- Let $f(x,y) = x^2+y^2$. Your goal is to find the smallest value of $f$ at points on the curve $g(x,y)=xy^2=16$.
- Draw the curve $xy^2=16$. Then add to your curve several level curves of $f$.
- Find $\vec \nabla f$ and $\vec \nabla g$.
- Solve the system $\nabla f=\lambda \nabla g$ together with $xy^2=16$. (Show the three equations you need to solve are $2x=\lambda y^2$, $2y=2\lambda xy$, and $xy^2=16$.)
- Find the volume of the box of largest volume that will fit inside the ellipsoid $\ds\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}=1.$
- Find the work done by $\vec F = (3,4,-2)$ on an object that moves from $(0,2,1)$ to $(3,1,-5)$ along a straight line.
- The work done by the nonconstant force $\vec F = (5y,-5x)$ around the circle $\vec r(t) = (2\cos t, 2\sin t)$ is given by the integral $$
\int_C\vec F\cdot d\vec r
=\int_C(M,N)\cdot (dx,dy)
=\int_C Mdx+Ndy
= \int_a^b \vec F(\vec r(t))\cdot \frac{d\vec r}{dt}dt
.
$$
- For the given vector field and curve, state $\vec F$, $\vec r$, $d\vec r$, $M$, $N$, $x$, $y$, $dx$, and $dy$.
- Fill in the appropriate pieces of the integral above, and then compute the integral.
- For the curve $\vec r(t) = (\cos t,\sin t)$ for $0\leq t\leq \pi$, and for the function $f(x,y) = x+y$, compute the integral $\int_Cf ds = \int_a^b f(\vec r(t))\left|\frac{d\vec r}{dt}\right|dt$. Clearly state each of $x$, $y$, $dx$, $dy$, and $ds$.
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