- Find the eigenvalues of the matrix $\begin{bmatrix}2&3\\5&4\end{bmatrix}$.
- Find the eigenvalues of the matrix $\begin{bmatrix}6&2\\2&3\end{bmatrix}$.
- Consider $f(x,y)=4-y^2$.
- Compute the derivative $Df(x,y)$.
- Draw several level curves of $f(x,y)$.
- 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?
- Give an equation of the tangent plane to the surface $xy^2-z^3=13$ at $(3,2,-1)$.
- Give an equation of the tangent plane to the surface $z=xy^2+4x$ at $(3,-2)$.
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