# BC Calculus Worksheets: Extra Multiple Choice Practice on Parametrics, Vectors, and Polar

• Class: 5H: BC Calculus<\li>
• Author: Peter Atlas<\li>
• Text: Calculus Finney, Demana, Waits, Kennedy<\li>

1. Find $$\displaystyle \frac{dy}{dx}$$ given $$x = t - \sin{t}$$ and $$y = 1 - \cos{t}$$
1. $$\displaystyle \frac{\sin{t}}{1 - \cos{t}}$$
2. $$\displaystyle \frac{1 - \cos{t}}{\sin{t}}$$
3. $$\displaystyle \frac{\sin{t}}{\cos{t} - 1}$$
4. $$\displaystyle \frac{1 - x}{y}$$
5. $$\displaystyle \frac{1 - \cos{t}}{t - \sin{t}}$$
2. Solution A
3. Find $$\displaystyle \frac{dy}{dx}$$ given $$x = \cos^3 {\theta}$$ and $$y = \sin^3 {\theta}$$
1. $$\displaystyle \tan^3 {\theta}$$
2. $$\displaystyle - \cot{\theta }$$
3. $$\displaystyle \cot{\theta}$$
4. $$\displaystyle - \tan{\theta }$$
5. $$\displaystyle - \tan^2{\theta}$$
4. Solution D
5. Find $$\displaystyle \frac{dy}{dx}$$ given $$\displaystyle x = 1 - e^{-t}$$ and $$\displaystyle y = t + e^{-t}$$
1. $$\displaystyle \frac{e^{-t}}{1 - e^{-t}}$$
2. $$\displaystyle e^{-t} - 1$$
3. $$\displaystyle e^t + 1$$
4. $$\displaystyle e^t - e^{-2t}$$
5. $$\displaystyle e^t - 1$$
6. Solution E
7. Find$$\displaystyle \frac{dy}{dx}$$ given $$\displaystyle x = \frac{1}{1 - t}$$ and $$y = 1 - \ln{(1 - t)}$$, given $$t < 1$$.
1. $$\displaystyle \frac{t}{1 - t}$$
2. $$\displaystyle 1 - t$$
3. $$\displaystyle \frac{1}{x}$$
4. $$\displaystyle \frac{(1 - t)^2}{t}$$
5. $$\displaystyle 1 + \ln {x}$$
8. Solution B
9. If $$\displaystyle x = t^2 - 1$$ and $$y = t^4 - 2t^3$$, then when $$\displaystyle t = 1, \frac{d^2y}{dx^2} =$$
1. 1
2. -1
3. 0
4. 3
5. $$\displaystyle \frac{1}{2}$$
10. Solution E
11. If $$\displaystyle x = e^{\theta}\cos{\theta}$$ and $$\displaystyle y = e^{\theta}\sin{\theta}$$, then when $$\displaystyle \theta = \frac{\pi}{2},\frac{dy}{dx} =$$
1. 1
2. 0
3. $$\displaystyle e^{\frac{\pi}{2}}$$
4. nonexistent
5. -1
12. Solution E
13. If $$\displaystyle x = \cos{t}$$ and $$y = \cos {(2t)}$$, then $$\displaystyle \frac{d^2y}{dx^2} =$$
1. $$\displaystyle 4\cos{t}$$
2. 4
3. $$\displaystyle \frac{4y}{x}$$
4. -4
5. $$\displaystyle -4\cot {t}$$
14. Solution B
15. Given a particle in a plane moves according to the equations $$\displaystyle x = 2t$$ and $$\displaystyle y = 4t - t^2,$$ the particle moves along
1. an ellipse
2. a circle
3. a hyperbola
4. a line
5. a parabola
16. Solution E
17. Given a particle in a plane moves according to the equations $$\displaystyle x = 2t$$ and $$\displaystyle y = 4t - t^2$$, the speed of the particle at any time $$t$$ is
1. $$\displaystyle \sqrt{ 6 - 2t}$$
2. $$\displaystyle 2\sqrt{t^2 - 4t + 5}$$
3. $$\displaystyle 2\sqrt{t^2 - 2t + 5}$$
4. $$\displaystyle \sqrt{8} \left| t - 2 \right|$$
5. $$\displaystyle 2(3 - t)$$
18. Solution B
19. Given a particle in a plane moves according to the equations $$\displaystyle x = 2t$$ and $$\displaystyle y = 4t - t^2$$, the minimum speed of the particle is
1. 2
2. $$\displaystyle 2\sqrt{2}$$
3. 0
4. 1
5. 4
20. Solution A
21. Given a particle in a plane moves according to the equations $$\displaystyle x = 2t$$ and $$\displaystyle y = 4t - t^2$$, the acceleration of the particle
1. depends on $$\displaystyle t$$
2. is always directed upward
3. is constant both in magnitude and in direction
4. never exceeds 1 in magnitude
5. is none of these
22. Solution C
23. Given $$\displaystyle \vec{R} = 3 \cos \left( {\frac{\pi}{3}}t \right) \vec{i} + 2 \sin \left( {\frac{\pi}{3}}t \right) \vec{j}$$ is the position vector $$x \vec{i} + y \vec{j}$$ from the origin to a moving point $$P(x, y)$$ at time $$t$$, a single equation in $$x$$ and $$y$$ for the path of the point is
1. $$\displaystyle x^2 + y^2 = 13$$
2. $$\displaystyle 9x^2 + 4y^2 = 36$$
3. $$\displaystyle 2x^2 + 3y^2 = 13$$
4. $$\displaystyle 4x^2 + 9y^2 = 1$$
5. $$\displaystyle 4x^2 + 9y^2 = 36$$
24. Solution E
25. Given $$\displaystyle \vec{R} = 3 \cos \left( {\frac{\pi}{3}}t \right) \vec{i} + 2 \sin \left( {\frac{\pi}{3}}t \right) \vec{j}$$ is the position vector $$x \vec{i} + y \vec{j}$$ from the origin to a moving point $$P(x, y)$$ at time $$t$$, when $$t = 3$$, the speed of the particle is
1. $$\displaystyle \frac{2\pi}{3}$$
2. 2
3. 3
4. $$\displaystyle \pi$$
5. $$\displaystyle \frac{\pi \sqrt{13}}{3}$$
26. Solution A
27. Given $$\displaystyle \vec{R} = 3 \cos \left( {\frac{\pi}{3}}t \right) \vec{i} + 2 \sin \left( {\frac{\pi}{3}}t \right) \vec{j}$$ is the position vector $$x \vec{i} + y \vec{j}$$ from the origin to a moving point $$P(x, y)$$ at time $$t$$, the magnitude of the acceleration when $$t = 3$$ is
1. 2
2. $$\displaystyle \frac{\pi ^2}{3}$$
3. 3
4. $$\displaystyle \frac{2\pi ^2}{9}$$
5. $$\displaystyle \pi$$
28. Solution B
29. Given $$\displaystyle \vec{R} = 3 \cos \left( {\frac{\pi}{3}}t \right) \vec{i} + 2 \sin \left( {\frac{\pi}{3}}t \right) \vec{j}$$ is the position vector $$\displaystyle x \vec{i} + y \vec{j}$$ from the origin to a moving point $$P(x, y)$$ at time $$t$$, at the point where $$\displaystyle t = \frac{1}{2}$$, the slope of the curve along which the particle moves is
1. $$\displaystyle \frac{-2\sqrt{3}}{9}$$
2. $$\displaystyle -\frac{\sqrt{3}}{2}$$
3. $$\displaystyle \frac{2}{\sqrt{3}}$$
4. $$\displaystyle -\frac{2\sqrt{3}}{3}$$
30. Solution D
31. A particle is moving on the curve $$\displaystyle y = 2x -\ln {x}$$ so that $$\displaystyle \frac{dx}{dt} = -2$$ at all times $$t$$. At the point (1, 2), $$\displaystyle \frac{dy}{dt}$$ is
1. 4
2. 2
3. -4
4. 1
5. -2
32. Solution E
33. The equation of the tangent to the curve with parametric equations $$\displaystyle x = 2t + 1, y = 3 - t^3$$ at the point where $$t = 1$$ is
1. $$\displaystyle 2x + 3y = 12$$
2. $$\displaystyle 3x + 2y = 13$$
3. $$\displaystyle 6x + y = 20$$
4. $$\displaystyle 3x - 2y = 5$$
5. none of these
34. Solution B
35. If $$\displaystyle x = 4 \cos {\theta}$$ and $$y = 3 \sin {\theta}$$, then $$\displaystyle \int_2^4 xy \, dx$$ is equivalent to
1. $$\displaystyle 48\int_{\frac{\pi}{3}}^0 \sin {\theta} \cos^2 {\theta} \, d\theta$$
2. $$\displaystyle 48\int_2^4 \sin^2 {\theta} \cos{\theta} \, d\theta$$
3. $$\displaystyle 36\int_2^4 \sin{\theta} \cos^2{\theta } \, d\theta$$
4. $$\displaystyle -48\int_0^{\frac{\pi}{3}} \sin {\theta} \cos^2 {\theta} \, d\theta$$
5. $$\displaystyle 48\int_0^{\frac{\pi}{3}} \sin^2 {\theta} \cos{\theta} \, d\theta$$
36. Solution E
37. A curve is defined by the parametric equations $$\displaystyle y = 2a \cos^2{\theta}$$ and $$x = 2a \tan{\theta}$$ , where $$0 \leq \theta \leq \pi$$. Then $$\displaystyle \int_0^{2a} y^2 \, dx$$ =
1. $$\displaystyle 4 a^2 \int_0^{\frac{\pi}{4}}\cos^4 {\theta} \, d\theta$$
2. $$\displaystyle 8 a^3 \int_\frac{\pi}{2}^{\pi }\cos^2 {\theta} \, d\theta$$
3. $$\displaystyle 8 a^3 \int_0^{\frac{\pi}{4}}\cos^2 {\theta} \, d\theta$$
4. $$\displaystyle 8 a^3 \int_0^{2a}\cos^2 {\theta} \, d\theta$$
5. $$\displaystyle 8 a^3 \int_0^{\frac{\pi}{4}}\sin {\theta} \cos^2 {\theta} \, d\theta$$
38. Solution C
39. A curve given parametrically by $$\displaystyle x = 1 - \cos{t}$$ and $$y = t - \sin{t}$$, where $$0 \leq t \leq \pi$$, then $$\displaystyle \int_0^{\frac{3}{2}} y \, dx =$$
1. $$\displaystyle \int_0^{\frac{3}{2}} \sin{t} (t - \sin{t} ) \, dt$$
2. $$\displaystyle \int_{\frac{2\pi}{3}}^{\pi} \sin{t} (t - \sin{t} ) dt$$
3. $$\displaystyle \int_0^{\frac{2\pi}{3}} (t - \sin{t} ) dt$$
4. $$\displaystyle \int_0^{\frac{2\pi}{3}} \sin{t} (t - \sin{t} ) dt$$
5. $$\displaystyle \int_0^{\frac{3}{2}} (t - \sin{t} ) dt$$
40. Solution D
41. The area enclosed by the ellipse with parametric equations $$\displaystyle x = 2 \cos {\theta}$$ and $$y = 3 \sin {\theta}$$ is
1. $$\displaystyle 6\pi$$
2. $$\displaystyle \frac{9\pi}{2}$$
3. $$\displaystyle 3\pi$$
4. $$\displaystyle \frac{3\pi}{2}$$
5. none of these
42. Solution A
43. The area enclosed by one loop of the cycloid with parametric equations $$\displaystyle x = \theta - \sin {\theta}$$ and $$\displaystyle y = 1 - \cos {\theta}$$ equals
1. $$\displaystyle \frac{3\pi}{2}$$
2. $$\displaystyle 3\pi$$
3. $$\displaystyle 2\pi$$
4. $$\displaystyle 6\pi$$
5. none of these
44. Solution B
45. The area enclosed by the hypocycloid with parametric equations $$\displaystyle x = \cos^3 {t}$$, and $$\displaystyle y = \sin^3 {t}$$ is given by
1. $$\displaystyle 3\int_{\frac{\pi}{2}}^0 \sin^4 {t} \cos^2 {t} \, dt$$
2. $$\displaystyle 4\int_0^{1} \sin^3 {t} \, dt$$
3. $$\displaystyle -4\int_{\frac{\pi}{2}}^0 \sin^6 {t} \, dt$$
4. $$\displaystyle 12\int_0^{\frac{\pi}{2}} \sin^4 {t} \cos^2 {t} \, dt$$
5. none of these
46. Solution D
47. Which expression represents the volume of the solid that results from rotating the cricle with parametric equations $$\displaystyle x = a \cos {\theta}, y = a \sin {\theta}\text{ }(a > 0)$$ about the line $$x = 2a$$?
1. $$\displaystyle 4\pi a^3 \int_0^{\pi} (2 \sin {\theta} - \sin^2 {\theta} ) \cos {\theta} \, d\theta$$
2. $$\displaystyle 4\pi a^3 \int_{\pi}^0 (2 \sin {\theta} - \cos {\theta} ) \, d\theta$$
3. $$\displaystyle -4\pi a^3 \int_{\pi }^0 (2 \sin {\theta} \cos {\theta} - \sin {\theta} \cos^2 {\theta} ) ,\ d\theta$$
4. $$\displaystyle 16\pi a^3 \int_0^{\frac{\pi}{2}} \, d\theta$$
5. $$\displaystyle 4\pi a^3 \int_0^{\pi} (2\sin ^2 {\theta} - \sin ^2 {\theta} \cos {\theta} ) \, d\theta$$
48. Solution E
49. Which expression represents the volume of the solid that results from rotating the curve with parametric equations $$\displaystyle x = \tan{\theta}$$, $$y = \cos ^2 {\theta}$$, and the lines $$x = 0, x = 1$$, and $$y = 0$$ about the x-axis?
1. $$\displaystyle \pi \int_0^{\frac{\pi}{4}} \cos ^4 {\theta} \, d\theta$$
2. $$\displaystyle \pi \int_0^{\frac{\pi}{4}} \cos ^2 {\theta} \sin {\theta} \, d\theta$$
3. $$\displaystyle \pi \int_0^{\frac{\pi}{4}} \cos ^2 {\theta} \, d\theta$$
4. $$\displaystyle \pi \int_0^{1} \cos ^2 {\theta} \, d\theta$$
5. $$\displaystyle \pi \int_0^{1} \cos ^4 {\theta} \, d\theta$$
50. Solution C
51. The length of one arch of the cycloid $$\displaystyle x = t - \sin{t} , y = 1 - \cos{t}$$ equals
1. $$\displaystyle 3\pi$$
2. 4
3. 16
4. 8
5. $$\displaystyle 2\pi$$
52. Solution D
53. The length of $$\displaystyle x = e^t \cos{t} , y = e^t \sin{t}$$ from $$t = 2$$ to $$t = 3$$ is equal to
1. $$\displaystyle \sqrt{2} e^2 \sqrt{e^2 - 1}$$
2. $$\displaystyle \sqrt{2}(e^3 - e^2)$$
3. $$\displaystyle 2(e^3 - e^2)$$
4. $$\displaystyle e^3 (\cos {3} + \sin {3}) - e^2(\cos {2} + \sin {2})$$
5. none of these
54. Solution B
55. The area enclosed by the four-leaved rose $$\displaystyle r = \cos {(2\theta )}$$ equals
1. $$\displaystyle \frac{\pi}{4}$$
2. $$\displaystyle \frac{\pi}{2}$$
3. $$\displaystyle \pi$$
4. $$\displaystyle 2\pi$$
5. $$\displaystyle \frac{\pi}{2} + \frac{1}{2}$$
56. Solution B
57. The area bounded by the small loop of the limacon $$\displaystyle r = 1 - 2 \sin {\theta}$$ is given by the definite integral
1. $$\displaystyle \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \left( \frac{1 - 2 \sin{\theta}}{2} \right) ^2 \, d\theta$$
2. $$\displaystyle \int_{\frac{7\pi}{6}}^{\frac{3\pi}{2}} (1 - 2 \sin{\theta} )^2 \, d\theta$$
3. $$\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} (1 - 2 \sin{\theta} )^2 \, d\theta$$
4. $$\displaystyle \int_0^{\frac{\pi}{6}} \left( \frac{1 - 2 \sin{\theta}}{2} \right) ^2 \, d\theta + \int_{\frac{5\pi}{6}}^{\pi} \left( \frac{1 - 2 \sin{\theta}}{2} \right) ^2 \, d\theta$$
5. $$\displaystyle \int_0^{\frac{\pi}{3}} (1 - 2 \sin{\theta} )^2 \, d\theta$$
58. Solution C
59. The rectangular equation of the curve given parametrically by $$\displaystyle x = 1 - \sin{t}$$ and $$\displaystyle y = 4 - 2 \cos{t}$$ is
1. $$\displaystyle 4(x - 1)^2 + (y - 4)^2 = 1$$
2. $$\displaystyle 4(x - 1)^2 + (y - 4)^2 = 4$$
3. $$\displaystyle (x - 1)^2 + (y - 4)^2 = 4$$
4. none of these
60. Solution B
61. The graph of the pair of parametric equations $$\displaystyle x = \sin{t} - 2, y = \cos^2 {t}$$ is
1. part of a circle
2. part of a parabola
3. a hyperbola
4. a line
5. a cycloid
62. Solution B
63. If $$\displaystyle x = 2 \sin {u}$$ and $$y = \cos {(2u)}$$, then a single equation in $$x$$ and $$y$$ is
1. $$\displaystyle x^2 + y^2 = 1$$
2. $$\displaystyle x^2 + 4y^2 = 4$$
3. $$\displaystyle x^2 + 2y = 2$$
4. $$\displaystyle x^2 + y^2 = 4$$
5. $$\displaystyle x^2 - 2y = 2$$
64. Solution C
65. The curve of the pair of parametric equations $$\displaystyle x = 2e^t , y = e^{-t}$$ is
1. a line
2. a parabola
3. a hyperbola
4. an ellipase
5. none of these
66. Solution C
67. The area bounded by the lemniscate with polar equation $$\displaystyle r^2 = 2 \cos 2{\theta} =$$
1. 4
2. 1
3. $$\displaystyle \frac{1}{2}$$
4. 2
5. none of these
68. Solution D
69. The area inside the circle $$\displaystyle r = 3 \sin {\theta}$$ and outside the cardioid $$r = 1 + \sin {\theta}$$ is given by
1. $$\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} 9 \sin^2 {\theta} - (1 + \sin {\theta} )^2 \, d\theta$$
2. $$\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} (2 \sin{\theta} - 1)^2 \, d\theta$$
3. $$\displaystyle \frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} 8 \sin^2 {\theta} - 1 \, d\theta$$
4. $$\displaystyle \frac{9\pi}{4} - \frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} (1 + \sin{\theta} )^2 d\theta$$
5. none of these
70. Solution A
71. The graph of the polar equation $$\displaystyle r = \theta$$, where $$\theta$$ is a real number, is
1. a circle
2. a hyperbolic spiral asymptotic to the line $$y = 1$$
3. a line of slope 1
4. a pair of lines passing through the origin
5. a double spiral which passes through the origin
72. Solution E
73. The graph of the polar equation $$\displaystyle r = \frac{1}{\sin {\theta} - 2 \cos {\theta}}$$ is
1. a line with slope 2
2. a line with slope 1
3. a circle
4. a parabola
5. a semicircle
74. Solution A