AP Calc AB Applications of Differentiation

Question 1

A 13-foot ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at a constant 2 ft/s. How fast is the top of the ladder sliding down the wall at the instant the bottom is 5 feet from the wall?

Accepted Answer

5/6 ft/s

Answer

From x² + y² = 169, x(dx/dt) + y(dy/dt) = 0. When x = 5, y = 12, so 5(2) + 12(dy/dt) = 0 and dy/dt = -5/6 ft/s, so the top slides down at 5/6 ft/s.

Question 2

Evaluate lim (x→0) (e^(2x) − 1 − 2x) / x².

Accepted Answer

2

Answer

Direct substitution gives 0/0, so apply L'Hospital's Rule twice. The first pass gives (2e^(2x) − 2)/(2x), still 0/0. The second gives 4e^(2x)/2, which tends to 2.

Question 3

Let f(x) = x³ − 3x on the interval [0, 2]. At which value of c in (0, 2) does the Mean Value Theorem guarantee that f'(c) equals the average rate of change of f on [0, 2]?

Accepted Answer

2/√3

Answer

The average rate of change is (f(2) − f(0))/2 = (2 − 0)/2 = 1. Setting f'(c) = 3c² − 3 = 1 gives c² = 4/3, so c = 2/√3, which lies in (0, 2).

Question 4

What is the absolute maximum value of f(x) = x³ − 6x² + 9x + 1 on the closed interval [0, 5]?

Accepted Answer

21

Answer

f'(x) = 3(x − 1)(x − 3), so the critical points are x = 1 and x = 3. Evaluating gives f(0) = 1, f(1) = 5, f(3) = 1, f(5) = 21, so the maximum is 21.

Question 5

A function f is twice differentiable with f''(x) = x(x − 2)². At which x-values does the graph of f have a point of inflection?

Accepted Answer

x = 0 only

Answer

An inflection point requires f'' to change sign. f'' is zero at x = 0 and x = 2, but (x − 2)² is never negative, so the sign of f'' changes only at x = 0.

AP Calc AB Practice Test

AP Calc AB Practice Test

AP Calc AB Applications of Differentiation