IB Maths: Analysis and Approaches chapter 5: Calculus MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

60 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.

60 questions · 6 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 the equation of the tangent to y = x2 + 3x at x = 1.

  1. A
    y = 5x + 4
  2. B
    y = 2x + 2
  3. C
    y = −⅕x + 4.2
  4. D
    y = 5x − 1
Show answer

Answer: D. At x = 1, y = 4 and dy/dx = 2x + 3 = 5. y − 4 = 5(x − 1) gives y = 5x − 1. y = −⅕x + 4.2 is the normal.

2. f′(x) = 3x2 − 2 and f(1) = 4. Find f(x).

  1. A
    x3 − 2x + 4
  2. B
    x3 − 2x + 3
  3. C
    6x − 2
  4. D
    x3 − 2x + 5
Show answer

Answer: D. f(x) = x3 − 2x + C, and f(1) = 1 − 2 + C = 4 gives C = 5.

3. Find ∫cos x dx.

  1. A
    −sin x + C
  2. B
    sin x + C
  3. C
    ½cos2x + C
  4. D
    −cos x + C
Show answer

Answer: B. The derivative of sin x is cos x, so ∫cos x dx = sin x + C. The minus sign belongs to ∫sin x dx = −cos x + C.

4. A curve has gradient dy/dx = 2e2x and passes through (0, 5). Find its equation.

  1. A
    y = e2x + 5
  2. B
    y = e2x + 4
  3. C
    y = 4e2x + 1
  4. D
    y = e2x + 3
Show answer

Answer: B. y = e2x + C, and at x = 0, e0 = 1, so 1 + C = 5 and C = 4. Forgetting that e0 = 1 gives C = 5.

5. A particle has displacement s = 5 sin t metres. Find its velocity when t = π/3.

  1. A
    2.5 m s−1
  2. B
    4.33 m s−1
  3. C
    −2.5 m s−1
  4. D
    5 m s−1
Show answer

Answer: A. v = 5 cos t, and 5 cos(π/3) = 5 × ½ = 2.5 m s−1. 4.33 is the displacement, 5 sin(π/3).

6. Find ∫ 4/x3 dx.

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

Answer: C. ∫4x−3 dx = 4x−2/(−2) + C = −2/x2 + C. The power goes up from −3 to −2, not down.

7. f(x) = x4 − 3x2. Find f″(x).

  1. A
    12x2 − 6
  2. B
    4x3 − 6x
  3. C
    12x2 − 6x
  4. D
    12x − 6
Show answer

Answer: A. f′(x) = 4x3 − 6x, then f″(x) = 12x2 − 6. 4x3 − 6x is only the first derivative.

8. f(x) = x3 + kx is increasing for all real x. Which condition on k is correct?

  1. A
    k ≤ 0
  2. B
    k > 3
  3. C
    k ≥ −3
  4. D
    k ≥ 0
Show answer

Answer: D. f′(x) = 3x2 + k must never be negative. Its least value is k (at x = 0), so k ≥ 0.

9. f″(x) > 0 for all x in an interval. Which statement must be true on that interval?

  1. A
    f is increasing
  2. B
    f′ is increasing
  3. C
    f is positive
  4. D
    f′ is positive
Show answer

Answer: B. f″ is the derivative of f′, so f″ > 0 means f′ is increasing (the graph of f is concave up). f itself may be decreasing.

10. f(x) = 4x3 − 5x2 + 7. Find f′(x).

  1. A
    12x2 − 10x
  2. B
    12x2 − 10x + 7
  3. C
    4x2 − 5x
  4. D
    12x3 − 10x2
Show answer

Answer: A. Multiply by the power and reduce it by one: 12x2 − 10x. The derivative of the constant 7 is 0.

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Ways to practise this chapter

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How many Calculus MCQs are there?

60, each with the answer and a short explanation, across 6 topics: SL5.1 to SL5.4 Introduction to differentiation, tangents and normals, SL5.6 Further differentiation, SL5.7, SL5.8 Second derivative, stationary points and optimisation, SL5.5, SL5.10 Anti-differentiation and indefinite integrals, SL5.5, SL5.11 Definite integrals and areas, SL5.9 Kinematics. Every chapter test draws ten of them, so you can take it many times.

Is the IB Maths: Analysis and Approaches 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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