Finding the rate of change in direction

How would I go about answering this question? Would anyone be able to point me in the right direction?

Find a unit vector in the direction in which $f$ increases most rapidly at $P$, and find the rate of change of $f$ at $P$ in that direction: $$f(x,y) = \sqrt>$$

2,076 14 14 silver badges 18 18 bronze badges asked Mar 24, 2014 at 1:33 user136954 user136954 313 1 1 gold badge 6 6 silver badges 13 13 bronze badges

$\begingroup$ Hint: read the section on "gradient" in your text. Maximum rate of increase is in the direction of gradient and the rate is the length or norm of gradient vector. $\endgroup$

Commented Mar 24, 2014 at 1:37

2 Answers 2

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I suppose we should point you in the direction of maximal increase?

At any rate, you're looking for the gradient of $f$; for any differentiable function $f$, $\nabla f$ (or grad$(f)$ if you prefer) points in the direction of maximal increase at any given point.

answered Mar 24, 2014 at 1:39 Ben Grossmann Ben Grossmann 228k 12 12 gold badges 172 172 silver badges 338 338 bronze badges $\begingroup$

Compute $\nabla f = \left(<\partial f\over \partial x>, <\partial f\over\partial y>\right)$. This is a vector that points in the direction of fastest increase of $f$. Evaluate it at point $P$. Then normalize the vector to unit length (because they asked for a unit vector). The rate of change of $f$ in that direction is the dot product of $\nabla f$ with the unit vector.

answered Mar 24, 2014 at 1:41 Jason Zimba Jason Zimba 2,595 12 12 silver badges 13 13 bronze badges

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calculus
vectors

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