- Consider $f(x,y)=x^2-y$.
- Compute the derivative $Df(x,y)$.
- Draw several level curves of $f(x,y)$, making sure your plot is relatively to scale (don't just plot some parabolas down without trying to make them pass through 5 points accurately).
- The derivative above is a vector field. Add the vector field plot to your contour plot.
- State a vector that is normal to the level curve of $f$ that passes through the point $(2,4)$?
- Repeat the above with $f(x,y) = x^2+y^2$.
- Repeat the above with $f(x,y) = 4-y^2$.
- What patterns do you see? State a vector that is normal to the level surface $x^2+y^2+z^2=9$ at the point $(-2,2,1)$.
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