# 11.3E: Exercises

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

$$m=0$$

3) $$x=4−3t, \quad y=−2+6t$$

4) $$x=−5t+7, \quad y=3t−1$$

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

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

Slope$$=0; y=8.$$

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

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

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= $$1$$

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

$$y=−\left(\frac{4}{e}\right)x+5$$

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 $$(1,1)$$

$$y=-2x+3$$

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.

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 $$x=\sin(t), \quad y=\cos(t)$$.

$$\dfrac{dy}{dx}=−\tan(t)$$

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 $$t=3$$.

$$\dfrac{dy}{dx}=\dfrac{3}{4}$$ 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 $$θ=\frac{π}{6}$$.

$$\dfrac{dy}{dx}=4, \quad \dfrac{d^2y}{dx^2}=−4\sqrt{3};$$ 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.

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) $$x=\sin(πt), \quad y=\cos(πt)$$

$$d^2y/dx^2 = −\sec^3(πt)$$

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) $$x=t(t^2−3), \quad y=3(t^2−3)$$

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

$$dy/dx = 1$$

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

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

$$dy/dx = 0$$

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

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

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 $$x=2t+\ln t, \quad y=2t−\ln t$$.

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 $$x=0$$.

$$1\text{ unit}^2$$

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 $$0≤θ≤\frac{π}{2}$$.

$$\frac{3π}{2}\text{ units}^2$$

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

$$6πa^2\text{ units}^2$$

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”)

$$2πab\text{ units}^2$$

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

$$s = \frac{1}{3}(2\sqrt{2}−1)$$ units

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

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

$$s = 7.075$$ 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)

$$s = 6a$$ 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: $$x=\sin^2t, \quad y=\cos^2t, \quad \text{for }0≤t≤3π$$.

$$6\sqrt{2}$$ 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) $$x=t^3, \quad y=t^2, \quad \text{for }0≤t≤1$$

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

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.)

$$59.101\text{ units}^2$$

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.

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