- 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 $(2,4)$?
- Imagine now that your graph is a topographical map. If you are standing at the point $(2,4)$ and walk one unit in the direction of the vector $(-3,4)$, will you be going uphill or downhill? Explain, and then state the slope of the mountain in the direction of $(-3,4)$.
- In which direction is the mountain the steepest? What is the slope in that direction?
- 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 the level surface $x^2+y^2+z^2=9$ at the point $(-2,2,1)$.
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