AP Calculus AB Practice Questions by Topic- Exam Prep

What This Guide Covers

If you need AP Calculus AB practice questions organized by topic, you're in the right place. This isn't a study guide full of theory. It's a breakdown of every major concept you'll face on the exam, paired with the types of questions that actually show up.

Work through each section. If you miss a question type, you know exactly where to focus.

Why Topic-Specific Practice Beats Random Problem Sets

Most students grind through practice tests and wonder why their scores plateau. Here's the problem: random practice doesn't expose gaps. You might nail derivatives but bomb related rates because you never isolated that weakness.

Practice questions by topic let you:

• Identify exactly which concepts need work
• Build confidence in each area before mixing topics
• Track improvement in specific skills over time
• Simulate the pressure of seeing only one topic at a time on the exam

The AP Calculus AB exam doesn't warn you before hitting you with limits. You need to be ready for anything.

AP Calculus AB Topics Breakdown

The College Board organizes the exam around three Big Ideas. Every question traces back to one of these:

• Limits — Understanding behavior at boundaries
• Derivatives — Rates of change and slopes
• Integrals — Accumulation and area under curves

Each section below targets a specific topic. Master these, and you master the exam.

Practice Questions: Limits and Continuity

Limits

Limits appear in roughly 10-12% of the multiple-choice section. They're the foundation everything else builds on.

Question Type 1: Direct substitution

Evaluate: lim(x→3) (x² - 9)/(x - 3)

Plug in x = 3 and you get 0/0. That's indeterminate. Factor the numerator: (x+3)(x-3)/(x-3) = x+3. Now substitute: 3+3 = 6. Answer: 6.

Question Type 2: Limits at infinity

Evaluate: lim(x→∞) (3x² + 2x)/(x² - 5)

Divide every term by x². You get (3 + 2/x)/(1 - 5/x²). As x→∞, the fractions with x in the denominator approach 0. Answer: 3/1 = 3.

Question Type 3: One-sided limits

Given f(x) = x² for x < 2, and f(x) = 4x - 4 for x ≥ 2. Find lim(x→2⁻) f(x).

From the left, f(x) = x². At x = 2, that's 4. Answer: 4.

Continuity

Continuity questions test whether you understand the three conditions: the function exists at the point, the limit exists, and they equal each other.

Question Type: Finding discontinuities

Where is f(x) = (x+2)/(x² - 4) discontinuous?

Set denominator = 0: x² - 4 = 0, so x = ±2. The function is discontinuous at x = -2 and x = 2. Check if either is removable. At x = -2, the factor (x+2) cancels with numerator, making a hole. At x = 2, it's a vertical asymptote.

Practice Questions: Derivatives

Derivative Rules

Derivatives make up the largest chunk of the exam—roughly 40-45% of multiple choice. You need every rule on lockdown.

Question Type 1: Power rule

Find d/dx [4x³ - 2x² + 7x - 5]

Bring down the exponent, reduce by 1. Answer: 12x² - 4x + 7.

Question Type 2: Product rule

Find d/dx [x² · sin(x)]

Product rule: f'g + fg'. Here: (2x · sin x) + (x² · cos x). That's your answer.

Question Type 3: Quotient rule

Find d/dx [(2x + 1)/(x² + 3)]

Quotient rule: (f'g - fg')/g². f = 2x+1, g = x²+3. f' = 2, g' = 2x.

Numerator: (2)(x²+3) - (2x+1)(2x) = 2x² + 6 - 4x² - 2x = -2x² - 2x + 6.

Denominator: (x²+3)². Simplify if possible.

Question Type 4: Chain rule

Find d/dx [sin(5x³)]

Outer function: sin(u), derivative: cos(u). Inner function: 5x³, derivative: 15x². Answer: cos(5x³) · 15x² = 15x²cos(5x³).

Applications of Derivatives

Question Type 1: Related rates

A ladder 10 feet long slides down a wall. The bottom moves away at 2 ft/s. How fast is the top falling when the bottom is 6 feet from the wall?

Set up: x² + y² = 100. Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0.

x = 6. Find y: 36 + y² = 100, so y = 8.

Plug in: 2(6)(2) + 2(8)(dy/dt) = 0 → 24 + 16(dy/dt) = 0 → dy/dt = -24/16 = -3/2 ft/s.

Question Type 2: Optimization

A farmer has 200 feet of fence. What's the largest area he can enclose in a rectangle?

Constraint: 2l + 2w = 200, so l + w = 100, w = 100 - l.

Area: A = l(100 - l) = 100l - l².

Take derivative: A' = 100 - 2l = 0 → l = 50. Then w = 50.

Maximum area: 50 × 50 = 2500 sq ft.

Question Type 3: Motion problems

An object's position is s(t) = t³ - 6t² + 9t. Find velocity when acceleration = 0.

Velocity: v(t) = s'(t) = 3t² - 12t + 9.

Acceleration: a(t) = v'(t) = 6t - 12. Set a(t) = 0 → t = 2.

Velocity at t = 2: v(2) = 3(4) - 12(2) + 9 = 12 - 24 + 9 = -3.

Practice Questions: Integrals

Antiderivatives and Basic Integration

Integration is the reverse of differentiation. If you know derivatives, integration is applying those rules backwards.

Question Type 1: Power rule for integration

∫(3x⁴ - 2x + 5) dx

Increase exponent by 1, divide by new exponent. Answer: (3/5)x⁵ - x² + 5x + C.

Question Type 2: u-substitution

∫2x(x² + 1)³ dx

Let u = x² + 1. Then du = 2x dx.

Integral becomes ∫u³ du = (1/4)u⁴ + C = (1/4)(x² + 1)⁴ + C.

Area Under the Curve

Question Type: Definite integrals

∫₀² (x² + 1) dx

Find antiderivative: (1/3)x³ + x. Evaluate from 0 to 2.

At x = 2: (1/3)(8) + 2 = 8/3 + 2 = 8/3 + 6/3 = 14/3.

At x = 0: 0.

Answer: 14/3.

Fundamental Theorem of Calculus

Question Type: FTC Part 1

If F(x) = ∫₀ˣ (t² + 1) dt, find F'(x).

By FTC Part 1, F'(x) = x² + 1. Done.

Question Type: FTC Part 2 combined with chain rule

If G(x) = ∫₁ˣ³ t² dt, find G'(x).

Here the upper limit is constant (x³), not x. Use chain rule: G'(x) = (x²) · 3x² = 3x⁴.

Practice Questions: Applications of Integration

Area Between Curves

Question Type: Two curves

Find area between y = x² and y = 2x from x = 0 to x = 2.

Top curve minus bottom curve: (2x - x²).

Area = ∫₀² (2x - x²) dx = [x² - (1/3)x³]₀² = (4 - 8/3) - 0 = 12/3 - 8/3 = 4/3.

Volume of Solids of Revolution

Question Type: Disk method

Find volume when y = √x from x = 0 to x = 4 rotates around the x-axis.

V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = π(16/2) = 8π.

How to Use These Practice Questions

Step 1: Timed Practice

Set a timer. Multiple choice: you have about 2 minutes per question. Free response: 15 minutes per problem. When time's up, stop. This builds exam-day stamina.

Step 2: Review Immediately

Don't save review for later. Check your answers right after each set. When you get something wrong, figure out why before moving on. Wrong habits calcify fast.

Step 3: Track Your Mistakes

Keep a log of every problem you miss. Write down:

• The topic
• Why you missed it (computation error, wrong rule, didn't understand the question)
• What you'll do differently

Step 4: Repeat Until Clean

Redo missed problems the next day. Then again in three days. Then a week later. Spaced repetition works. Cramming doesn't.

Best Resources for AP Calculus AB Practice

Common Mistakes That Cost Points

• Forgetting the constant of integration (C) in indefinite integrals. The College Board will mark this wrong every time.
• Not writing units on free response questions when applicable. Sometimes they don't require it, but when they do, missing it costs you.
• Rushing through calculator active sections on the free response. You have 15 minutes per problem. Use them.
• Misidentifying the variable of integration in u-substitution. Check your substitution twice.
• Not showing work on free response. The graders need to see your reasoning. A correct answer with no work gets partial credit at best.

What the Free Response Actually Tests

The free response section has 6 problems. They mix calculator and non-calculator questions. Each problem tests multiple concepts.

You need to:

• Set up integrals and derivatives correctly
• Show your work step by step
• Include proper units when the problem asks
• Answer the question asked, not a related one

Graders look for three things: correct setup, correct execution, and correct interpretation of the result. Blow any of these and you lose points.

Final Advice

You don't need to enjoy calculus to pass the AP exam. You need to know the rules, practice applying them under pressure, and avoid stupid mistakes.

Work through problems until the process is automatic. When you see a related rates problem, your first move should be drawing a diagram and writing the constraint equation. When you see an area problem, your first move should be finding intersection points.

Build the habits now. The exam won't wait for you to figure things out.