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



  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