AP Calculus AB: Mean Value Theorem and Extreme Value Theorem — Drill 1

About This Drill

AP Calculus AB: Mean Value Theorem and Extreme Value Theorem — Drill 1 is a Math practice drill covering Mean Value Theorem and Extreme Value Theorem. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice applying the Mean Value Theorem and Extreme Value Theorem to determine the existence of guaranteed values and absolute extrema on closed intervals. These AP Calculus AB topics appear regularly on both sections of the AP exam.

Questions in This Drill

1. Let \( f(x) = x^2 - 4x \) on the interval \( [1, 5] \). What value of \( c \) in \( (1, 5) \) is guaranteed by the Mean Value Theorem?
2. The function \( h(x) = x^3 - 3x \) is continuous on \( [-2, 2] \). What are the absolute maximum and absolute minimum values of \( h \) on this interval?
3. Which of the following functions satisfies the conditions of the Mean Value Theorem on \( [-1, 1] \)?
4. The table below shows selected values of a continuous function \( f \) on the closed interval \( [-1, 3] \).
   
   

   x	−1	0	1	2	3
   f(x)	4	−1	6	3	−2

   
   Based on the table and the Extreme Value Theorem, which of the following must be true?
5. A car travels along a straight road. Its position (in miles) at time \( t \) (in hours) is given by a differentiable function \( s(t) \). If \( s(0) = 10 \) and \( s(4) = 90 \), which of the following must be true?