Related Rates Test – AP Calculus AB Math (Printable PDF)

Question 1 Direction change (2 points)

A particle moves along a line with position s(t) = t³ − 12t + 5 for t ≥ 0. At what time does the particle change direction?

1. 1
2. 2
3. 4
4. 6

Question 2 L’Hôpital multiple choice (2 points)

What is lim_x→0 e^x − 1 − xx²?

1. 0
2. 1/2
3. 1
4. The limit does not exist

Question 3 Rectangle related rates (3 points)

The length of a rectangle increases at 3 in/s while its width decreases at 2 in/s. At the moment the length is 10 in and the width is 6 in:

1. Find the rate of change of the area. (1 point)
2. Is the area increasing or decreasing? Justify. (1 point)
3. Find the rate of change of the diagonal. (1 point)

Question 4 Linearization of a square root (3 points)

Let f(x) = √x.

1. Write the equation of the tangent line to f at x = 49. (1 point)
2. Use it to approximate √50. (1 point)
3. Is your approximation greater or less than √50? Explain. (1 point)

Question 5 Velocity graph analysis (3 points)

The velocity v(t) of a particle, in m/s, is graphed for 0 ≤ t ≤ 8. It consists of line segments.

1. At what time is the particle momentarily at rest? (1 point)
2. Find the acceleration at t = 5. (1 point)
3. At t = 5 is the particle speeding up or slowing down? Explain. (1 point)

Question 6 Marginal analysis (3 points)

The cost in dollars of making x backpacks is C(x) = 0.01x² + 5x + 300.

1. Find C′(x). (1 point)
2. Find C′(200) and interpret it with units. (1 point)
3. Compute C(201) − C(200) and compare. (1 point)

Question 7 L’Hôpital with a logarithm (2 points)

Evaluate lim_x→∞ ln x√x and show the form that justifies your method.

Question 8 Sliding ladder test (2 points)

A 10-foot ladder leans against a wall. The top slides down at 1.2 ft/s. How fast is the base moving away from the wall when the top is 6 feet above the ground?