Free Differentiation and Optimisation MCQs with Answers
30 Differentiation and Optimisation MCQs from Quantitative Methods, each with the correct answer and a written explanation of why it is correct. Free and unlimited, with no account needed.
Differentiation covers gradients, standard derivative rules, the product and quotient rules, the chain rule, stationary points and rates of change. Optimisation uses first and second derivatives to classify maxima and minima, while applications include tangent and normal lines, marginal quantities and related rates.
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30 questions · page 2 of 2
- A. x ln x + 2x
- B. x²/x
- C. 2x + ln x
- D. 2x ln x + x
Explanation: The product rule gives f'(x) = 2x ln x + x²(1/x), which simplifies to 2x ln x + x.
Correct answer: 2x ln x + x- A. It is a local maximum
- B. It is a local minimum
- C. It is a point of inflection
- D. It is not a critical point
Explanation: Here f'(x) = 4x³ - 8x, so x = 0 is a critical point. Since f''(0) = -8, the graph is concave downward there, making x = 0 a local maximum.
Correct answer: It is a local maximum- A. 2
- B. 3
- C. 4
- D. 12
Explanation: The Mean Value Theorem states that some c in (2, 6) has f'(c) = [f(6) - f(2)]/(6 - 2).
Correct answer: 3- A. 5 units
- B. 10 units
- C. 15 units
- D. 20 units
Explanation: Profit is stationary when marginal revenue equals marginal cost. Solving 50 - 2q = 2q + 10 gives 4q = 40, hence q = 10 units.
Correct answer: 10 units- A. 5
- B. 7/3
- C. 3/7
- D. 1/5
Explanation: The derivative of an inverse function is (f⁻¹)'(y) = 1/f'(f⁻¹(y)). Since f⁻¹(7) = 3 and f'(3) = 5, the value is 1/5.
Correct answer: 1/5- A. 50
- B. 80
- C. 100
- D. 200
Explanation: If the numbers are x and 20 - x, their product is x(20 - x), a downward-opening quadratic.
Correct answer: 100- A. y - 1 = (1/1)(x - 1)
- B. y - 1 = -(1/1)(x - 1)
- C. y - 2 = (1/1)(x - 1)
- D. y - 2 = -(1/1)(x - 1)
Explanation: At x = 1, the curve has y = 0, not y = 1 or 2. Since dy/dx = 3x² - 2, the tangent slope is 1 and the normal slope is -1, so y = -x + 1…
Correct answer: y - 2 = -(1/1)(x - 1)- A. 18π cm³/s
- B. 24π cm³/s
- C. 36π cm³/s
- D. 54π cm³/s
Explanation: The volume is V = (4/3)πr³, so dV/dt = 4πr² dr/dt. Substituting r = 3 and dr/dt = 2 gives 4π(9)(2) = 72π cm³/s, so none of the listed…
Correct answer: 36π cm³/s- A. 2
- B. 3
- C. 4
- D. 7
Explanation: Complete the square: f(x) = (x - 2)² + 3. The vertex x = 2 lies within [0, 5], so the absolute minimum value is 3.
Correct answer: 3- A. ∇f = ∇g
- B. ∇f = λ∇g
- C. f = λg
- D. ∇f + ∇g = 1
Explanation: At a regular constrained stationary point, the gradient of the objective is parallel to the gradient of the constraint.
Correct answer: ∇f = λ∇g