BC Calculus: Midyear Review Worksheet 4

Questions have been taken from Lifshitz, Maxine, Calculus AB/BC Preparing for the Advanced Placement Examinations; Amsco, 2004

Calculator Inactive.

  1. If \( f(x) = 2 \sin ^2 {(5x)}, f'(x) = \)
    1. \( 10 \sin{(10x)}\)
    2. \( 20 \sin{(5x)}\)
    3. \( 10 \sin{(5x)}\)
    4. \( 4 \cos{(5x)}\)
    5. \( 20 \cos{(5x)} \)
    2. Solution A
  2. Find the x-intercept of the line tangent to \(y = \ln{ (\ln{ x})}\) at \(x = e\)
    1. \(x = -1\)
    2. \(x = 0\)
    3. \(x = 1\)
    4. \(x = e\)
    5. \( \displaystyle x = -\frac{1}{e} \)
    3. Solution D
  3. \(a(t) = 2e^t\) and \(v(1) = 2\). Find \(v(t)\)
    1. \(2(e^t - e + 1)\)
    2. \(2(e^{t - 1} + 1)\)
    3. \(2t + 1\)
    4. \(2e^t + e\)
    5. \(2e^t \)
    4. Solution A
  4. If \( \displaystyle \lim_{x \to - \infty} \frac{\ln{ (1 - x)}}{x^2 + 1} =\)
    1. \(-\infty\)
    2. -1
    3. 0
    4. 1
    5. \(\infty\)
    5. Solution C
  5. \( \displaystyle y = \frac{e^{2x - 1}}{x}\) has
    1. a relative maximum at \( \displaystyle x = \frac{1}{2}\)
    2. a horizontal asymptote at \( y = 0\)
    3. a vertical asymptote at \( x = 0\)
    1. I only
    2. I and II
    3. I and III
    4. II and III
    5. I, II and III
    6. Solution D
  6. Given the piecewise function defined as \( \displaystyle f(x) = \begin{cases} x^2 + 2, & \text{if } x < 1 \\ 2x + 1, & \text{if }x \geq 1 \end{cases}\), which of the following is true?
    1. \( f(x)\) is not continuous at \(x = 1\).
    2. \( f(x)\) is continuous but not idfferentiable at \( x = 1\).
    3. \( f(x)\) is differentiable but not continuous at \( x = 1\).
    4. \( f(x)\) is continuous and differentiable at \( x = 1\).
    5. \( f(2) = 6\)
    7. Solution D
  7. \( \displaystyle \int_1^e x \ln{x} \, dx =\)
    1. \( \displaystyle \frac{e^2+ 1}{4}\)
    2. \(2e^2 - 1\)
    3. \( \displaystyle \frac{e^2 - e - 1}{2}\)
    4. \(2e^2 + 1\)
    5. undefined
    8. Solution A
  8. Write an integral that represents the length of one arch of \(y = \sin{ x}\).
    1. \( \displaystyle \int_0^{\frac{\pi}{2}} \sqrt{1 + \cos^2{x}} \, dx =\)
    2. \( \displaystyle \int_0^{\pi} \sqrt{1 + \cos^2{x}} \, dx =\)
    3. \( \displaystyle \int_0^{\frac{\pi}{2}} ( 1 + \cos{ x}) \, dx =\)
    4. \( \displaystyle \int_0^{\frac{\pi}{2}} \sin^2{x} \, dx =\)
    5. \( \displaystyle \int_0^{\frac{\pi}{2}} \sqrt{1 + \sin^2{x}} \, dx =\)
    9. Solution B
  9. For what values of \(a\) and \(c\) is the piecewise function defined by \( \displaystyle f(x) = \begin{cases} x + c, & \text{if } x > 2 \\ ax^2, & \text{if }x \leq 2 \end{cases}\) differentiable at x = 2?
    1. \( \displaystyle a = \frac{1}{2}, c= 0\)
    2. \( \displaystyle a = \frac{1}{4}, c = -1\)
    3. \(a = 1, c = 6\)
    4. \(a = 0, c = -2\)
    5. no solution
    10. Solution B
  10. The slope field below depicts a certain differential equation. Which of the following choices could be a solution to that equation?
    1. \(y = \ln{x}\)
    2. \(y = e^{-x}\)
    3. \(y = \sin{x}\)
    4. \(y = e^x\)
    5. \(y = x^{2\sin{ x}}\)
    11. Solution B
  11. \( \displaystyle h(x) = \frac{f(x)}{\left( g(x) \right)^2}\). If \( \displaystyle f'(x) = g(x), g'(x) = \frac{1}{f(x)}, g(x) > 0\) for all real \(x\), and \(f(x) \neq 0\) for all real \(x\), then \(h'(x) =\)
    1. \( \displaystyle \frac{2}{\left(g(x) \right)^3} - \frac{1}{g(x)}\)
    2. \( \displaystyle \frac{1}{g(x)} - \frac{2}{\left(g(x)\right)^3} \)
    3. \( \displaystyle \left(g(x)\right)^2 - 2\)
    4. \( \displaystyle \frac{1}{g(x)} - \frac{2f(x)}{\left(g(x)\right)^3} \)
    5. 0
    12. Solution B
  12. The maximum value of \( \displaystyle f(x) = \frac{\sqrt{x - 3}}{ x}\) occurs at \(x =\)
    1. -6
    2. \( \displaystyle \frac{\sqrt{3}}{3}\)
    3. 3
    4. 6
    5. 7
    13. Solution D
  13. \(y' = 2y + 5\). Find \(y''\) in terms of \(y\).
    1. 0
    2. 2
    3. \(4y + 5\)
    4. \(4y + 10\)
    5. \(4y + 20\)
    14. Solution D
  14. For what value of \(c\) does \( \displaystyle y = cx + \frac{3}{x}\) have a relative minimum at \(x = 2\)?
    1. \( \displaystyle -\frac{2}{3} \ln {2}\)
    2. 0
    3. \( \displaystyle \frac{3}{8}\)
    4. \( \displaystyle \frac{1}{2}\)
    5. \( \displaystyle \frac{3}{4}\)
    15. Solution E
  15. \(f(x) = 3(x - 2)^2 + 6(x - 2) + 1\). Find the equation of the line tangent to \(f(x)\) at \(x = 2\).
    1. \(6x - y = 11\)
    2. \(y = 0\)
    3. \(6x - y = 12\)
    4. \(6x - y = 13\)
    5. \(7x - y = 13\)
    16. Solution A
  16. Which of the following formulas could be used to calculate the average rate of change of \(f\) on the closed interval [0, 4]?
    1. \( \displaystyle \frac{f(4) - f(0)}{2}\)
    2. \( \displaystyle \frac{f(0) + f(4)}{2}\)
    3. \( \displaystyle \frac{f(4) - f(0)}{4 }\)
    4. \( \displaystyle \frac{f(0) + f(4)}{4}\)
    5. \( \displaystyle \frac{f'(4) - f'(0)}{4}\)
    17. Solution C
  17. \( \displaystyle f(x) = \frac{x}{x - 3}\). If \(f^{-1}(x)\) is the inverse of \(f(x)\), find the derivative of \(f^{-1}(x)\) at \(x = -2\)
    1. -3
    2. \( \displaystyle -\frac{1}{3}\)
    3. \( \displaystyle \frac{1}{3}\)
    4. 1
    5. 3
    18. Solution B
  18. \( f(x) = 3x^3 - 4\). Find \( \displaystyle \lim_{x \to 2} \frac{f(x) - f(2)}{x - 2}\)
    1. 18
    2. 20
    3. 24
    4. 32
    5. 36
    19. Solution E
  19. Given the graph of \(y = f(x)\) shown below, if \( \displaystyle g(x) = f\left(\frac{1}{x}\right) \), find the value of \( \displaystyle \lim_{x \to 0^+} g(x)\)
    1. -1
    2. 0
    3. 1
    4. \(\infty\)
    5. does not exist
    20. Solution C
  20. \(y = \ln{ (\cos^2{x})}. y' =\)
    1. \(-2 \tan{ x}\)
    2. \(\sec^2 {x}\)
    3. \(2 \sec {x}\)
    4. \(2 \tan{ x}\)
    5. \(-2 \sin{x} \cos{ x}\)
    21. Solution A
  21. Write the equation of the line perpendicular to the tangent of the curve represented by the equation \(y = e^{x+1}\) at \(x = 0\)
    1. \( \displaystyle y = -\frac{x}{e}\)
    2. \( \displaystyle y = -\frac{x}{e} + e\)
    3. \(y = ex + e\)
    4. \( \displaystyle y = \frac{x}{e} + e\)
    5. \(y = ex \)
    22. Solution B
  22. Find the area bounded by the graph of \( \displaystyle y = \frac{x}{x^2 - 1}\) and the x-axis on the interval \( \displaystyle \left[-\frac{1}{2}, \frac{1}{2} \right]\).
    1. \( \displaystyle \ln{ \left(\frac{3}{4}\right)}\)
    2. 0
    3. \( \displaystyle \frac{1}{2}\ln{ \left(\frac{4}{3}\right)}\)
    4. \( \displaystyle \ln{ \left(\frac{4}{3}\right)}\)
    5. \(\ln {2}\)
    23. Solution D
  23. \(F(x)\) is the antiderivative of \(f(x), F(5) = 7\), and \( \displaystyle \int_2^5 f(x) \, dx = 9\). Find \(F(2)\)
    1. -16
    2. -7
    3. -2
    4. 2
    5. 16
    24. Solution C