11.2E: Exercises for Section 11.2

In exercises 1 - 4, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

1) $x=3+t,\quad y=1−t$

2) $x=8+2t, \quad y=1$

Answer

$m=0$

3) $x=4−3t, \quad y=−2+6t$

4) $x=−5t+7, \quad y=3t−1$

Answer

$m= -\frac{3}{5}$

In exercises 5 - 9, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

5) $x=3\sin t,\quad y=3\cos t, \quad \text{for }t=\frac{π}{4}$

6) $x=\cos t, \quad y=8\sin t, \quad \text{for }t=\frac{π}{2}$

Answer

Slope$=0; y=8.$

7) $x=2t, \quad y=t^3, \quad \text{for } t=−1$

8) $x=t+\dfrac{1}{t}, \quad y=t−\dfrac{1}{t}, \quad \text{for }t=1$

Answer

Slope is undefined; $x=2$.

9) $x=\sqrt{t}, \quad y=2t, \quad \text{for }t=4$

In exercises 10 - 13, find all points on the curve that have the given slope.

10) $x=4\cos t, \quad y=4\sin t,$ slope = $0.5$

Solution

$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} = \dfrac{4\cos t}{-4\sin t} = - \cot t.$
Setting this derivative equal to $0.5,$ we obtain the equation, $\tan t = -2.$
$\tan t = -2 \implies \dfrac{y}{x} = -2 \implies y = -2x.$
Note also that this pair of parametric equations represents the circle $x^2 + y^2 = 16.$
By substitution, we find that this curve has a slope of $0.5$ at the points:
$\left(\frac{4\sqrt{5}}{5},\frac{−8\sqrt{5}}{5}\right)$ and $\left(\frac{-4\sqrt{5}}{5},\frac{8\sqrt{5}}{5}\right).$

11) $x=2\cos t, \quad y=8\sin t,$ slope= $−1$

12) $x=t+\dfrac{1}{t}, \quad y=t−\dfrac{1}{t},$ slope= $1$

Answer

No points possible; undefined expression.

13) $x=2+\sqrt{t}, \quad y=2−4t,$ slope= $0$

In exercises 14 - 16, write the equation of the tangent line in Cartesian coordinates for the given parameter $t$.

14) $x=e^{\sqrt{t}}, \quad y=1−\ln t^2, \quad \text{for }t=1$

Answer

$y=−\left(\frac{4}{e}\right)x+5$

15) $x=t\ln t, \quad y=\sin^2t, \quad \text{for }t=\frac{π}{4}$

16) $x=e^t, \quad y=(t−1)^2,$ at $(1,1)$

Answer

$y=-2x+3$

17) For $x=\sin(2t), \quad y=2\sin t$ where $0≤t<2π.$ Find all values of $t$ at which a horizontal tangent line exists.

18) For $x=\sin(2t), \quad y=2\sin t$ where $0≤t<2π$. Find all values of $t$ at which a vertical tangent line exists.

Answer

A vertical tangent line exists at $t = \frac{π}{4},\frac{5π}{4},\frac{3π}{4},\frac{7π}{4}$

19) Find all points on the curve $x=4\cos(t), \quad y=4\sin(t)$ that have the slope of $\frac{1}{2}$.

20) Find $\dfrac{dy}{dx}$ for $x=\sin(t), \quad y=\cos(t)$.

Answer

$\dfrac{dy}{dx}=−\tan(t)$

21) Find the equation of the tangent line to $x=\sin(t), \quad y=\cos(t)$ at $t=\frac{π}{4}$.

22) For the curve $x=4t, \quad y=3t−2,$ find the slope and concavity of the curve at $t=3$.

Answer

$\dfrac{dy}{dx}=\dfrac{3}{4}$ and $\dfrac{d^2y}{dx^2}=0$, so the curve is neither concave up nor concave down at $t=3$. Therefore the graph is linear and has a constant slope but no concavity.

23) For the parametric curve whose equation is $x=4\cos θ, \quad y=4\sin θ$, find the slope and concavity of the curve at $θ=\frac{π}{4}$.

24) Find the slope and concavity for the curve whose equation is $x=2+\sec θ, \quad y=1+2\tan θ$ at $θ=\frac{π}{6}$.

Answer

$\dfrac{dy}{dx}=4, \quad \dfrac{d^2y}{dx^2}=−4\sqrt{3};$ the curve is concave down at $θ=\frac{π}{6}$.

25) Find all points on the curve $x=t+4, \quad y=t^3−3t$ at which there are vertical and horizontal tangents.

26) Find all points on the curve $x=\sec θ, \quad y=\tan θ$ at which horizontal and vertical tangents exist.

Answer

No horizontal tangents. Vertical tangents at $(1,0)$ and $(−1,0)$.

In exercises 27 - 29, find $d^2y/dx^2$.

27) $x=t^4−1, \quad y=t−t^2$

28) $x=\sin(πt), \quad y=\cos(πt)$

Answer

$d^2y/dx^2 = −\sec^3(πt)$

29) $x=e^{−t}, \quad y=te^{2t}$

In exercises 30 - 31, find points on the curve at which tangent line is horizontal or vertical.

30) $x=t(t^2−3), \quad y=3(t^2−3)$

Answer

Horizontal $(0,−9)$;
Vertical $(±2,−6).$

31) $x=\dfrac{3t}{1+t^3}, \quad y=\dfrac{3t^2}{1+t^3}$

In exercises 32 - 34, find $dy/dx$ at the value of the parameter.

32) $x=\cos t,y=\sin t, \quad \text{for }t=\frac{3π}{4}$

Answer

$dy/dx = 1$

33) $x=\sqrt{t}, \quad y=2t+4,t=9$

34) $x=4\cos(2πs), \quad y=3\sin(2πs), \quad \text{for }s=−\frac{1}{4}$

Answer

$dy/dx = 0$

In exercises 35 - 36, find $d^2y/dx^2$ at the given point without eliminating the parameter.

35) $x=\frac{1}{2}t^2, \quad y=\frac{1}{3}t^3, \quad \text{for }t=2$

36) $x=\sqrt{t}, \quad y=2t+4, \quad \text{for }t=1$

Answer

$d^2y/dx^2 = 4$

37) Find intervals for $t$ on which the curve $x=3t^2, \quad y=t^3−t$ is concave up as well as concave down.

38) Determine the concavity of the curve $x=2t+\ln t, \quad y=2t−\ln t$.

Answer

Concave up on $t>0$.

39) Sketch and find the area under one arch of the cycloid $x=r(θ−\sin θ), \quad y=r(1−\cos θ)$.

40) Find the area bounded by the curve $x=\cos t, \quad y=e^t, \quad \text{for }0≤t≤\frac{π}{2}$ and the lines $y=1$ and $x=0$.

Answer

$1\text{ unit}^2$

41) Find the area enclosed by the ellipse $x=a\cos θ, \quad y=b\sin θ, \quad \text{for }0≤θ<2π.$

42) Find the area of the region bounded by $x=2\sin^2θ, \quad y=2\sin^2θ\tan θ$, for $0≤θ≤\frac{π}{2}$.

Answer

$\frac{3π}{2}\text{ units}^2$

In exercises 43 - 46, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

43) $x=2\cot θ, \quad y=2\sin^2θ, \quad \text{for }0≤θ≤π$

44) [T] $x=2a\cos t−a\cos(2t), \quad y=2a\sin t−a\sin(2t), \quad \text{for }0≤t<2π$

Answer

$6πa^2\text{ units}^2$

45) [T] $x=a\sin(2t), \quad y=b\sin(t), \quad \text{for }0≤t<2π$ (the “hourglass”)

46) [T] $x=2a\cos t−a\sin(2t), \quad y=b\sin t, \quad \text{for }0≤t<2π$ (the “teardrop”)

Answer

$2πab\text{ units}^2$

In exercises 47 - 52, find the arc length of the curve on the indicated interval of the parameter.

47) $x=4t+3, \quad y=3t−2, \quad \text{for }0≤t≤2$

48) $x=\frac{1}{3}t^3, \quad y=\frac{1}{2}t^2, \quad \text{for }0≤t≤1$

Answer

$s = \frac{1}{3}(2\sqrt{2}−1)$ units

49) $x=\cos(2t), \quad y=\sin(2t), \quad \text{for }0≤t≤\frac{π}{2}$

50) $x=1+t^2, \quad y=(1+t)^3, \quad \text{for }0≤t≤1$

Answer

$s = 7.075$ units

51) $x=e^t\cos t, \quad y=e^t\sin t, \quad \text{for }0≤t≤\frac{π}{2}$ (express answer as a decimal rounded to three places)

52) $x=a\cos^3θ, \quad y=a\sin^3θ$ on the interval $[0,2π)$ (the hypocycloid)

Answer

$s = 6a$ units

53) Find the length of one arch of the cycloid $x=4(t−\sin t), \quad y=4(1−\cos t).$

54) Find the distance traveled by a particle with position $(x,y)$ as $t$ varies in the given time interval: $x=\sin^2t, \quad y=\cos^2t, \quad \text{for }0≤t≤3π$.

Answer

$6\sqrt{2}$ units

55) Find the length of one arch of the cycloid $x=θ−\sin θ, \quad y=1−\cos θ$.

56) Show that the total length of the ellipse $x=4\sin θ, \quad y=3\cos θ$ is $\displaystyle L=16∫^{π/2}_0\sqrt{1−e^2\sin^2θ}\,dθ$, where $e=\frac{c}{a}$ and $c=\sqrt{a^2−b^2}$.

57) Find the length of the curve $x=e^t−t, \quad y=4e^{t/2}, \quad \text{for }−8≤t≤3.$

In exercises 58 - 59, find the area of the surface obtained by rotating the given curve about the $x$-axis.

58) $x=t^3, \quad y=t^2, \quad \text{for }0≤t≤1$

Answer

$\dfrac{2π(247\sqrt{13}+64)}{1215}\text{ units}^2$

59) $x=a\cos^3θ, \quad y=a\sin^3θ, \quad \text{for }0≤θ≤\frac{π}{2}$

60) [T] Use a CAS to find the area of the surface generated by rotating $x=t+t^3, \quad y=t−\frac{1}{t^2}, \quad \text{for }1≤t≤2$ about the $x$-axis. (Answer to three decimal places.)

Answer

$59.101\text{ units}^2$

61) Find the surface area obtained by rotating $x=3t^2, \quad y=2t^3, \quad \text{for }0≤t≤5$ about the $y$-axis.

62) Find the area of the surface generated by revolving $x=t^2, \quad y=2t, \quad \text{for }0≤t≤4$ about the $x$-axis.

Answer

$\frac{8π}{3}(17\sqrt{17}−1) \text{ units}^2$

63) Find the surface area generated by revolving $x=t^2, \quad y=2t^2, \quad \text{for }0≤t≤1$ about the $y$-axis.