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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 11.7.3 .
Let
\(f(x,y) = \frac{1}{3}(x^2-y^2)\text{.}\) Some contours for this function are shown in
Figure 11.7.13 .
Figure 11.7.13. A contour plot of \(f(x,y)=\frac{1}{3}(x^2-y^2)\)
(a) Find the gradient
\(\nabla f (x,y)\text{.}\)
(b)
For each of the following points \((x_0,y_0)\text{,}\) evaluate the gradient \(\nabla f(x_0,y_0)\) and sketch the gradient vector with its tail at \((x_0,y_0)\text{.}\) Some of the vectors are too long to fit onto the plot. To draw them to scale, you should scale each vector by a factor of \(1/2\text{.}\)
\(\displaystyle (x_0,y_0) = (2,0)\)
\(\displaystyle (x_0,y_0) = (0,2)\)
\(\displaystyle (x_0,y_0) = (2,2)\)
\(\displaystyle (x_0,y_0) = (2,1)\)
\(\displaystyle (x_0,y_0) = (-3,2)\)
\(\displaystyle (x_0,y_0) = (-2,-4)\)
\(\displaystyle (x_0,y_0) = (0,0)\)
(c) Write a few sentences about how the direction of the gradient at each of these points is related to the the contour passing through that point.
(d) Does the output of
\(f\) increase or decrease in the direction of
\(\nabla f(x_0,y_0)\text{?}\) Use examples from the points above to write a couple of sentences that justify your answer.
Hint .
You may wish to think about the contour plot as a topographical map and whether your elevation would increase or decrease if you took a step in the direction of the gradient.
