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Prepare for the timed AP Calculus AB test.

Your answers are held until submission. The final review identifies the units that need another pass.

AP Calculus AB coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

01

Limits and Continuity

Limit estimates, properties, asymptotes, continuity, IVT

Goal: recognise a limits and continuity problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Differentiation: Definition and Properties

Derivative definitions, rules, tangent lines

Goal: recognise a differentiation: definition and properties problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Composite, Implicit, and Inverse Differentiation

Chain rule, implicit and inverse derivatives

Goal: recognise a composite, implicit, and inverse differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Contextual Applications of Differentiation

Motion, rates, related rates, local linearity

Goal: recognise a contextual applications of differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Analytical Applications of Differentiation

MVT, extrema, monotonicity, concavity, optimization

Goal: recognise a analytical applications of differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Integration and Accumulation of Change

FTC, antiderivatives, substitution, Riemann sums

Goal: recognise a integration and accumulation of change problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Differential Equations

Slope fields, separation, exponential models

Goal: recognise a differential equations problem from its wording, carry out the governing method, and check that the result is reasonable.

08

Applications of Integration

Average value, area, volume, accumulation

Goal: recognise a applications of integration problem from its wording, carry out the governing method, and check that the result is reasonable.

AP Calculus AB questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 8 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Find the limit of (1 - cos x) / x^2 as x approaches 0.

AP Calculus AB · Limits and Continuity

  1. 1/2
  2. 0
  3. 1
  4. the limit does not exist
Show the worked answer
  1. Direct substitution gives 0/0, so more work is needed.
  2. Apply L'Hospital's rule once: the derivatives give sin x / (2x).
  3. That is still 0/0, so apply it again: cos x / 2.
  4. At x = 0 this equals 1/2.

Answer: 1/2

Question 2: What is the derivative of f(x) = the square root of x, evaluated at x = 9?

AP Calculus AB · Differentiation: Definition and Properties

  1. 1/6
  2. 1/3
  3. 6
  4. 3/2
Show the worked answer
  1. Write the function as x to the power 1/2.
  2. The power rule gives f'(x) = (1/2) x to the power -1/2, which is 1 divided by (2 times the square root of x).
  3. At x = 9 the square root is 3.
  4. So f'(9) = 1 / (2 x 3) = 1/6.

Answer: 1/6

Question 3: If y = (2x + 1)^5, what is dy/dx at x = 0?

AP Calculus AB · Composite, Implicit, and Inverse Differentiation

  1. 10
  2. 5
  3. 2
  4. 80
Show the worked answer
  1. Use the chain rule: bring down the 5, drop the power by one, then multiply by the derivative of the inside.
  2. dy/dx = 5(2x + 1)^4 x 2 = 10(2x + 1)^4.
  3. At x = 0 the inside is 1, and 1 to any power is 1.
  4. So dy/dx = 10.

Answer: 10

Question 4: A particle moves with position s(t) = t^3 - 6t^2 + 9t. At what times is it at rest?

AP Calculus AB · Contextual Applications of Differentiation

  1. t = 1 and t = 3
  2. t = 0 and t = 3
  3. t = 2 only
  4. t = 3 and t = 9
Show the worked answer
  1. At rest means velocity is zero, and velocity is the derivative of position.
  2. v(t) = 3t^2 - 12t + 9.
  3. Factor out 3: 3(t^2 - 4t + 3) = 3(t - 1)(t - 3).
  4. So v = 0 at t = 1 and t = 3.

Answer: t = 1 and t = 3

Question 5: For f(x) = x^3 - 3x^2, where is the local minimum and what is its value?

AP Calculus AB · Analytical Applications of Differentiation

  1. at x = 2, value -4
  2. at x = 0, value 0
  3. at x = 1, value -2
  4. at x = 2, value 4
Show the worked answer
  1. f'(x) = 3x^2 - 6x = 3x(x - 2), so the critical points are x = 0 and x = 2.
  2. f''(x) = 6x - 6. At x = 2 this is 6, which is positive, so the curve bends upward there.
  3. A positive second derivative at a critical point means a local minimum.
  4. f(2) = 8 - 12 = -4.

Answer: at x = 2, value -4

Question 6: Evaluate the integral of 2x + 1 over the square root of x, from x = 1 to x = 4.

AP Calculus AB · Integration and Accumulation of Change

  1. 17
  2. 15
  3. 20
  4. 13
Show the worked answer
  1. Rewrite the second term as x to the power -1/2, whose antiderivative is 2 times the square root of x.
  2. So the antiderivative of the whole thing is x^2 + 2 times the square root of x.
  3. At x = 4: 16 + 2(2) = 20.
  4. At x = 1: 1 + 2(1) = 3. Subtract: 20 - 3 = 17.

Answer: 17

Question 7: Solve dy/dx = 2xy with y(0) = 3, then find y(1) to three decimal places.

AP Calculus AB · Differential Equations

  1. 8.155
  2. 6.000
  3. 3.000
  4. 5.437
Show the worked answer
  1. Separate: dy / y = 2x dx.
  2. Integrating gives ln|y| = x^2 + C, so y = A e^(x^2).
  3. y(0) = 3 means A = 3, so y = 3 e^(x^2).
  4. At x = 1: 3e = 3 x 2.71828 = 8.155.

Answer: 8.155

Question 8: What is the area of the region between y = 2x and y = x^2?

AP Calculus AB · Applications of Integration

  1. 4/3
  2. 8/3
  3. 2
  4. 4
Show the worked answer
  1. Find where the curves meet: 2x = x^2 gives x = 0 and x = 2.
  2. Between those points the line is above the parabola, so integrate 2x - x^2.
  3. The antiderivative is x^2 - x^3 / 3.
  4. At x = 2: 4 - 8/3 = 4/3. At x = 0 it is 0, so the area is 4/3.

Answer: 4/3

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed AP Calculus AB questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Limits and ContinuityFind the limit of (1 - cos x) / x^2 as x approaches 0.1/2
2Differentiation: Definition and PropertiesWhat is the derivative of f(x) = the square root of x, evaluated at x = 9?1/6
3Composite, Implicit, and Inverse DifferentiationIf y = (2x + 1)^5, what is dy/dx at x = 0?10
4Contextual Applications of DifferentiationA particle moves with position s(t) = t^3 - 6t^2 + 9t. At what times is it at rest?t = 1 and t = 3
5Analytical Applications of DifferentiationFor f(x) = x^3 - 3x^2, where is the local minimum and what is its value?at x = 2, value -4
6Integration and Accumulation of ChangeEvaluate the integral of 2x + 1 over the square root of x, from x = 1 to x = 4.17
7Differential EquationsSolve dy/dx = 2xy with y(0) = 3, then find y(1) to three decimal places.8.155
8Applications of IntegrationWhat is the area of the region between y = 2x and y = x^2?4/3

How the generated AP Calculus AB sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this AP Calculus AB course test

What does this page cover?

It covers all 8 AP Calculus AB units listed above.

When can I see correct answers and explanations?

Submit the full test first. The result screen then shows every response, the correct answer, and the explanation so you can review each miss.

How long is each section of the AP Calculus AB exam?

Multiple choice: 1 hour 40 minutes, 42 questions. Free response: 1 hour 30 minutes, 6 questions.

Which AP Calculus AB sections allow a graphing calculator?

Rules change from section to section. Multiple choice, Part A: no calculator. Multiple choice, Part B: graphing calculator required. Free response, Part A: graphing calculator required. Free response, Part B: no calculator. Practise each section under its own rule.

Which AP Calculus AB units does this test cover?

All eight: Limits and Continuity, Differentiation: Definition and Properties, Composite, Implicit, and Inverse Differentiation, Contextual Applications of Differentiation, Analytical Applications of Differentiation, Integration and Accumulation of Change, Differential Equations, Applications of Integration. Study first: Analytical Applications of Differentiation (15–20%) and Integration and Accumulation of Change (15–20%) tie for the largest published weighting.

What should I check first when I review a missed AP Calculus AB question?

Justification questions want the sign chart, not just its conclusion. Say where f' changes from positive to negative, then name the maximum.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.