2026 AP® Calculus AB Free-Response Questions

PREDICTED EXAM – EDUCATIONAL ANALYSIS

© 2026 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of College Board. Visit College Board on the web: collegeboard.org.
AP Central is the official online home for the AP Program: apcentral.collegeboard.org.

AP® Calculus AB 2026 Free-Response Questions

CALCULUS AB

SECTION II, Part A

Time—30 minutes
2 Questions

A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.

GO ON TO THE NEXT PAGE.

1. A coastal city is implementing a carbon reduction program to combat climate change. The rate at which carbon dioxide is being removed from the atmosphere through the city’s green initiatives is modeled by the differentiable function R(t), where R(t) is measured in tons per week and t is measured in weeks since the program began. Selected values of R(t) are given in the table below.

(a) Use the data in the table to estimate R'(10). Using correct units, interpret the meaning of your answer in the context of this problem.

(b) The total amount of carbon dioxide removed from the atmosphere over the first 20 weeks is given by ∫₀²⁰ R(t) dt. Use a left Riemann sum with the five subintervals indicated by the data in the table to approximate this value. Show your work and indicate units.

(c) For 12 ≤ t ≤ 24, the rate of carbon removal can be modeled by the function G(t) = 95 – 45e^(-0.15(t-12)), where G(t) is measured in tons per week. Using this model, find the average rate of carbon removal over the interval 12 ≤ t ≤ 20. Show the setup for your calculation.

(d) It can be shown that G'(t) = 6.75e^(-0.15(t-12)). For the model in part (c), is the rate of carbon removal increasing at an increasing rate or increasing at a decreasing rate for 12 < t < 24? Give a reason for your answer.

2. Two drones are flying along parallel paths for a surveillance mission. Drone A moves along the x-axis with velocity v_A(t) = 3t² – 12t + 15 meters per second, and Drone B moves along a path parallel to the x-axis with velocity v_B(t) = 8cos(0.5t) + 2 meters per second, where t is measured in seconds for 0 ≤ t ≤ 10.

At time t = 0, both drones are at position x = 0.

(a) Find the positions of Drone A and Drone B at time t = 6 seconds. Show the setup for your calculations.

(b) Find all times t in the open interval 0 < t < 10 when Drone A is momentarily at rest. For each such time, determine whether Drone A is speeding up or slowing down at that instant. Justify your answers.

(c) The region R is bounded by the graphs of y = v_A(t) and y = v_B(t) from t = 2 to t = 8. Find the area of region R. Show the setup for your calculation.

(d) At time t = 4 seconds, a signal is transmitted between the drones. If the signal travels at a constant speed and the vertical separation between the drone paths is 50 meters, find the rate at which the distance between the drones is changing at the instant t = 4 seconds. Show your work and indicate units.

END OF PART A

CALCULUS AB

SECTION II, Part B

Time—1 hour
4 Questions

NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS.

3. The water level in a tidal estuary is modeled by a function H(t) that satisfies the differential equation dH/dt = (1/3)cos(πt/6)(15 – H), where H(t) is the height of the water in feet above mean sea level and t is the time in hours after midnight. At t = 0, the water level is H(0) = 12 feet.

(a) A portion of the slope field for the differential equation is shown below. Sketch the solution curve that passes through the point (0, 12).

[Slope field diagram would be provided]

(b) Find H'(0), the rate of change of the water level at midnight. Using correct units, interpret the meaning of your answer in the context of this problem.

(c) Use separation of variables to find the particular solution H(t) to the differential equation dH/dt = (1/3)cos(πt/6)(15 – H) with the initial condition H(0) = 12.

(d) For 0 ≤ t ≤ 12, find the time when the water level reaches its maximum height. Justify your answer.

4. Let f be a continuous function defined on the closed interval [-4, 8]. The graph of f’, the derivative of f, consists of line segments and a semicircle as shown in the figure below. It is known that f(2) = 5.

[Graph of f’ would be provided showing:

• Linear decrease from (-4,3) to (-2,0)

• Semicircle from (-2,0) to (2,0) with center at (0,0) and radius 2

• Linear increase from (2,0) to (6,4)

• Linear decrease from (6,4) to (8,0)]

(a) Find the values of f(-2) and f(6).

(b) For what values of x, if any, does the graph of f have a point of inflection in the open interval (-4, 8)? Give a reason for your answer.

(c) Let g(x) = ∫₂ˣ f'(t) dt. Find the absolute maximum value of g(x) on the interval [-4, 8]. Justify your answer.

(d) Find the limit: lim[x→2] [f(x) – 3x + 1]/(x² – 4x + 4), or show that the limit does not exist.

5. Consider the functions p(x) = x³ – 6x² + 9x + 1 and q(x) = 2x + 5. Let h(x) be the function defined by h(x) = p(q(x)).

(a) Find h'(x). Show your work.

(b) The table below gives values of r(x) and r'(x) at selected values of x.

Let k(x) = r(p(x)). Find k'(3).

(c) Consider the function m(x) = ∫₁ˣ p'(t) dt. Find the value of x for which m(x) is minimized on the interval [0, 4]. Justify your answer.

(d) A particle moves along the x-axis so that its velocity at time t ≥ 0 is given by v(t) = p'(t). At time t = 1, the particle’s position is x = 10. Find the position of the particle at time t = 4.

6. Consider the curve defined by the equation x²y + y³ = 8 + 2x.

(a) Show that dy/dx = (2 – 2xy)/(x² + 3y²).

(b) Write an equation for the line tangent to the curve at the point (2, 1).

(c) Find the coordinates of any points on the curve where the tangent line is horizontal. If no such points exist, explain why.

(d) A particle moves along the curve. At the instant when the particle is at the point (2, 1), the x-coordinate is increasing at a rate of 3 units per second. At that same instant, find the rate of change of the distance from the particle to the origin. Show your work and indicate units.

STOP

END OF EXAM

ANALYSIS AND PREDICTIONS SUMMARY

This predicted 2026 AP Calculus AB exam incorporates several key trends identified from the 2021-2025 analysis:

Part A (Calculator Required):

• Q1: Environmental science application (carbon removal) featuring Riemann sums, rate interpretation, and exponential modeling

• Q2: Drone surveillance context with particle motion, optimization, area calculation, and related rates

Part B (No Calculator):

• Q3: Tidal estuary differential equation with slope fields and separation of variables

• Q4: Graph analysis of derivative function with Fundamental Theorem of Calculus applications

• Q5: Function composition and chain rule with table analysis and optimization

• Q6: Implicit differentiation with tangent lines and related rates

Key Features Consistent with Historical Patterns:

• Real-world environmental contexts reflecting current issues (climate change, technology)

• Strong emphasis on computational skills in calculator section

• Traditional calculus theory in non-calculator section

• Consistent question positioning patterns observed 2021-2025

• Integration of multiple mathematical concepts within single problems

Emerging Trends Incorporated:

• Environmental/climate science applications

• Technology/surveillance contexts

• More sophisticated modeling scenarios

• Integration of sustainability themes