BC Calculus: Midyear Review Worksheet

Questions have been taken from the 2004 Advanced Placement Calculus BC Test

The following should be done without a calculator .

  1. \( \displaystyle \lim_{x \to \infty} \frac{20x^2 - 13x + 5}{5 - 4x^3} = \)
    1. -5
    2. \( \infty\)
    3. 0
    4. 5
    5. 1
    2. Solution C
  2. \( \displaystyle \lim_{x \to \frac{\pi}{2}} \frac{\cos {x}}{x - \frac{\pi}{2} } = \)
    1. -1
    2. 1
    3. 0
    4. \( \infty\)
    5. none of these
    3. Solution A
  3. \( \displaystyle \lim_{x \to 0} x \sin { \left(\frac{1}{x} \right)} =\)
    1. 1
    2. 0
    3. \( \infty \)
    4. -1
    5. none of these
    4. Solution B
  4. \( \displaystyle \lim_{h \to 0} \frac{\ln{(2 + h)} - \ln {2}}{h} = \)
    1. 0
    2. \( \ln {2}\)
    3. \( \displaystyle \frac{1}{2}\)
    4. \( \displaystyle \frac{1}{\ln {2}} \)
    5. \( \infty \)
    5. Solution C
  5. If \( \displaystyle y = \frac{x - 3}{2 - 5x}\) then \( \displaystyle \frac{dy}{dx} =\)
    1. \( \displaystyle \frac{17 - 10x}{(2 - 5x)^2} \)
    2. \( \displaystyle \frac{13}{(2 - 5x)^2} \)
    3. \( \displaystyle \frac{x - 3}{(2 - 5x)^2} \)
    4. \( \displaystyle \frac{17}{(2 - 5x)^2} \)
    5. \( \displaystyle -\frac{13}{(2 - 5x)^2} \)
    6. Solution E
  6. If \( \displaystyle f(x) = x \cos {\left(\frac{1}{x}\right)}\), then \( \displaystyle f' \left( \frac{2}{\pi} \right) = \)
    1. \( \displaystyle \frac{\pi}{2}\)
    2. \( \displaystyle -\frac{2}{\pi} \)
    3. -1
    4. \( \displaystyle - \frac{\pi}{2}\)
    5. 1
    7. Solution A
  7. If \( \displaystyle xy^2 - 3x + 4y - 2 = 0\) and \(y\) is a differentiable function of \(x\), then \( \displaystyle \frac{dy}{dx} = \)
    1. \( \displaystyle -\frac{1 + y^2}{2xy}\)
    2. \( \displaystyle \frac{3}{2y+4}\)
    3. \( \displaystyle \frac{3}{2xy+4}\)
    4. \( \displaystyle \frac{3 - y^2}{2xy + 4}\)
    5. \( \displaystyle \frac{5 - y^2}{2xy + 4}\)
    8. Solution D
  8. If \( \displaystyle x = \sqrt{1 - t^2}\) and \(y = \sin^{-1}{t}\), then \( \displaystyle \frac{dy}{dx} = \)
    1. \( \displaystyle -\frac{\sqrt{1 - t^2}}{t}\)
    2. \( -t\)
    3. \( \displaystyle \frac{t}{1 - t^2}\)
    4. 2
    5. \( \displaystyle -\frac{1}{t}\)
    9. Solution E
  9. If \(y\) is a differentiable function of \(x\), then the derivative of \( \sin^2{(x + y)}\) with respect to \(x\) is
    1. \( \displaystyle 2\sin{(x + y)} \frac{dy}{dx} \)
    2. \( \displaystyle \cos^2{(x + y)} \left(1 + \frac{dy}{dx} \right) \)
    3. \( \displaystyle \sin { \left( 2(x + y) \right)} \left(1 + \frac{dy}{dx} \right) \)
    4. \( \displaystyle \cos^2{(x + y)}\)
    5. \( \displaystyle 2\sin{(x + y)}\)
    10. Solution C
  10. The equation of the tangent to the curve \(y = e^x\ln{x}\), where \(x = 1\) is
    1. \( y = ex \)
    2. \( \displaystyle y = e^x + 1\)
    3. \( y = e(x - 1)\)
    4. \( y = ex + 1\)
    5. \( y = x - 1 \)
    11. Solution C
  11. If differentials are used for computation then the cube root of 63 is approximately equal, to the nearest hundredth, to
    1. 4.00
    2. 3.98
    3. 3.93
    4. 3.80
    5. 3.88
    12. Solution B
  12. The hypotenuse AB of a right triangle ABC is 5 ft, and one leg, AC, is decreasing at a rate of 2 ft/sec. The rate, in square feet per second, at which the area is changing when AC = 3 is
    1. \( \displaystyle \frac{25}{4} \)
    2. \( \displaystyle \frac{7}{4} \)
    3. \( \displaystyle -\frac{3}{2} \)
    4. \( \displaystyle -\frac{7}{4} \)
    5. \( \displaystyle -\frac{7}{2} \)
    13. Solution D
  13. The derivative of a function \(f\) is given for all \(x\) by \(f'(x) = x^2(x + 1)^3(x - 4)^2\). The set of \(x\) for which \(f\) has a relative maximum is
    1. {0, -1, 4}
    2. {-1}
    3. {0, 4}
    4. {1}
    5. none of these
    14. Solution E
  14. If the displacement from the origin of a particle on a line is given by \(s = 3 + (t - 2)^4\), then the number of times the particle reverses direction is
    1. 0
    2. 1
    3. 2
    4. 3
    5. none of these
    15. Solution B
  15. The maximum value of the function \( f(x) = xe^{-x}\) is
    1. \( \displaystyle \frac{1}{e}\)
    2. \( e\)
    3. 1
    4. -1
    5. none of these
    16. Solution A
  16. A rectangle of perimeter 18 in. is rotated about one of its sides to generate a right circular cylinder. The rectangle which generates the cylinder of largest volume has an area, in square inches, of
    1. 14
    2. 20
    3. \( \displaystyle \frac{81}{4}\)
    4. 18
    5. \( \displaystyle \frac{77}{4}\)
    17. Solution D
  17. If \(f'(x)\) exists on the closed interval \([a, b]\), then it follows that
    1. \(f(x)\) is constant on \([a, b]\)
    2. \( \displaystyle \exists c \in (a, b) \mid f'(c) = 0\)
    3. the function has a maximum value on the open interval \((a, b)\)
    4. the function has a minimum value on the open interval \((a, b)\)
    5. the Mean Value Theorem applies to the function.
    18. Solution E
  18. \( \displaystyle \int_{1}^2 (3x - 2)^3\, dx = \)
    1. \( \displaystyle \frac{16}{3}\)
    2. \( \displaystyle \frac{63}{4}\)
    3. \( \displaystyle \frac{13}{3}\)
    4. \( \displaystyle \frac{85}{4}\)
    5. none of these
    19. Solution D
  19. \( \displaystyle \int x \cos {\left( x^2 \right) } \, dx =\)
    1. \( \displaystyle \sin {\left( x^2 \right)} + C\)
    2. \( \displaystyle 2 \sin {\left( x^2 \right)} + C\)
    3. \( \displaystyle -\frac{1}{2} \sin { \left( x^2 \right)} + C\)
    4. \( \displaystyle \frac{1}{4} \cos^2 {\left( x^2 \right)} + C \)
    5. \( \displaystyle \frac{1}{2} \sin {\left( x^2 \right)} + C\)
    20. Solution E
  20. \( \displaystyle \int_{0}^1\frac{e^x}{(3 - e ^x)^2} \, dx = \)
    1. \( 3 ln (e - 3) \)
    2. 1
    3. \( \displaystyle \frac{1}{3 - e}\)
    4. \( \displaystyle \frac{e - 2}{3 - e} \)
    5. none of these
    21. Solution E
  21. \( \displaystyle \int_{-1}^1 \left( 1 - |x| \right) \, dx = \)
    1. 0
    2. \( \displaystyle \frac{1}{2}\)
    3. 1
    4. 2
    5. none of these
    22. Solution C
  22. The general solution of the differential equation \( \displaystyle \frac{dy}{dx} = \frac{1 - 2x}{y}\) is a family of
    1. lines
    2. circles
    3. hyperbolas
    4. parabolas
    5. ellipses
    23. Solution E
  23. If \( F'(x) = G'(x)\) for all \(x\) and \(k\) is a constant, then it is necessary that
    1. \( F(x) = G(x) + k\)
    2. \( F(x) = G(x)\)
    3. \( F(k) = G(k)\)
    4. \( F(x) = kG(x)\)
    5. \( F(x) = G(x + k)\)
    24. Solution A
  24. The area enclosed by the curve of \(y^2 = x^3\) and the line segment joining (1, 1) and (4, -8) is given by
    1. \( \displaystyle 2 \int_{0}^1 x^{\frac{3}{2}} \, dx + \int_{1}^4 (-3x + 4 - x^{\frac{3}{2}}) \, dx\)
    2. \( \displaystyle 2 \int_{0}^1 x^{\frac{3}{2}} \, dx + \int_{1}^4 (4 - 3x + x^{\frac{3}{2}}) \, dx \)
    3. \( \displaystyle \int_{0}^4 (4 - 3x + x^{\frac{3}{2}}) \, dx \)
    4. \( \displaystyle 2 \int_{0}^1 x^{\frac{3}{2}} \, dx + \int_{-8}^1 ( \frac{4 - y}{3} - y ^{\frac{2}{3}}) \, dy \)
    5. \( \displaystyle \int_{1}^4 (x^{\frac{3}{2}} + 3x - 4) \, dx \)
    25. Solution B
  25. The first-quadrant area bounded below by the x-axis and laterally by the curves \(y = x^2\) and \(y = 4 - x^2\) equals
    1. \( 6 - 2 \sqrt{3}\)
    2. \( \displaystyle \frac{10}{3}\)
    3. \( \displaystyle \frac{8}{3}(2 - \sqrt{2}) \)
    4. \( \displaystyle \frac{4\sqrt{2}}{3}\)
    5. none of these
    26. Solution C
  26. Find the area bounded by \( \displaystyle y = \frac{4}{1 + x^2}, y = 4, x = 1\), and \(x = -1\).
    1. \( \displaystyle 4 - \frac{4}{\pi}\)
    2. \( 8 - 2 \pi \)
    3. \( 8 - \pi \)
    4. \( \displaystyle 8 - \frac{\pi}{2}\)
    5. \( 2 \pi - 4\)
    27. Solution B
  27. If \(f(x)\) and \(g(x)\) are both continuous functions on the closed interval \([a, b]\), if \(f(a) = g(a)\) and \(f(b) = g(b)\), and if, further, \(f(x) > g(x) \forall x \in (a, b)\), then it follows that
    1. \( \displaystyle \int_{a}^b f(x) \, dx \geq 0 \)
    2. \( \displaystyle \int_{a}^b \left| g(x) \right| \, dx > \int_{a}^b \left| f(x) \right| \, dx\)
    3. \( \displaystyle \int_{a}^b \left| f(x) \right| \, dx > \int_{a}^b \left| g(x) \right| \, dx\)
    4. \( \displaystyle \int_{a}^b f(x) \, dx > \int_{a}^b g(x) \, dx\)
    5. none of the preceeding is necessarily true
    28. Solution D
  28. A particle moves along a line with velocity, in feet per second, \(v = t ^2 - t\). The total distance, in feet, traveled from \(t = 0\) to \(t = 2\) equals
    1. \( \displaystyle \frac{1}{3}\)
    2. \( \displaystyle \frac{2}{3}\)
    3. 2
    4. 1
    5. \( \displaystyle \frac{4}{3}\)
    29. Solution D
  29. If \(f'(x) = h(x)\) and \(g(x) = x^3\), then \( \displaystyle \frac{d}{dx} f(g(x)) = \)
    1. \( h(x^3)\)
    2. \( 3x^2h(x)\)
    3. \( h'(x)\)
    4. \( 3x^2h(x^3)\)
    5. \( x^3h(x^3)\)
    30. Solution D
  30. If \( \displaystyle \frac{dy}{dx} = y \tan {x}\) and \(y = 3\) when \(x = 0\), then, when \( \displaystyle x = \frac{\pi }{3}, y = \)
    1. \( \ln {\sqrt{3}} \)
    2. \( \ln {3} \)
    3. \( \displaystyle \frac{3}{2} \)
    4. \( \displaystyle \frac{3\sqrt{3}}{2} \)
    5. 6
    31. Solution E