Calculus and Optimization GATE Practice Set
Mixed GATE DA style questions on limits, continuity, differentiability, derivatives, Taylor series, maxima and minima, convexity and numerical optimization, with answers and explanations.
These questions mix every topic in this track, in the GATE formats MCQ, MSQ and NAT. Each numerical answer is recomputed by a program when the site is built.
Limits and continuity
[Limits] What is limx→0 (sin 5x)/(tan 2x)?
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Answer: 2.5sin 5x ≈ 5x and tan 2x ≈ 2x near 0, so the limit is 5/2 = 2.5.
[L'Hôpital] What is limx→0 (x − sin x)/x³? Give the answer to four decimal places.
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Answer: 0.1667sin x = x − x³/6 + …, so x − sin x = x³/6 − …, and the limit is 1/6 ≈ 0.1667. (L'Hôpital three times gives cos x / 6 → 1/6.)
[Limits at infinity] What is limx→∞ (4x³ − x + 2)/(2x³ + 5x²)?
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Answer: 2The degrees are equal, so the limit is the ratio of the leading coefficients: 4/2 = 2.
[Differentiability] At which of these points is f(x) = |x² − 4| not differentiable? (Select all that apply.)
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Answer: A, Bx² − 4 changes sign at x = ±2 with non-zero slope (±4), so the absolute value creates corners there. At x = 0, f(x) = 4 − x² is smooth (its derivative is 0). At x = 4 it is x² − 4, also smooth.
Derivatives and Taylor series
[Chain rule] f(x) = ln(1 + e^(2x)). What is f′(0)?
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Answer: 1f′(x) = 2e^(2x)/(1 + e^(2x)). At 0: 2 × 1/(1 + 1) = 1.
[Taylor] What is the coefficient of x² in the Maclaurin series of f(x) = cos(3x)?
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Answer: -4.5cos u = 1 − u²/2 + …, and with u = 3x the x² term is −9x²/2. The coefficient is −4.5.
[Taylor] The Maclaurin series of f is 2 + 3x − x² + 4x³ + …. What is f‴(0)?
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Answer: 24The coefficient of x³ is f‴(0)/3!, so f‴(0) = 4 × 6 = 24.
Maxima, minima and optimization
[Extrema] What is the minimum value of f(x) = x + 4/x for x > 0?
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Answer: 4f′(x) = 1 − 4/x² = 0 at x = 2, and f″(x) = 8/x³ > 0 there. The minimum is f(2) = 2 + 2 = 4.
[Global extrema] What is the absolute maximum of f(x) = 2x³ − 9x² + 12x on [0, 3]?
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Answer: 9f′(x) = 6x² − 18x + 12 = 6(x − 1)(x − 2), so the critical points are 1 and 2. f(0) = 0, f(1) = 5, f(2) = 4, f(3) = 54 − 81 + 36 = 9. The absolute maximum is 9, at the endpoint x = 3.
[Second derivative test] f(x) = x³ − 3x² + 4. Which statements are true? (Select all that apply.)
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Answer: A, B, Df′ = 3x² − 6x = 3x(x − 2) gives critical points 0 and 2. f″ = 6x − 6: f″(0) = −6 (maximum), f″(2) = 6 (minimum). f″ changes sign at x = 1, the inflection point.
[Gradient descent] Gradient descent on f(x) = 2x² − 8x with learning rate 0.1 starts at x₀ = 0. What is x₂?
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Answer: 1.28f′(x) = 4x − 8. x₁ = 0 − 0.1 × (−8) = 0.8. x₂ = 0.8 − 0.1 × (3.2 − 8) = 0.8 + 0.48 = 1.28. (The minimum is at x = 2.)
[Newton's method] Newton's method for the root of g(x) = x² − 5 starts at x₀ = 2. What is x₁?
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Answer: 2.25x₁ = x₀ − g(x₀)/g′(x₀) = 2 − (4 − 5)/4 = 2 + 0.25 = 2.25 (√5 ≈ 2.236).