AP Calculus AB 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

A pump removes water from a basement at a rate of R(t) gallons per minute, where t is the time in minutes. What are the units of the area of the region between the graph of R(t) and the t-axis over a given time interval?

A)gallonsB)minutesC)gallons per minuteD)gallons per minute squared

All Questions (11)

A) gallons

B) minutes

C) gallons per minute

D) gallons per minute squared

Correct Answer: A

The unit for the area under a rate of change graph is the unit for the rate of change multiplied by the unit for the independent variable. In this case, the rate's unit is (gallons/minute) and the independent variable's unit is (minutes). Therefore, the resulting unit is (gallons/minute) * (minutes) = gallons. [cite: 2621]

The function v(t) represents the velocity of a car in meters per second. What is the correct interpretation of the area of the region between the graph of v(t) and the t-axis from t = 10 seconds to t = 60 seconds?

A) The car's average velocity between 10 and 60 seconds.

B) The car's total change in position between 10 and 60 seconds.

C) The car's acceleration at t = 60 seconds.

D) The car's instantaneous velocity at t = 60 seconds.

Correct Answer: B

Velocity is the rate of change of position. The area of the region between the graph of a rate of change function (velocity) and the x-axis gives the accumulation of change (change in position) over that interval. [cite: 2617, 2618]

The graph shows the constant rate, r(t), in items per hour, at which a machine produces items over a 4-hour period. How many total items are produced during this period?

A) 20

B) 40

C) 60

D) 80

Correct Answer: D

The total number of items produced is the accumulation of change, which is the area of the region under the rate graph. This region is a rectangle with a height of 20 (items/hour) and a width of 4 (hours). The area can be evaluated using geometry: Area = height × width = 20 items/hour × 4 hours = 80 items. [cite: 2618, 2619]

The graph shows the rate, r(t) in liters per second, at which oil is leaking from a tank. The rate decreases linearly over 10 seconds. What is the total amount of oil that has leaked in the first 10 seconds?

A) 25 liters

B) 50 liters

C) 100 liters

D) 200 liters

Correct Answer: B

The total amount of leaked oil is the accumulation of change, found by calculating the area of the region under the rate graph. The region is a triangle with a base of 10 seconds and a height of 10 liters/second. Using geometry, the area is (1/2) * base * height = (1/2) * 10 s * 10 L/s = 50 liters. [cite: 2618, 2619]

The graph shows the velocity v(t) of a particle moving along a line. The rate is positive for 0 < t < 2 and negative for 2 < t < 4. What is the physical interpretation of the area of the region between the graph and the t-axis for the interval 2 < t < 4?

A) The total distance the particle traveled.

B) The particle's displacement in the positive direction.

C) The particle's displacement in the negative direction.

D) The particle's final position.

Correct Answer: C

When the rate of change (velocity) is negative over an interval, the accumulated change (displacement) is negative. This means the particle moved in the negative direction during that time. The area represents this negative displacement. [cite: 2617, 2620]

The graph shows the rate of change of a city's population, P'(t), in hundreds of people per year over 6 years. What is the net change in the city's population from t=0 to t=6?

A) 100 people

B) 200 people

C) 300 people

D) 400 people

Correct Answer: A

The net change is the total accumulation, found by summing the areas between the graph and the axis. The area from t=0 to t=4 is a trapezoid: (1/2)(4+2)(2) = 6. The area from t=4 to t=6 is a triangle below the axis: (1/2)(2)(-5) = -5. The net change is 6 + (-5) = 1. Since the units are in hundreds of people, the net change is 1 * 100 = 100 people. [cite: 2618, 2619, 2620]

Let R(t) be the rate at which a company's revenue is changing, in thousands of dollars per month. The area of the region between the graph of R(t) and the t-axis from t=2 to t=8 is calculated to be -45. What is the correct interpretation of this value?

A) The company's revenue at month 8 was $45,000.

B) The company's revenue decreased by $45,000 per month between month 2 and month 8.

C) The company's total revenue decreased by $45,000 between month 2 and month 8.

D) The company lost a total of $45 between month 2 and month 8.

Correct Answer: C

The area under the rate of change graph represents the total accumulated change over the interval. Since the rate R(t) is in thousands of dollars per month and the interval is in months, the area represents the total change in revenue in thousands of dollars. A negative value indicates a decrease. Therefore, the revenue decreased by 45 thousand dollars, or $45,000. [cite: 2617, 2620, 2621]

The rate of snow accumulation on a driveway is modeled by a function S(t), in inches per hour. The area of the region between the graph of S(t) and the t-axis from t=0 to t=6 is 8. The area of the region between the graph of S(t) and the t-axis from t=6 to t=10 is -3. If there were 2 inches of snow on the driveway at t=0, how many inches of snow are on the driveway at t=10?

A) 5 inches

B) 7 inches

C) 11 inches

D) 13 inches

Correct Answer: B

The total accumulation of change from t=0 to t=10 is the sum of the areas: 8 + (-3) = 5 inches. This represents the net change in the amount of snow. To find the final amount, we add this net change to the initial amount: Initial Amount + Net Change = 2 inches + 5 inches = 7 inches. [cite: 2617, 2618, 2620]

A runner's speed increases linearly from 0 m/s to 6 m/s over the first 3 seconds of a race. For the next 5 seconds, the runner maintains a constant speed of 6 m/s. What is the total distance the runner traveled in the first 8 seconds?

A) 30 m

B) 36 m

C) 39 m

D) 48 m

Correct Answer: C

The total distance is the accumulation of change of the speed. We can find this by calculating the area under the speed-time graph using geometry. The first 3 seconds form a triangle with area (1/2) * base * height = (1/2) * 3s * 6m/s = 9m. The next 5 seconds form a rectangle with area base * height = 5s * 6m/s = 30m. The total distance is the sum of these areas: 9m + 30m = 39m. [cite: 2618, 2619]

Let W'(t) be the rate, in degrees Celsius per minute, at which the temperature of water in a pot is changing. If W'(t) is positive for the entire interval 0 ≤ t ≤ 5, what can be concluded about the temperature of the water?

A) The temperature of the water was positive over the interval.

B) The temperature of the water increased over the interval.

C) The temperature of the water was constant over the interval.

D) The rate of heating was constant over the interval.

Correct Answer: B

If the rate of change is positive over an interval, then the accumulated change is positive. In this context, a positive accumulated change in temperature means the temperature increased from t=0 to t=5. [cite: 2620]

A particle moves along the x-axis with a velocity v(t) given by the graph. If the particle's position at t=0 is x=10, what is its position at t=7?

A) 4

B) 10

C) 14

D) 16

Correct Answer: C

The particle's final position is its initial position plus the total displacement (net accumulated change). The displacement is the net area under the velocity graph. Area from t=0 to t=4 is a trapezoid: (1/2)(4+2)(4) = 12. Area from t=4 to t=7 is a trapezoid below the axis: (1/2)(3+1)(-4) = -8. The net displacement is 12 + (-8) = 4. The final position is Initial Position + Displacement = 10 + 4 = 14. [cite: 2617, 2618, 2619, 2620]