AP Calculus BC Practice Quiz: Exploring Accumulations of Change

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 11 questions to check your progress.

Question 1 of 11

The function R(t) represents the rate, in gallons per minute, at which water is leaking from a tank at time t, in minutes. What is the meaning of the area of the region between the graph of R(t) and the t-axis over the interval 0 ≤ t ≤ 5?

A)The total amount of water, in gallons, that has leaked from the tank during the first 5 minutes.B)The rate at which water is leaking, in gallons per minute, at t=5 minutes.C)The average rate at which water is leaking, in gallons per minute, during the first 5 minutes.D)The change in the rate of leakage, in gallons per minute per minute, during the first 5 minutes.

All Questions (11)

A) The total amount of water, in gallons, that has leaked from the tank during the first 5 minutes.

B) The rate at which water is leaking, in gallons per minute, at t=5 minutes.

C) The average rate at which water is leaking, in gallons per minute, during the first 5 minutes.

D) The change in the rate of leakage, in gallons per minute per minute, during the first 5 minutes.

Correct Answer: A

The area of the region between the graph of a rate of change function and the x-axis gives the accumulation of change. The unit for this area is the unit for the rate (gallons/minute) multiplied by the unit for the independent variable (minutes), which results in gallons. Therefore, the area represents the total gallons leaked. [cite: 2617, 2618, 2621]

A car travels at a constant velocity of 50 miles per hour for 3 hours. The graph of the car's velocity versus time is a horizontal line at v=50. Using geometry, what is the total distance the car traveled?

A) 50 miles

B) 100 miles

C) 150 miles

D) 53 miles

Correct Answer: C

The accumulation of change can be evaluated using geometry. The graph of the rate (velocity) versus time forms a rectangle with a height of 50 miles/hour and a width of 3 hours. The area, which represents the total distance, is length × width = 50 miles/hour × 3 hours = 150 miles. [cite: 2619, 2618]

The rate at which a factory produces phones, P'(t), increases linearly from 0 phones per hour at t=0 to 200 phones per hour at t=4 hours. The graph of P'(t) is a line segment from (0,0) to (4, 200). How many phones are produced in total during this 4-hour period?

A) 200

B) 400

C) 800

D) 1600

Correct Answer: B

The total number of phones produced is the accumulation of change, which is the area under the rate graph. The region is a triangle with a base of 4 hours and a height of 200 phones/hour. The area is calculated using the formula for a triangle: (1/2) × base × height = (1/2) × 4 hours × 200 phones/hour = 400 phones. [cite: 2619, 2618]

The velocity of a particle, v(t), is shown in the graph. For 0 ≤ t ≤ 2, the graph is a triangle above the t-axis with a maximum value of 4 m/s. For 2 ≤ t ≤ 4, the graph is a triangle below the t-axis with a minimum value of -4 m/s. What is the net displacement of the particle from t=0 to t=4?

A) -4 m

B) 0 m

C) 4 m

D) 8 m

Correct Answer: B

Net displacement is the total accumulated change. From t=0 to t=2, the rate is positive, and the area is (1/2) * 2 * 4 = 4 m. From t=2 to t=4, the rate is negative, and the area is (1/2) * 2 * (-4) = -4 m. The net displacement is the sum of these accumulated changes: 4 m + (-4 m) = 0 m. [cite: 2618, 2620, 2619]

The function C'(t) gives the rate of consumption of electricity in kilowatts per hour. The variable t is measured in hours. What are the units for the accumulated change in electricity consumption, represented by the area under the graph of C'(t)?

A) kilowatts

B) hours

C) kilowatts per hour

D) kilowatts per hour squared

Correct Answer: A

The unit for the area of a region defined by a rate of change is the unit for the rate of change multiplied by the unit for the independent variable. In this case, it is (kilowatts / hour) × (hour) = kilowatts. [cite: 2621]

The graph of a rocket's vertical velocity, v(t) in meters/second, is shown for the first 10 seconds of its flight. The velocity increases linearly from v(0)=50 m/s to v(10)=150 m/s. What is the total vertical distance the rocket traveled during this interval?

A) 500 m

B) 750 m

C) 1000 m

D) 1500 m

Correct Answer: C

The total distance is the area under the velocity graph. This area forms a trapezoid. The area can be calculated using geometry: Area = (1/2) * (base1 + base2) * height. Here, the 'bases' are the parallel vertical sides, and the 'height' is the horizontal time interval. Area = (1/2) * (v(0) + v(10)) * (10 - 0) = (1/2) * (50 + 150) * 10 = (1/2) * 200 * 10 = 1000 meters. [cite: 2619, 2618]

The rate of change of the temperature of a chemical solution, R(t), is measured in degrees Celsius per second. For the interval 0 ≤ t ≤ 10, the graph of R(t) is entirely below the t-axis. Which of the following best describes the meaning of the area between the graph and the t-axis?

A) The total increase in temperature of the solution after 10 seconds.

B) The temperature of the solution at t=10 seconds.

C) The average temperature of the solution over the 10-second interval.

D) The total decrease in temperature of the solution after 10 seconds.

Correct Answer: D

Since the rate of change R(t) is negative (the graph is below the t-axis), the accumulated change will be negative. This represents a decrease in the original quantity (temperature). The area represents the magnitude of this total decrease in degrees Celsius over the 10-second interval. [cite: 2620, 2617]

The graph shows the velocity of a toy car. From t=0 to t=2 seconds, velocity increases linearly to 6 cm/s. From t=2 to t=5, velocity is constant at 6 cm/s. From t=5 to t=7, velocity decreases linearly to 0 cm/s. What is the total distance the car traveled?

A) 24 cm

B) 30 cm

C) 36 cm

D) 42 cm

Correct Answer: C

The total distance is the total area under the velocity graph, which can be found using geometry. The area is composed of a triangle, a rectangle, and another triangle. Area 1 (0-2s) = (1/2) * base * height = (1/2) * 2 * 6 = 6 cm. Area 2 (2-6s) = width * height = 4 * 6 = 24 cm. Area 3 (6-8s) = (1/2) * base * height = (1/2) * 2 * 6 = 6 cm. The total distance is the sum of these areas: 6 + 24 + 6 = 36 cm. [cite: 2618, 2619]

The rate of change of the number of visitors at a museum, V'(t), is strictly positive for the entire time the museum is open, from t=0 (9 AM) to t=8 (5 PM). Which statement must be true?

A) The number of visitors in the museum at 5 PM is zero.

B) The number of visitors in the museum was constant all day.

C) There were more visitors in the museum at 5 PM than there were at 9 AM.

D) The rate at which visitors entered the museum decreased throughout the day.

Correct Answer: C

If the rate of change V'(t) is positive over an interval, then the accumulated change is positive. This means the total number of visitors has increased from the beginning of the interval to the end. Therefore, the number of visitors at t=8 (5 PM) must be greater than the number of visitors at t=0 (9 AM). [cite: 2620]

At time t=0 hours, a storage tank holds 1000 liters of oil. For the next 6 hours, oil is pumped into the tank at a constant rate of 50 liters per hour. The graph of the rate is a horizontal line. How many liters of oil are in the tank at t=6 hours?

A) 50 liters

B) 300 liters

C) 1050 liters

D) 1300 liters

Correct Answer: D

The amount of oil at t=6 is the initial amount plus the accumulated change. The accumulated change is the area under the rate graph from t=0 to t=6. This area is a rectangle with height 50 liters/hour and width 6 hours. Area = 50 * 6 = 300 liters. The total amount is the initial amount + accumulated amount = 1000 + 300 = 1300 liters. [cite: 2618, 2619]

The graph of f'(x) represents the rate of growth of a plant in centimeters per day. The area of the region between the graph of f'(x) and the x-axis from x=10 to x=30 is calculated to be 8. What is the correct interpretation of this value?

A) The plant's height at day 30 is 8 centimeters.

B) The plant grew a total of 8 centimeters between day 10 and day 30.

C) The plant's rate of growth on day 30 was 8 centimeters per day.

D) The plant's average height between day 10 and day 30 was 8 centimeters.

Correct Answer: B

The area under the graph of a rate of change function, f'(x), gives the total accumulated change in the original function, f(x), over that interval. The units are the rate's units (cm/day) multiplied by the independent variable's units (day), resulting in cm. Therefore, the value 8 represents the total growth in centimeters over the interval from day 10 to day 30. [cite: 2617, 2618, 2621]