GATE AG 2019 • Q54
NAT • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Vector differential operators – del, gradient
Question Statement:
The directional derivative of $f(x,y,z) = xy^2 + yz^3$ at the point $(2,-1,1)$ in the direction of vector $\hat{i} + 2\hat{j} + 2\hat{k}$ is ______
Official Answer Key:
-3.70
Step-by-Step Solution & Explanation
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) = (y², 2xy+z³, 3yz²).
At (2,−1,1): ∇f = ((−1)², 2(2)(−1)+1³, 3(−1)(1)²) = (1, −4+1, −3) = (1, −3, −3).
Unit vector along (1,2,2): |v| = √(1+4+4) = 3, so û = (1/3, 2/3, 2/3).
Directional derivative = ∇f · û = 1(1/3) + (−3)(2/3) + (−3)(2/3) = 1/3 − 2 − 2 = 1/3 − 4 = −3.667 ≈ −3.67. Answer: within range −3.70 to −3.60.
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