Practice question · Multiple choice
For a function of several variables the gradient is a vector of partial derivatives. Why does that vector point in the direction of steepest ascent rather than in some other direction?
Hints
- The directional derivative in direction u is the dot product of the gradient with u. When is a dot product largest?
- Ask what that implies about which u gives the greatest rate of increase.
Show the answer
D. Because a dot product is largest when the two vectors are aligned.
Why
The rate of change in a unit direction u is ∇f · u, and a dot product is largest when the vectors align, so steepest ascent is along the gradient and steepest descent along its negative. The gradient is defined as the vector of partials; the steepest-ascent property is a theorem. It is purely local, which is why practitioners restart from several points.
Practise Multivariable Calculus and Gradients
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