Rapid Recall
- Find the critical points of the function $f(x,y)=x^2+4xy+y^2-3x$.
Solution
There is only one critical point. It is $(-1/2, 1)$. Just find where the first derivative is equal to zero (the zero matrix). .
- For the function $f(x,y)=x^2+4xy+y^2-3x$, compute the second derivative.
Solution
The solution is $$D^2f(x,y) = \begin{bmatrix}\begin{matrix}2\\4\end{matrix}&\begin{matrix}4\\2\end{matrix}\end{bmatrix}.$$
- For the function $f(x,y)=x^2+4xy+y^2-3x$, determine the location of any maxes, mins, or saddles, and classify each location appropriately using eigenvalues.
Solution
The only critical point is $(-1/2,1)$. The eigenvalues of $D^2f(-1/2,1)$ are $-2$ and $6$. Because these differ in sign, the point $(-1/2,1)$ is the location of a saddle point.
Group problems
- A rover travels along the line $2x+3y=6$. The surrounding terrain has elevation $z=x^2+4y$. The rover reaches a local minimum along this path.
- State the function $f$ to optimize, and the constraint $g=c$.
- Compute $\vec \nabla f$ and $\vec \nabla g$.
- Write the system of equations that results from $\vec \nabla f=\lambda\vec \nabla g$ together with $g(x,y) = 6$.
- Solve the system above (you should get $x=4/3$ and $y=10/9$).
- Consider the function $f(x,y)= 2x^2+3xy+4y^2-5x+2y$.
- Find all critical points of $f$.
- Determine the eigenvalues of the second derivative at each critical point. Do we have a max, min, or saddle?
- Consider the function $f(x,y,z) = 3xy+z^2$. We'll be analyzing the surface at the point $P=(1,-3,2)$.
- If $dx=0.1$, $dy=0.2$ and $dz=0.3$, then what is $df$ at $P$.
- Give an equation of the tangent plane to the level surface of $f$ that passes through $(1,2,-3)$.
- Give an equation of the tangent plane to the level surface of $f$ that passes through $(a,b,c)$.
- Give an equation of the tangent plane to $xy+z^2=7$ at the point $P=(-3,-2,1)$.
- Give an equation of the tangent plane to $z=f(x,y)=xy^2$ at the point $P=(4,-1,f(4,-1))$.
- Find the directional derivative of $f(x,y)=xy^2$ at $P=(4,-1)$ in the direction $(-3,4)$.
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