Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These 20 problems cover AP Calculus AB Unit 4: interpreting derivatives with units, position, velocity and acceleration, related rates, local linearization and L’Hôpital’s Rule. Early problems build vocabulary; later ones ask for a full related-rates setup.
The last six problems are multi-step situations such as a walking shadow, a radar station and a particle that changes direction.
Standards: AP Calculus AB Unit 4 · Level: Grade 12 AP Calculus AB
Problem 1 Interpreting marginal cost ★★★
The cost, in dollars, of baking q cupcakes is C(q), and C′(200) = 1.85. Explain what this value means in context, with units.
Problem 2 Velocity and acceleration at an instant ★★★
A cart’s position in feet is s(t) = t3 − 9t2 + 15t. Find its velocity, acceleration and speed at t = 2.
Problem 3 Direction of motion ★★★
A particle on a line has position s(t) = t3 − 6t2 + 9t for t ≥ 0. Find when it moves right and when it moves left.
Problem 4 Acceleration from a table ★★★
The velocity v(t) of a robot, in meters per second, is recorded. Estimate its acceleration at t = 3.5 using the two nearest times, with units.
| t (s) | 0 | 2 | 5 | 9 |
|---|---|---|---|---|
| v(t) | 3 | 7 | 16 | 20 |
Problem 5 Speeding up or slowing down ★★★
A particle has velocity v(t) = t2 − 8t + 12. Decide whether it is speeding up or slowing down at t = 1 and at t = 7.
Problem 6 Linearization of a cube root ★★★
Write the local linearization of f(x) = ∛x at x = 27 and use it to approximate ∛29. Is the estimate too high or too low?
Problem 7 Linear approximation of an exponential ★★★
Use the tangent line to g(x) = e5x at x = 0 to approximate g(0.02). Is the approximation an underestimate or an overestimate?
Problem 8 L’Hôpital’s Rule once ★★★
Evaluate limx→0 e4x − 1sin 6x.
Problem 9 L’Hôpital’s Rule twice ★★★
Evaluate limx→∞ x2e3x.
Problem 10 Growing circle ★★★
A circular oil spill’s radius grows at 0.4 cm/s. How fast is the area increasing when the radius is 15 cm?
Problem 11 Inflating a balloon ★★★
A spherical balloon is filled with air at 90 in3/s. How fast is the radius increasing when the radius is 3 inches? (Volume of a sphere: V = (4/3)πr3.)
Problem 12 Sliding ladder ★★★
A 17-foot ladder leans against a wall. Its base slides away at 1.5 ft/s. How fast is the top sliding down the wall when the base is 8 feet from the wall?
Problem 13 Marginal cost and actual cost ★★★
The cost, in dollars, of producing x skateboards is C(x) = 0.02x2 + 4x + 500. Find C′(100), interpret it, and compare it with the actual cost of the 101st skateboard.
Problem 14 Filling a conical tank ★★★
A tank shaped like an upside-down cone has radius 6 feet and height 12 feet. Water flows in at 20 ft3/min. How fast is the depth rising when the depth is 4 feet?
Problem 15 Relative growth of a population ★★★
A town’s population is P(t) = 8,000e0.04t, with t in years. Find P′(t), the rate of change at t = 5, and the relative rate P′/P.
Problem 16 Shadow of a person walking ★★★
A 6-foot person walks away from a 15-foot lamp post at 4 ft/s. How fast is the tip of the shadow moving along the ground, and how fast is the shadow’s length growing?
Problem 17 Distance traveled by a particle ★★★
A particle’s position is s(t) = t3 − 12t2 + 36t for 0 ≤ t ≤ 7. Find the total distance traveled and the net displacement.
Problem 18 Trigonometric limit with two passes ★★★
Evaluate limx→0 1 − cos 5xx2 and name each form that appears.
Problem 19 Intervals of speeding up ★★★
A particle’s position is s(t) = 2t3 − 15t2 + 36t for 0 ≤ t ≤ 4. On which intervals is the particle speeding up?
Problem 20 Distance from a radar station ★★★
A plane flies horizontally at an altitude of 4 miles and a speed of 500 mph, moving directly away from a radar station. How fast is the distance from the station to the plane increasing when the plane is 3 miles of ground distance from the station?
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