Allied & ICAR Exam Bank • ALLIED_2642NAT • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Total differentiation
Question Statement:
Let $f(x, y) = x^2 + 3 x y + y^2$. The directional derivative of $f$ at the point $(1, 1)$ in the direction of the unit vector $\vec{u} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}$ is $k \sqrt{2}$. The value of $k$ is:
Answer Key:
5
Step-by-Step Solution & Explanation
$\nabla f = (2x + 3y) \hat{i} + (3x + 2y) \hat{j}$. At $(1, 1)$, $\nabla f = 5 \hat{i} + 5 \hat{j}$. Directional derivative is $\nabla f \cdot \vec{u} = 5(1/\sqrt{2}) + 5(1/\sqrt{2}) = 10 / \sqrt{2} = 5 \sqrt{2}$. Hence $k = 5$.