- Find the directional derivative of $f(x,y)=x^2-y$ at the point $(-1,2)$ in the direction $(3,4)$.
- Find the directional derivative of $f(x,y,z)=x^2+y^2-z^2$ at the point $(-1,2,-2)$ in the direction $(3,4,0)$.
- Find a vector orthogonal to the surface $1=x^2+y^2-z^2$ at the point $(-1,2,-2)$. Use your answer to give an equation of the tangent plane to the surface at that point. (Hint: How is the gradient related to level surfaces?)
- Give an equation of the tangent plane to the surface $x^2+y^2+z^2=25$ at the point $(-3,0,4)$.
- Find the eigenvalues of the following matrices: \(\begin{bmatrix}2&4\\3&1\end{bmatrix}\), \(\begin{bmatrix}6&2\\2&3\end{bmatrix}\), \(\begin{bmatrix}2&4\\4&2\end{bmatrix}\)
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