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MathematicsGrade 12English

AP Mathematics Calculus AB Section I Multiple Choice 2023 Sample

This paper tests the core calculus skills that separate high scorers from the rest: can you reason through a limit without a calculator, differentiate a messy composite function in your head, and connect the derivative to real motion and optimization? Section I is 45 minutes of pure algebraic and conceptual problem-solving—no calculator, no second chances—and this sample mirrors the board's favourite question patterns and gotchas. You'll encounter limit definitions, chain rule traps, implicit differentiation, and contextual rate problems that demand more than formula-plugging. Perfect for Grade 12 students in AP Calculus AB who want to build confidence before sitting the real exam. By the end, you'll spot the derivative-vs-rate trick from across the room and know exactly which integration rule is hiding inside the word problem.

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  1. Question 1

    What is lim⁡x→2x2−4x−2\lim_{x \to 2} \frac{x^2 - 4}{x - 2}?

  2. Question 2

    A function f(x)={x2if x<12xif x≥1f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x & \text{if } x \geq 1 \end{cases} is continuous at x=1x = 1 if and only if the value of lim⁡x→1−f(x)\lim_{x \to 1^-} f(x) equals which of the following?

  3. Question 3

    If f(x)=3x2−5x+7f(x) = 3x^2 - 5x + 7, what is f′(x)f'(x)?

  4. Question 4

    Use the product rule to find the derivative of f(x)=(2x3)(5x2+1)f(x) = (2x^3)(5x^2 + 1).

  5. Question 5

    What is the derivative of f(x)=sin⁡(3x2)f(x) = \sin(3x^2) with respect to xx?

  6. Question 6

    If x2+y2=25x^2 + y^2 = 25, find dydx\frac{dy}{dx} by implicit differentiation.

  7. Question 7

    True or False: If f′(x)>0f'(x) > 0 on an interval, then f(x)f(x) is decreasing on that interval.

  8. Question 8

    A particle moves along a line with position function s(t)=t3−6t2+9ts(t) = t^3 - 6t^2 + 9t. What is the velocity at t=2t = 2?

  9. Question 9

    A rectangular box has a fixed volume of 64 cubic units. If the base has dimensions xx by xx, and the height is hh, what is the surface area as a function of xx alone?

  10. Question 10

    To minimize the surface area from the previous question, you would set dSdx\frac{dS}{dx} equal to zero and solve for xx. Which of the following is the critical point?

  11. Question 11

    What is lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} for f(x)=5x2f(x) = 5x^2?

  12. Question 12

    The graph of f(x)=x4−4x2f(x) = x^4 - 4x^2 has a local maximum at which xx-value?

  13. Question 13

    If f(x)=ln⁡(x2+1)f(x) = \ln(x^2 + 1), what is f′(x)f'(x)?

  14. Question 14

    A water tank is being filled at a rate of 10 cubic meters per minute. If the depth of water in the tank increases at a rate of 0.5 meters per minute, what is the cross-sectional area of the tank?