IB Maths: Applications and Interpretation chapter 5: Calculus MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

42 exam-style questions on Calculus, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

42 questions · 5 topics · 10 per test

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What chapter 5 covers

Ten Calculus questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

1. Find dy/dx for y = (x2 + 1)/x.

  1. A
    2x
  2. B
    1 + x−2
  3. C
    2x − x−2
  4. D
    1 − x−2
Show answer

Answer: D. Divide first: y = x + x−1, so dy/dx = 1 − x−2. You cannot differentiate the top and bottom separately.

2. Find ∫ (4x3 − 2x + 5) dx.

  1. A
    x4 − x2 + 5x + C
  2. B
    12x2 − 2 + C
  3. C
    x4 − x2 + C
  4. D
    4x4 − 2x2 + 5x + C
Show answer

Answer: A. Integrate term by term: 4x4/4 − 2x2/2 + 5x + C. The constant 5 integrates to 5x, not 0.

3. How can the accuracy of a trapezoidal rule estimate usually be improved?

  1. A
    Use fewer, wider intervals
  2. B
    Use only the end points of the interval
  3. C
    Use more intervals of smaller width
  4. D
    Double the final answer
Show answer

Answer: C. Smaller intervals make the chords follow the curve more closely, so the error shrinks.

4. Find the gradient of y = x3 − 4x at x = 2.

  1. A
    8
  2. B
    0
  3. C
    12
  4. D
    4
Show answer

Answer: A. dy/dx = 3x2 − 4, so at x = 2 the gradient is 12 − 4 = 8. 0 is the y-value at x = 2, not the gradient.

5. The normal to y = x3 at the point (1, 1) meets the x-axis at P. Find the coordinates of P.

  1. A
    (0.667, 0)
  2. B
    (−2, 0)
  3. C
    (4, 0)
  4. D
    (1.33, 0)
Show answer

Answer: C. The gradient at x = 1 is 3, so the normal has gradient −1/3: y − 1 = −(x − 1)/3. Setting y = 0 gives x = 4. (0.667, 0) is where the tangent meets the axis.

6. Find ∫ 3x−2 dx.

  1. A
    3x−1 + C
  2. B
    −x−3 + C
  3. C
    −3x−1 + C
  4. D
    −6x−3 + C
Show answer

Answer: C. The new power is −2 + 1 = −1, so 3x−1/(−1) = −3x−1 + C. −6x−3 is the derivative.

7. Find the area of the region enclosed by the curve y = x(6 − x) and the x-axis.

  1. A
    72
  2. B
    108
  3. C
    18
  4. D
    36
Show answer

Answer: D. The curve meets the x-axis at x = 0 and x = 6. Area = ∫ from 0 to 6 of (6x − x2) dx = 108 − 72 = 36.

8. A farmer uses 120 m of fencing for three sides of a rectangle, with a river as the fourth side. Find the maximum area enclosed.

  1. A
    900 m2
  2. B
    1600 m2
  3. C
    1800 m2
  4. D
    3600 m2
Show answer

Answer: C. With two sides x, A = x(120 − 2x); A′ = 120 − 4x = 0 at x = 30, so A = 30 × 60 = 1800 m2. 1600 m2 assumes three equal sides of 40 m.

9. Given f′(x) = 6x − 4 and f(1) = 5, find f(x).

  1. A
    f(x) = 3x2 − 4x + 5
  2. B
    f(x) = 6x2 − 4x + 3
  3. C
    f(x) = 3x2 − 4x + 4
  4. D
    f(x) = 3x2 − 4x + 6
Show answer

Answer: D. f(x) = 3x2 − 4x + C and f(1) = 3 − 4 + C = 5, so C = 6. C is not simply f(1).

10. Use the trapezoidal rule with 4 intervals to estimate the area under y = x2 from x = 0 to x = 2.

  1. A
    2.67
  2. B
    5.50
  3. C
    1.75
  4. D
    2.75
Show answer

Answer: D. h = 0.5 and y = 0, 0.25, 1, 2.25, 4: area ≈ ½ × 0.5 × [0 + 4 + 2(0.25 + 1 + 2.25)] = 2.75. 2.67 is the exact integral, and 5.50 leaves out the ½.

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How many Calculus MCQs are there?

42, each with the answer and a short explanation, across 5 topics: SL5.1, SL5.2 The derivative as a rate of change; increasing and decreasing functions, SL5.3, SL5.4 Differentiating axⁿ; tangents and normals, SL5.6, SL5.7 Stationary points and optimisation, SL5.5 Integration and areas, SL5.8 The trapezoidal rule. Every chapter test draws ten of them, so you can take it many times.

Is the IB Maths: Applications and Interpretation chapter 5 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

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