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AP · · Calculus AB

Free AP Calculus AB
Practice Paper

8 mixed-difficulty practice questions in the style of real AP papers — answers, mark-scheme-style explanations, and the official exam structure all on one page.

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What the real AP Calculus AB paper looks like

Paper 1
Objective and short-answer questions covering the breadth of the specification.
Paper 2
Structured and extended-response questions testing depth of understanding.
Total exam time varies by board and tier — typically 2.5 to 3.5 hours.
Grading: See the official board specification for the current grading scale.

Mini practice paper: 8 questions

Mixed-difficulty questions from across the Calculus AB syllabus. Tap "Show answer" after each to check yourself.

Q1 · Difficulty 1/3

lim(x→2) (x³) = ?

  1. 6
  2. 8
  3. 12
  4. 4
Show answer & explanation
✓ Answer: B — 8
2³ = 8 by substitution.
Q2 · Difficulty 1/3

lim(x→∞) (1/x) = ?

  1. 0
  2. 1
  3. ∞
  4. −1
Show answer & explanation
✓ Answer: A — 0
As x grows large, 1/x approaches 0.
Q3 · Difficulty 1/3

lim(x→4) √x = ?

  1. 2
  2. 4
  3. 8
  4. 16
Show answer & explanation
✓ Answer: A — 2
√4 = 2 by direct substitution.
Q4 · Difficulty 2/3

If lim(x→a⁻) f(x) ≠ lim(x→a⁺) f(x), then:

  1. f is continuous at a
  2. the two-sided limit does not exist
  3. f(a) = 0
  4. f is differentiable at a
Show answer & explanation
✓ Answer: B — the two-sided limit does not exist
Unequal one-sided limits mean the two-sided limit DNE.
Q5 · Difficulty 2/3

lim(x→3) (x² − 9)/(x − 3) = ?

  1. 0
  2. 3
  3. 6
  4. undefined
Show answer & explanation
✓ Answer: C — 6
Factor: (x−3)(x+3)/(x−3) = x+3 → at x=3 gives 6.
Q6 · Difficulty 2/3

A vertical asymptote of f(x) = 1/(x − 5) occurs at:

  1. x = 0
  2. x = 5
  3. x = −5
  4. y = 5
Show answer & explanation
✓ Answer: B — x = 5
The denominator is zero at x = 5, giving a vertical asymptote.
Q7 · Difficulty 3/3

A removable discontinuity (hole) occurs when:

  1. the limit does not exist
  2. the limit exists but ≠ f(a) or f(a) is undefined
  3. the function is linear
  4. there is a vertical asymptote
Show answer & explanation
✓ Answer: B — the limit exists but ≠ f(a) or f(a) is undefined
A removable discontinuity has a limit that differs from (or is missing) the function value.
Q8 · Difficulty 3/3

The Intermediate Value Theorem requires the function to be:

  1. differentiable
  2. continuous on a closed interval
  3. linear
  4. increasing
Show answer & explanation
✓ Answer: B — continuous on a closed interval
IVT applies to functions continuous on [a, b].
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AP Calculus AB FAQ

What does the AP Calculus AB exam look like?
The AP Calculus AB exam is structured across 2 components. Paper 1: Objective and short-answer questions covering the breadth of the specification. Paper 2: Structured and extended-response questions testing depth of understanding. Total exam time varies by board and tier — typically 2.5 to 3.5 hours.
Can I download a free AP Calculus AB past paper?
Real AP past papers are published directly by AP on their official website. Kramizo doesn't redistribute copyrighted past papers, but we do generate free AI-written practice papers in the exact same style — same command words, same difficulty tier, same mark conventions. Use this practice paper as warm-up, then time yourself on official past papers before exam day.
How is AP Calculus AB graded?
See the official board specification for the current grading scale. Kramizo's practice questions are tagged with difficulty 1-3 mapping roughly to the lower, middle, and top grade boundaries you'll encounter in the real exam.