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GATE 2019 AE – Question 13

Engineering Mathematics · Calculus: Functions of several variables, partial derivatives, gradient, divergence, curl and directional derivatives · 1 mark · Multiple choice

For $f(x,y)=x^2+y^2$, the unit direction of fastest increase at (3,4) is

  1. $(4\mathbf i-3\mathbf j)/5$
  2. $(3\mathbf i-4\mathbf j)/5$
  3. $(3\mathbf i+4\mathbf j)/5$
  4. $(4\mathbf i+3\mathbf j)/5$

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Correct answer: (C) $(3\mathbf i+4\mathbf j)/5$

Explanation

The gradient is (2x,2y), hence (6,8) at the specified point. Divide by its magnitude 10 to obtain (3/5,4/5), C. A unit directional derivative is largest when its direction is parallel to the gradient.

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