Rapid Recall
- Find the critical points of the function $f(x,y)=x^3+3x^2+y^2+10y$.
Solution
There are two critical points, namely $(0,-5)$ and $(-2,-5)$. Note $\vec \nabla f(x,y) = (3x^2+6x,2y+10)$ is the zero matrix precisely when $x=0$ or $x=-2$, and $y=-5$.
- For the function $f(x,y)=x^3+3x^2+y^2+10y$, compute the second derivative.
Solution
Note $\frac{\partial}{\partial x} \vec\nabla f= (6x+6,0)$ and $\frac{\partial}{\partial y}\vec\nabla f = (0,2)$. Placing these vectors into the columns of a matrix gives us $$D^2f(x,y) = \begin{bmatrix}\begin{matrix}6x+6\\0\end{matrix}&\begin{matrix}0\\2\end{matrix}\end{bmatrix}.$$
- For the function $f(x,y)=x^3+3x^2+y^2+10y$, determine the location of any maxes, mins, or saddles, and classify each location appropriately using eigenvalues.
Solution
At each critical point, we need to (1) evaluate the second derivative, (2) compute the eigenvalues, and (3) classify the point using the eigenvalues.
- At the point $(0,-5)$, the second derivative is $D^2f(0,-5) = \begin{bmatrix}\begin{matrix}6\\0\end{matrix}&\begin{matrix}0\\2\end{matrix}\end{bmatrix}$. The eigenvalues are 6 and 2 (gradient point outwards from the point), which means at $(0,-5)$ we have a minimum.
- At the point $(-2,-5)$, the second derivative is $D^2f(-2,-5) = \begin{bmatrix}\begin{matrix}-6\\0\end{matrix}&\begin{matrix}0\\2\end{matrix}\end{bmatrix}$. The eigenvalues are -6 and 2 (gradient point outwards from the point in one direction, and inwards in another), which means at $(-2,-5)$ we have a saddle point.
Group problems
- Consider the function $f(x,y)= x^3+3xy-y^3$. This function has two critical points, namely $(0,0)$ and $(1,-1)$.
- Compute the gradient $\vec \nabla f(x,y)$.
- Verify that both $(0,0)$ and $(1,-1)$ are a critical points by computing both $\vec \nabla f(0,0)$ and $\vec \nabla f(1,-1)$. (What value should obtain, and do you obtain it?)
- Compute the second derivative $D^2f(x,y)$. The compute both $D^2f(0,0)$ and $D^2f(1,-1)$.
- Classify each critical point as a maximum, minimum, or saddle point, by computing the eigenvalues at that point.
- Two objects lie on the $x$-axis. The first object has a mass of 2 kg and is located at the point $x=-1$ (or rather its center of mass is at that point). The second object has a mass of 3 kg and is located at the point $x=4$. Find the center-of-mass of the combined system.
- Find the directional derivative of $f(x,y)=xy^2$ at $P=(4,-1)$ in the direction $(-3,4)$.
- 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 $(1,-3,2)$.
- Give an equation of the tangent plane to the level surface of $f$ that passes through $(a,b,c)$.
If you finish as a group, then open up the Mathematica notebook on problem 33 or 36, and use them to re answer question 1, and the rapid recall.
Desmos Graph for 5.1
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