Section 13.7 : Directional Derivatives
6. Find the maximum rate of change of \(f\left( {x,y} \right) = \sqrt {{x^2} + {y^4}} \) at \(\left( { - 2,3} \right)\) and the direction in which this maximum rate of change occurs.
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First, we’ll need the gradient and its value at \(\left( { - 2,3} \right)\).
\[\nabla f = \left\langle {\frac{x}{{\sqrt {{x^2} + {y^4}} }},\frac{{2{y^3}}}{{\sqrt {{x^2} + {y^4}} }}} \right\rangle \hspace{0.5in}\nabla f\left( { - 2,3} \right) = \left\langle {\frac{{ - 2}}{{\sqrt {85} }},\frac{{54}}{{\sqrt {85} }}} \right\rangle \]Show Step 2
Now, by the theorem in class we know that the direction in which the maximum rate of change at the point in question is simply the gradient at \(\left( { - 2,3} \right)\), which we found in the previous step. So, the direction in which the maximum rate of change of the function occurs is,
\[\nabla f\left( { - 2,3} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\left\langle {\frac{{ - 2}}{{\sqrt {85} }},\frac{{54}}{{\sqrt {85} }}} \right\rangle }}\]Show Step 3
The maximum rate of change is simply the magnitude of the gradient in the previous step. So, the maximum rate of change of the function is,
\[\left\| {\nabla f\left( { - 2,3} \right)} \right\| = \sqrt {\frac{4}{{85}} + \frac{{2916}}{{85}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\sqrt {\frac{{584}}{{17}}} = 5.8611}}\]