BC Calculus: Midyear Review Worksheet 3

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

A Graphing Calculator is required for some of these questions.

  1. Which of the following statements is true about the graph of \(f(x)\) below:
    1. \( \displaystyle \lim_{x \to 5} f(x) \) exists
    2. \( \displaystyle \lim_{x \to 1} f(x) \) exists
    3. \( \displaystyle \lim_{x \to 5} f(x) = f(5) \)
    4. \( \displaystyle \lim_{x \to 1} f(x) = f(1) \)
    5. \( \displaystyle \frac{f(5) - f(1)}{5 - 1} = f'(c)\)
    2. Solution A
  2. How many points of inflection are there for the function \(y = x + \cos{ (2x)}\) on the interval \([0, \pi]\)?
    1. 0
    2. 1
    3. 2
    4. 3
    5. 4
    3. Solution C
  3. The graph of \(f(x)\) consists of two segments, one joining the origin to the point (1, 1), and a second joining the point (1, 1) to the point (2, 1). If \(F'(x) = f(x)\), and \(F(0) = -3\), then \(F(2) = \)
    1. -4.5
    2. -1.5
    3. 1.5
    4. 3
    5. 4.5
    4. Solution B
  4. If \( \displaystyle \lim_{h \to 0} \frac{f(3 + h) - f(3)}{h} = 0\), then which of the following must be true?
    1. \(f\) has a derivative at \(x = 3\)
    2. \(f\) is continuous at \(x = 3\)
    3. \(f\) has a critical value at \(x = 3\)
    1. I only
    2. II only
    3. I and II
    4. I and III
    5. I, II, and III
    5. Solution E
  5. Consider the function \(y = x^3 - x^2 -1\). For what value(s) of \(x\) is the slope of the tangent equal to 5?
    1. -1 only
    2. \( \displaystyle \frac{5}{3}\) only
    3. -1 and \( \displaystyle \frac{5}{3}\)
    4. \( \displaystyle \frac{1}{3}\)
    5. 2.219
    6. Solution C
  6. A pebble thrown into a pond creates circular ripples such that the rate of change of the circumference is \(12\pi\) cm/sec. How fast is the area of the ripple changing when the radius is 3 cm?
    1. \(6\pi \frac{\text{cm}^2}{\text{sec}}\)
    2. \(2\pi \frac{\text{cm}^2}{\text{sec}} \)
    3. \(12\pi \frac{\text{cm}^2}{\text{sec}}\)
    4. \(36\pi \frac{\text{cm}^2}{\text{sec}} \)
    5. \(6 \frac{\text{cm}^2}{\text{sec}}\)
    7. Solution D
  7. If \(y = x^2 + 1\), what is the smallest positive value of \(x\) such that \(\sin {y}\) is a relative maximum?
    1. 0.756
    2. 0.841
    3. 1
    4. 1.463
    5. 1.927
    8. Solution A
  8. Find the area in the first quadrant bounded by \( y = 2 \cos{ x}, y = 3 \tan {x}\), and the y-axis.
    1. 0.347
    2. 0.374
    3. 0.432
    4. 0.568
    5. 1.040
    9. Solution D
  9. \(f'(x) = x^3(x - 2)^4(x - 3)^2. f(x)\) has a relative maximum at
    1. 0
    2. 2
    3. 2 and 3
    4. 0 and 3
    5. There is no relative maximum.
    10. Solution E
  10. Find the average rate of change of \(f(x) = \sec {x}\) on the interval \([0, \frac{\pi}{3}]\)
    1. 0.396
    2. 0.955
    3. 1.350
    4. 1.910
    5. undefined
    11. Solution B
  11. \( \displaystyle a(t) = \frac{5t^2+ 1}{5t}\) and \(v(1) = 1\). Find \(v(2)\)
    1. 1.139
    2. 2.10
    3. 2.139
    4. 2.639
    5. undefined
    12. Solution D
  12. Use the table shown to approximate the area under the curve of \(y = f(x)\) using trapezoids:
    \(x\) \(y\)
    0
    1
    1
    2
    3
    4
    4
    1
    1. 5.5
    2. 8
    3. 10
    4. 11
    5. 20
    13. Solution C
  13. Given the graph of \(y = f(x)\) below, which of the following statements is true?
    1. \( \displaystyle \lim_{x \to 1} f(x) = 3 \)
    2. \( \displaystyle \lim_{x \to 1^+} f(x) = 3\)
    3. \(f'(1) = 3\)
    4. \(f(1) = 3\)
    5. The average rate of change of \(f(x)\) on [1, 3] is \(f'(2) \)
    14. Solution E
  14. The position of a particle on a line is given by \(x(t) = t^3 - t, t \geq 0\). Find the distance traveled by the particle in the first two seconds.
    1. 0.385
    2. 3.385
    3. 6
    4. 6.385
    5. 6.770
    15. Solution E
  15. \( \displaystyle \int_a^b \left| f(x) \right| \, dx = p\) and \( \left| \int_a^b f(x) \, dx \right| = q\). Which of the following must be true?
    1. \(p = q\)
    2. \(p\geq q\)
    3. \(p \leq q\)
    4. \(p > q\)
    5. \(p < q\)
    16. Solution B