Recall that average value is given by the formula $\bar f = \dfrac{\int_C fds}{\int_C ds}$, and mass is given by the formula $m=\int_C \delta ds$ where $\delta$ is the mass per unit length. Also, recall that a little bit of length is equal to the speed times a little bit of time, ie $ds = \left|\dfrac{d\vec r}{dt}\right|dt$.
- Set up an integral formula to find the average value of $f(x,y,z)=x^2+y^2+z^2$ along three coils of the helix $\vec r(t) = (4\cos t,4\sin t,3t)$ starting at $t=0$.
- Consider a wire that lies along the right half of the circle of radius 5 centered at the origin.
- Give a parametrization for the right half of the circle.
- The centroid of the wire is the $(\bar x,\bar y)$ location that gives the average $x$ value and $y$ value for points along the wire. Without any integrals, what is $\bar y$?
- Set up a formula to compute $\bar x$.
- Compute the integral from the previous part.
- For the function $f(x,y)=3x^2+5xy+y^2$, compute both $Df$ and $D^2f$.
- Find a function $f(x,y)$ so that $Df(x,y) = \begin{bmatrix}2x+3y&3x+4y\end{bmatrix}.$
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