Parametric Curves and Their Applications: Get Stronger Answer Key

Fundamentals of Parametric Equations

  1. A parabola open to the right with (−1, 0) being the point furthest the left with arrow going from the bottom through (−1, 0) and up.
    orientation: bottom to top
  2. A straight line passing through (0, −3) and (6, 0) with arrow pointing up and to the right.
    orientation: left to right
  3. Half a parabola starting at the origin and passing through (2, 2) with arrow pointed up and to the right.
  4. A curve going through (1, 0) and (0, 3) with arrow pointing up and to the left.
  5. A graph with asymptotes at the x and y axes. There is a portion of the graph in the third quadrant with arrow pointing down and to the right. There is a portion of the graph in the first quadrant with arrow pointing down and to the right.
  6. An ellipse with minor axis vertical and of length 8 and major axis horizontal and of length 12 that is centered at the origin. The arrows go counterclockwise.
  7. An ellipse in the fourth quadrant with minor axis horizontal and of length 4 and major axis vertical and of length 6. The arrows go clockwise.
  8. A graph with asymptotes at y = x and y = −x. The first part of the graph occurs in the second and third quadrants with vertex at (−1, 0). The second part of the graph occurs in the first and fourth quadrants with vertex as (1, 0).
    Asymptotes are [latex]y=x[/latex] and [latex]y=\text{-}x[/latex]
  9. A curve starting slightly above the origin and increasing to the right with arrow pointing up and to the right.
  10. A curve with asymptote being the y axis. The curve starts in the fourth quadrant and increases rapidly through (1, 0) at which point is increases much more slowly.
  11. [latex]y=\dfrac{\sqrt{x+1}}{2}[/latex]; domain: [latex]x\in \left[1,\infty \right)[/latex].
  12. [latex]\dfrac{{x}^{2}}{16}+\dfrac{{y}^{2}}{9}=1[/latex]; domain [latex]x\in \left[-4,4\right][/latex].
  13. [latex]y=3x+2[/latex]; domain: all real numbers.
  14. [latex]{\left(x - 1\right)}^{2}+{\left(y - 3\right)}^{2}=1[/latex]; domain: [latex]x\in \left[0,2\right][/latex].
  15. [latex]y=\sqrt{{x}^{2}-1}[/latex]; domain: [latex]x\in \left[-\infty,1\right][/latex].
  16. [latex]{y}^{2}=\dfrac{1-x}{2}[/latex]; domain: [latex]x\in \left[2,\infty \right)\cup \left(\text{-}\infty ,-2\right][/latex].
  17. [latex]y=\text{ln}x[/latex]; domain: [latex]x\in \left(1,\infty \right)[/latex].
  18. [latex]y=\text{ln}x[/latex]; domain: [latex]x\in \left(0,\infty \right)[/latex].
  19. [latex]{x}^{2}+{y}^{2}=4[/latex]; domain: [latex]x\in \left[-2,2\right][/latex].
  20. line
  21. parabola
  22. circle
  23. ellipse
  24. hyperbola
  25. A graph starting at (−6, 0) increasing rapidly to a sharp point at (−3, 2) and then decreasing rapidly to the origin. The graph is symmetric about the y axis, so the graph increases rapidly to (3, 2) before decreasing rapidly to (6, 0).
    The equations represent a cycloid.
  26. A graph starting at roughly (−6, 0) increasing to a rounded point and then decreasing to roughly (0, −0.5). The graph is symmetric about the y axis, so the graph increases to a rounded point before decreasing to roughly (6, 0).

Calculus with Parametric Curves

  1. [latex]0[/latex]
  2. [latex]\dfrac{-3}{5}[/latex]
  3. [latex]\text{Slope}=0[/latex]; [latex]y=8[/latex].
  4. Slope is undefined; [latex]x=2[/latex].
  5. [latex]\tan{t}=\left(-2\right)[/latex] [latex]\left(\dfrac{4}{\sqrt{5}},\dfrac{-8}{\sqrt{5}}\right),\left(\dfrac{4}{\sqrt{5}},\dfrac{-8}{\sqrt{5}}\right)[/latex].
  6. No points possible; undefined expression.
  7. [latex]y=\text{-}\left(\dfrac{4}{e}\right)x+5[/latex]
  8. [latex]y=-2x + 3[/latex]
  9. [latex]\dfrac{\pi }{4},\dfrac{5\pi }{4},\dfrac{3\pi }{4},\dfrac{7\pi }{4}[/latex]
  10. [latex]\dfrac{dy}{dx}=\text{-}\tan\left(t\right)[/latex]
  11. [latex]\dfrac{dy}{dx}=\dfrac{3}{4}[/latex] and [latex]\dfrac{{d}^{2}y}{d{x}^{2}}=0[/latex], so the curve is neither concave up nor concave down at [latex]t=3[/latex]. Therefore the graph is linear and has a constant slope but no concavity.
  12. [latex]\dfrac{dy}{dx}=4,\dfrac{{d}^{2}y}{d{x}^{2}}=-6\sqrt{3}[/latex]; the curve is concave down at [latex]\theta =\dfrac{\pi }{6}[/latex].
  13. No horizontal tangents. Vertical tangents at [latex]\left(1,0\right),\left(-1,0\right)[/latex].
  14. [latex]\text{-}{\sec}^{3}\left(\pi t\right)[/latex]
  15. Horizontal [latex]\left(0,-9\right)[/latex]; vertical [latex]\left(\pm2,-6\right)[/latex].
  16. [latex]1[/latex]
  17. [latex]0[/latex]
  18. [latex]4[/latex]
  19. Concave up on [latex]t>0[/latex].
  20. [latex]\dfrac{e\frac{1}{2}-1}{2}[/latex]
  21. [latex]\dfrac{3\pi }{2}[/latex]
  22. [latex]6\pi {a}^{2}[/latex]
  23. [latex]2\pi ab[/latex]
  24. [latex]\dfrac{1}{3}\left(2\sqrt{2}-1\right)[/latex]
  25. [latex]7.075[/latex]
  26. [latex]6a[/latex]
  27. [latex]\dfrac{2\pi \left(247\sqrt{13}+64\right)}{1215}[/latex]
  28. [latex]59.101[/latex]
  29. [latex]\dfrac{8\pi }{3}\left(17\sqrt{17}-1\right)[/latex]