Find an equation of the tangent line to the Beziercurve in Exercise 67 at $t=\frac{1}{3}$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
07:53

Problem 69

Refer to the Bezier curve defined by Eqs. (9) and (10).
Find and plot the Bezier curve $c(t)$ with control points $$P_{0}=(3,2), \quad P_{1}=(0,2), \quad P_{2}=(5,4), \quad P_{3}=(2,4)$$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
05:24

Problem 70

Refer to the Bezier curve defined by Eqs. (9) and (10).
Show that a cubic Bezier curve is tangent to the segment $\overline{P_{2} P_{3}}$ at $P_{3} .$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:03

Problem 71

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 $\frac{a b}{16}$ from the origin.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
07:27

Problem 72

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
06:43

Problem 73

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
05:39

Problem 74

Find the equation of the tangent line at $t=\frac{\pi}{4}$ to the cycloid generated by the unit circle with parametric equation (6).

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:37

Problem 75

Find the points with a horizontal tangent line on the cycloid with parametric equation (6).

Carson Merrill
Carson Merrill
Numerade Educator
04:10

Problem 76

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 21. Assume the generating circle of the cycloid has radius 1.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:29

Problem 77

A curtate cycloid (Figure 22$)$ 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
01:53

Problem 78

Use a computer algebra system to explore what happens when $h>R$ in the parametric equations of Exercise 77. Describe the result.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
06:26

Problem 79

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 23). Show further that $t=y /(x+1)$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
07:44

Problem 80

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 24).
(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 horizontal or vertical tangent.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
05:49

Problem 81

Use the results of Exercise 80 to show that the asymptote of the folium is the line $x+y=-a .$Hint. Show that $$\lim _{t \rightarrow-1}(x+y)=-a.$$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:18

Problem 82

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
03:37

Problem 83

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
08:47

Problem 84

The second derivative of $y=x^{2}$ is $d y^{2} / d^{2} x=2 .$ Verify that Eq. (13) 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 $d y^{2} / d^{2} x=2.$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
05:34

Problem 85

use Eq .(13) to find $d^{2} y / d x^{2}.$
$x=t^{3}+t^{2}, \quad y=7 t^{2}-4, \quad t=2$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:35

Problem 86

use Eq .(13) to find $d^{2} y / d x^{2}.$
$x=s^{-1}+s, \quad y=4-s^{-2}, \quad s=1$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:36

Problem 87

use Eq .(13) to find $d^{2} y / d x^{2}.$
$x=8 t+9, \quad y=1-4 t, \quad t=-3$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
04:37

Problem 88

use Eq .(13) to find $d^{2} y / d x^{2}.$
$x=\cos \theta, \quad y=\sin \theta, \quad \theta=\frac{\pi}{4}$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
04:22

Problem 89

Use Eq. (13) to find the $t$-intervals on which $c(t)=\left(t^{2}, t^{3}-4 t\right)$ is concave up.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
05:24

Problem 90

Use Eq. (13) to find the $t$-intervals on which $c(t)=\left(t^{2}, t^{4}-4 t\right)$ is concave up.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
06:12

Problem 91

Calculate the area under $y=x^{2}$ over $[0,1]$ using Eq. (11) with the parametrizations $\left(t^{3}, t^{6}\right)$ and $\left(t^{2}, t^{4}\right).$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:10

Problem 92

What does Eq. (11) say if $c(t)=(t, f(t)) ?$

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
03:41

Problem 93

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. (11).
(b) Find the area under the curve by expressing $y$ as a function of $x$ and finding the area using the standard method.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
04:17

Problem 94

Compute the area under the parametrized curve $c(t)=\left(e^{t}, t\right)$ for $0 \leq t \leq 1$ using Eq (11).

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:59

Problem 95

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. (11).

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
04:18

Problem 96

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 (11).

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
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Problem 97

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. (11).

Susan Hallstrom
Susan Hallstrom
Numerade Educator
06:09

Problem 98

Prove the following generalization of Exercise 97: 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 Mockensturm
Eric Mockensturm
Numerade Educator
02:27

Problem 99

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 $d y / d x .$ Dragram showing that the ratio in the limit is the slope of a secant line.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:46

Problem 100

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 Mockensturm
Eric Mockensturm
Numerade Educator
07:59

Problem 101

In Exercise 59 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 100 satisfies this differential equation. Note that the derivative on the left is taken with respect to $x,$ not $t$ .

Linda Hand
Linda Hand
Numerade Educator
01:07

Problem 102

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. (11).

Carson Merrill
Carson Merrill
Numerade Educator
06:12

Problem 103

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator