- 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.
- 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 $(a,b,c)$.
- 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 $(-3,4)$.
Let's spend some time, with our computers, applying the second derivative test and Lagrange multipliers. I'll put some problems up from the text, and we can work with those. Here's what to do with each problem.
- Identify the thing to optimize and label it $f$. If relevant, identify the constraint $g=c$.
- Plug the applicable functions into the correct notebook, and evaluate. Then adjust the bounds of your graph, if needed, so that you can see a visual of the solution.
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