15 solved GATE DA questions on Calculus & Optimization, drawn from 3 exam years and grouped by year. Every question shows the official answer and a step-by-step solution.
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State the domain, continuity, or differentiability condition a formula requires before substituting. Optimization problems need both a critical-point calculation and a classification step. These selected foundations connect single-variable calculus, multivariable derivatives, and constrained optimization as they appear in the GATE DA syllabus. They are not a complete syllabus.
Our study notes and original examples support the PYQs below; they are not official exam questions or a replacement for the current syllabus.
A critical point of f(x) occurs where f′(x) = 0 or f′(x) is undefined. Not every critical point is an extremum: the second derivative test classifies it when f″ is nonzero. For f″(c) > 0 the point is a local minimum; for f″(c) < 0 a local maximum. When f″(c) = 0 the test is inconclusive.
Check yourself: Did you check the second derivative before declaring a minimum or maximum?
A definite integral computes net signed area between the curve and the x-axis. Substitution and integration by parts require matching differentials. Verify that the integrand is defined and continuous on the interval; split the integral at any discontinuity and check convergence for improper integrals.
Check yourself: Is the integrand continuous on the closed interval, including both endpoints?
The gradient ∇f = (∂f/∂x₁, …, ∂f/∂xₙ) points in the direction of steepest ascent. At a local extremum of an unconstrained differentiable function, the gradient is zero. The Hessian matrix of second partial derivatives classifies critical points: positive-definite implies a local minimum, negative-definite a local maximum.
Check yourself: Have you computed all second partial derivatives, including the cross terms?
To optimize f subject to g(x) = 0, solve ∇f = λ∇g together with the constraint. Each solution is a candidate; evaluate f at all candidates and any boundary points to determine the optimum. Multiple constraints require one multiplier per constraint equation.
Check yourself: Did you substitute back into the constraint to confirm feasibility?
Gradient descent updates x ← x − α∇f(x), where α is the learning rate. Each step moves in the steepest-descent direction. A step size that is too large can overshoot; too small converges slowly. Convergence to a global minimum requires convexity and an appropriate step size; non-convex problems may converge to local minima.
Check yourself: Did the objective value decrease after the update?
Original teaching example · not a PYQ
Original mini-example: minimize f(x, y) = x² + y² subject to x + y = 4 using Lagrange multipliers.
The minimum value is f(2, 2) = 8.
Separate original check: find all critical points of g(x) = x³ − 3x + 2 and classify each as a local maximum, local minimum, or neither.
x = −1 is a local maximum with g(−1) = 4; x = 1 is a local minimum with g(1) = 0.
g′(x) = 3x² − 3 = 3(x² − 1) = 0 gives x = ±1. g″(x) = 6x. At x = 1: g″(1) = 6 > 0, so local minimum. At x = −1: g″(−1) = −6 < 0, so local maximum. Values: g(1) = 1 − 3 + 2 = 0; g(−1) = −1 + 3 + 2 = 4.
Supplemental reading, not an official GATE reading list or an endorsement of these notes.
Free supplemental lectures, problem sets, and solutions covering partial derivatives, gradients, optimization, and Lagrange multipliers.
Free supplemental materials on limits, differentiation, integration, and single-variable optimization with worked examples.
This page shows 15 recent questions from the released archive, newest first. For older questions and complete papers, browse all GATE DA papers. Questions can carry more than one subject tag; counts are not marks weightage.
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