AP Calculus AB: Curve Sketching and Connecting f, f’, f” — Drill 1

About This Drill

AP Calculus AB: Curve Sketching and Connecting f, f’, f” — Drill 1 is a Math practice drill covering Curve Sketching. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice reading graphs of f, f', and f'' to determine increasing/decreasing behavior, concavity, and extrema. These AP Calculus AB skills are central to Unit 5 and appear on both the multiple-choice and free-response sections.

Questions in This Drill

1. The graph of \( f' \) is positive on \( (-2, 1) \), equals zero at \( x = 1 \), and is negative on \( (1, 4) \). Which of the following correctly describes \( f \)?
2. The function \( f \) is continuous and twice differentiable. \( f''(x) > 0 \) on \( (0, 3) \) and \( f''(x) < 0 \) on \( (3, 6) \). Which of the following must be true?
3. A function \( f \) is continuous on \( [-3, 3] \) with the following properties: \( f'(x) 0 \) on \( (0, 3) \), and \( f''(x) < 0 \) on \( (-3, 3) \). Which of the following correctly describes the graph of \( f \) on \( [-3, 3] \)?
4. The graph of \( f \) is concave up on \( (-1, 2) \) and concave down on \( (2, 5) \). Which of the following must be true?
5. The table below gives values of \( f'(x) \) at selected points for a twice-differentiable function \( f \) whose derivative \( f' \) is strictly increasing on \( [0, 4] \).
   
   

   x	0	1	2	3	4
   f'(x)	−3	−1	0	2	5

   
   Which of the following conclusions is supported by this data?