
# 4.2E:

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## Calculus of Parametric Curves

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

1) $$\displaystyle x=3+t,y=1−t$$

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

Solution: 0

3) $$\displaystyle x=4−3t,y=−2+6t$$

4) $$\displaystyle x=−5t+7,y=3t−1$$

Solution: $$\displaystyle \frac{−3}{5}$$

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

5) $$\displaystyle x=3sint,y=3cost,t=\frac{π}{4}$$

6) $$\displaystyle x=cost,y=8sint,t=\frac{π}{2}$$

Solution: $$\displaystyle Slope=0; y=8.$$

7) $$\displaystyle x=2t,y=t^3,t=−1$$

8) $$\displaystyle x=t+\frac{1}{t},y=t−\frac{1}{t},t=1$$

Solution: Slope is undefined; $$\displaystyle x=2$$.

9) $$\displaystyle x=\sqrt{t},y=2t,t=4$$

For the following exercises, find all points on the curve that have the given slope.

10) $$\displaystyle x=4cost,y=4sint,$$ slope = 0.5

Solution: $$\displaystyle t=arctan(−2); (\frac{4}{\sqrt{5}},\frac{−8}{\sqrt{5}})$$.

11) $$\displaystyle x=2cost,y=8sint,slope=−1$$

12) $$\displaystyle x=t+\frac{1}{t},y=t−\frac{1}{t},slope=1$$

Solution: No points possible; undefined expression.

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

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

14) $$\displaystyle x=e^{\sqrt{t}},y=1−lnt^2,t=1$$

Solution: $$\displaystyle y=−(\frac{2}{e})x+3$$

15) $$\displaystyle x=tlnt,y=sin^2t,t=\frac{π}{4}$$

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

Solution: $$\displaystyle y=2x−7$$

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

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

Solution: $$\displaystyle \frac{π}{4},\frac{5π}{4},\frac{3π}{4},\frac{7π}{4}$$

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

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

Solution: $$\displaystyle \frac{dy}{dx}=−tan(t)$$

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

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

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

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

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

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

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

26) Find all points on the curve $$\displaystyle x=secθ,y=tanθ$$ at which horizontal and vertical tangents exist.

Solution: No horizontal tangents. Vertical tangents at $$\displaystyle (1,0),(−1,0)$$.

For the following exercises, find $$\displaystyle d^2y/dx^2$$.

27)  $$\displaystyle x=t^4−1,y=t−t^2$$

28) $$\displaystyle x=sin(πt),y=cos(πt)$$

Solution: $$\displaystyle −sec^3(πt)$$

29) $$\displaystyle x=e^{−t},y=te^{2t}$$

For the following exercises, find points on the curve at which tangent line is horizontal or vertical.

30) $$\displaystyle x=t(t^2−3),y=3(t^2−3)$$

Solution: Horizontal $$\displaystyle (0,−9)$$; vertical $$\displaystyle (±2,−6).$$

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

For the following exercises, find $$\displaystyle dy/dx$$ at the value of the parameter.

32) $$\displaystyle x=cost,y=sint,t=\frac{3π}{4}$$

Solution: 1

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

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

Solution: 0

For the following exercises, find $$\displaystyle d^2y/dx^2$$ at the given point without eliminating the parameter.

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

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

Solution: 4

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

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

Solution: Concave up on $$\displaystyle t>0$$.

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

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

Solution: 1

41) Find the area enclosed by the ellipse $$\displaystyle x=acosθ,y=bsinθ,0≤θ<2π.$$

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

Solution: $$\displaystyle \frac{3π}{2}$$

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

43) $$\displaystyle x=2cotθ,y=2sin^2θ,0≤θ≤π$$

44) [T] $$\displaystyle x=2acost−acos(2t),y=2asint−asin(2t),0≤t<2π$$

Solution: $$\displaystyle 6πa^2$$

45) [T] $$\displaystyle x=asin(2t),y=bsin(t),0≤t<2π$$ (the “hourglass”)

46) [T] $$\displaystyle x=2acost−asin(2t),y=bsint,0≤t<2π$$ (the “teardrop”)

Solution: $$\displaystyle 2πab$$

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

47) $$\displaystyle x=4t+3,y=3t−2,0≤t≤2$$

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

Soluton: $$\displaystyle \frac{1}{3}(2\sqrt{2}−1)$$

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

50) $$\displaystyle x=1+t^2,y=(1+t)^3,0≤t≤1$$

Solution: 7.075

51) $$\displaystyle x=e^tcost,y=e^tsint,0≤t≤\frac{π}{2}$$ (express answer as a decimal rounded to three places)

52) $$\displaystyle x=acos^3θ,y=asin^3θ$$ on the interval $$\displaystyle [0,2π)$$ (the hypocycloid)

Solution: $$\displaystyle 6a$$

53) Find the length of one arch of the cycloid $$\displaystyle x=4(t−sint),y=4(1−cost).$$

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

Solution: $$\displaystyle 6\sqrt{2}$$

55) Find the length of one arch of the cycloid $$\displaystyle x=θ−sinθ,y=1−cosθ$$.

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

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

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

58) $$\displaystyle x=t^3,y=t^2,0≤t≤1$$

Solution: $$\displaystyle \frac{2π(247\sqrt{13}+64)}{1215}$$

59) $$\displaystyle x=acos^3θ,y=asin^3θ,0≤θ≤\frac{π}{2}$$

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

Solution: 59.101

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

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

Solution: $$\displaystyle \frac{8π}{3}(17\sqrt{17}−1)$$

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