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AP Calculus AB/BC MCQs

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AP Calculus AB/BC practice questions by unit

Fiveable has 16,707 AP Calculus AB/BC multiple-choice practice questions across 10 units. Practice one unit or topic at a time.

  • Unit 1 – Limits and Continuity: 2,259 multiple-choice questions
  • Unit 2 – Fundamentals of Differentiation: 1,325 multiple-choice questions
  • Unit 3 – Composite, Implicit, and Inverse Functions: 737 multiple-choice questions
  • Unit 4 – Contextual Applications of Differentiation: 1,042 multiple-choice questions
  • Unit 5 – Analytical Applications of Differentiation: 2,161 multiple-choice questions
  • Unit 6 – Integration and Accumulation of Change: 2,239 multiple-choice questions
  • Unit 7 – Differential Equations: 1,688 multiple-choice questions
  • Unit 8 – Applications of Integration: 1,805 multiple-choice questions
  • Unit 9 – Parametric Equations, Polar Coordinates, and Vector–Valued Functions (BC Only): 1,317 multiple-choice questions
  • Unit 10 – Infinite Sequences and Series (BC Only): 2,134 multiple-choice questions

Sample AP Calculus AB/BC practice question

Consider lim⁡x→0x+4−2x\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}. Substituting x=0x = 0 yields 00\frac{0}{0}. Which algebraic technique transforms this expression into an equivalent form that allows direct limit evaluation?

Multiplying by the conjugate x+4+2x+4+2\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2} yields xx(x+4+2)=1x+4+2\frac{x}{x(\sqrt{x+4}+2)} = \frac{1}{\sqrt{x+4}+2}

Correct answer

Factoring out x\sqrt{x} from the numerator simplifies to x(1+4/x−2/x)x\frac{\sqrt{x}(\sqrt{1+4/x}-2/\sqrt{x})}{x}

Dividing numerator and denominator by x\sqrt{x} yields 1+4/x−2/x1\frac{\sqrt{1+4/x}-2/\sqrt{x}}{1}, which approaches −∞-\infty

Using the identity a−b=a−b2a+b\sqrt{a} - b = \frac{a - b^2}{\sqrt{a}+b} directly gives xx+4+2\frac{x}{\sqrt{x+4}+2} without further simplification

Explanation

Rationalizing the numerator by multiplying by the conjugate eliminates the radical. The numerator becomes (x+4)−4=x(x+4) - 4 = x, which cancels the denominator's xx. The resulting equivalent form 1x+4+2\frac{1}{\sqrt{x+4}+2} is continuous at x=0x = 0, yielding limit 14\frac{1}{4}.

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