AP Calculus AB Practice Questions After a Low Practice Score (2026)

After a low AP Calculus AB practice score, do not immediately take another full exam. Solve a small set that separates concept, setup, algebra, interpretation, and communication. The five original questions below cover distinct AB skills and include answers and explanations. Use the results to select one repair target before returning to released College Board questions.

For official practice, scoring guidelines, and student samples, use AP Central's AP Calculus AB past exam questions. These original problems are diagnostic practice, not retired exam questions.

Question 1: derivative from a table

The differentiable functions f and g satisfy:

x	f(x)	f'(x)	g(x)	g'(x)
2	5	-1	3	4

Let h(x) = f(g(x)). Find h'(2) if g(2) = 3, f'(3) = 6, and g'(2) = 4.

Answer: 24.

Explanation: The chain rule gives h'(2) = f'(g(2))g'(2) = f'(3)(4) = 6·4 = 24. The value f(2) is irrelevant. If you multiplied f'(2) by g'(2), the issue is not arithmetic; it is evaluating the outer derivative at the inner function's output.

Question 2: accumulation and the Fundamental Theorem

Define

F(x) = ∫₁ˣ (t² - 4) dt for 0 < x < 4.

1. Find F'(3).
2. At what value of x in the interval does F change from decreasing to increasing?

Answer: F'(3) = 5, and the change occurs at x = 2.

Explanation: By the Fundamental Theorem of Calculus, F'(x) = x² - 4. Therefore F'(3) = 9 - 4 = 5. The derivative is negative for 0 < x < 2 and positive for 2 < x < 4, so F changes from decreasing to increasing at 2. Calculating the integral formula first is possible but unnecessary.

Question 3: slope field thinking without a picture

A function y = f(x) satisfies dy/dx = x - y and passes through (1, 2).

1. What is the slope of the solution curve at (1, 2)?
2. Use the tangent line to approximate f(1.1).

Answer: The slope is -1, and f(1.1) ≈ 1.9.

Explanation: Substitute the point into the differential equation: 1 - 2 = -1. The tangent line at (1, 2) is y - 2 = -1(x - 1). At x = 1.1, the tangent-line value is 2 - 0.1 = 1.9. This is a local linear approximation, not the exact solution unless separately proven.

Question 4: related rates with a defined positive rate

A circular ripple expands so that its radius increases at 0.5 meter per second. Find the rate of change of the area when the radius is 6 meters.

Answer: 6π square meters per second.

Explanation: Begin with A = πr². Differentiate with respect to time:

dA/dt = 2πr · dr/dt.

Substitute r = 6 and dr/dt = 0.5:

dA/dt = 2π(6)(0.5) = 6π.

The units must be area per time. A response of 6π meters per second loses the distinction between radius and area rates.

Question 5: justify an absolute maximum

Let p(x) = x³ - 3x² + 1 on the closed interval [0, 3]. Find the absolute maximum value of p on the interval and justify your answer.

Answer: The absolute maximum value is 1, occurring at x = 0.

Explanation:

p'(x) = 3x² - 6x = 3x(x - 2).

Critical points in [0, 3] are x = 0 and x = 2; also test both endpoints. Evaluate:

• p(0) = 1
• p(2) = 8 - 12 + 1 = -3
• p(3) = 27 - 27 + 1 = 1

The maximum value is 1, reached at both x = 0 and x = 3. If the prompt asks for the maximum value, report 1; if it asks where it occurs, report both endpoints. This wording distinction is a common source of lost credit.

Convert results into a repair prescription

Missed question	Likely gap	Next set
1	Chain rule evaluation	Six composition derivatives from formulas and tables
2	FTC and derivative sign	Accumulation functions plus increase/decrease analysis
3	Differential equation interpretation	Slope and Euler/tangent-line approximation practice
4	Related-rates model or units	Three geometry models with labeled units
5	Closed-interval method and prompt wording	Extrema problems requiring values and locations

Do not count only correct answers. A correct guess with no explanation belongs in the repair column. An algebra error after a correct calculus setup needs a different intervention from choosing the wrong theorem.

A three-session recovery sequence

Session 1: Re-solve the missed question from a blank page and explain every setup. Then complete four narrow questions of the same type.

Session 2: Mix that skill with two stronger AB topics. Remove labels so you must recognize which method applies.

Session 3: Complete one released free-response question or an AP Classroom assignment that includes the skill. Score the setup, notation, justification, and numerical result separately.

Use the AP Calculus AB practice-test guide to select appropriate official material, locate prerequisites with AP Calculus AB units and topics, and schedule the recovery through the AP Calculus AB study plan. A low total becomes useful when it produces one precise mathematical repair.