Question Bank • QB_EXP_EM_GEN_002 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus: Sequences, Infinite Series & Convergence

Subtopic: Ratio test

Question Statement:

The value of the directional derivative of $\phi(x, y, z) = x^2 y + y^2 z$ at point $P(1, 1, 1)$ in the direction of vector $\vec{v} = 2\hat{i} + \hat{j} + 2\hat{k}$ is ________ (round off to two decimal places).

Answer Key:

3.00

Step-by-Step Solution & Explanation

1. Compute gradient $\nabla \phi = (2xy)\hat{i} + (x^2 + 2yz)\hat{j} + (y^2)\hat{k}$. 2. Evaluate at $P(1, 1, 1)$: $\nabla \phi(1, 1, 1) = 2\hat{i} + 3\hat{j} + 1\hat{k}$ 3. Unit vector in direction of $\vec{v}$: $\hat{u} = \frac{2\hat{i} + \hat{j} + 2\hat{k}}{\sqrt{2^2 + 1^2 + 2^2}} = \frac{2\hat{i} + \hat{j} + 2\hat{k}}{3}$ 4. Directional derivative: $D_{\hat{u}} \phi = \nabla \phi \cdot \hat{u} = \frac{2(2) + 3(1) + 1(2)}{3} = \frac{4 + 3 + 2}{3} = \frac{9}{3} = 3.00$
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