Directional Derivatives and the Gradient Vector
This lab is meant to give you an intuitive idea of what directional derivatives and the gradient vector represent, and the relationship between them.
Directional Derivatives
We start with a function of two variables 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 = 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 .
f := (x, y) -> 2*x*y^2 - y^3 + x^2*y + x^3 + 4;
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
Here are the partials:
fx := D[1](f);
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
fy := D[2](f);
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
Here's a contour plot and the corresponding surface. You should have written the 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 values of the level sets on your handout.
with(plots): contour := contourplot(f(x, y), x=-1..1, y=-1..1, axes=boxed, filled=true, contours=12, coloring=[white, blue], scaling=constrained):
Warning, the name changecoords has been redefined
display(contour);
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surf := contourplot3d(f(x, y), x=-1..1, y=-1..1, axes=boxed, filled=true, coloring=[white, blue], contours=12):
display(surf);
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