- The surface area of a sphere is $\sigma = 4\pi r^2$. Use differentials to estimate the increase in surface area if the radius changes from $2$ cm to $2.03$ cm.
- Compute $d(uv)$ in terms of $u,v, du,dv$. Then solve for $udv$ and integrate to explain where integration by parts comes from.
- Compute the matrix product $\begin{bmatrix}2&3\\1&-2\end{bmatrix}\begin{bmatrix}0&2&-1\\4&5&-2\end{bmatrix}$.
- The volume of a cylinder is $V=\pi r^2 h$. Compute $dV$ in terms of $r, h, dr, dh$. Then write your answer as the matrix product $$ dV = \begin{bmatrix}?&?\end{bmatrix}\begin{bmatrix}dr\\dh\end{bmatrix}.$$
- If $A=xy$, find $dA$ in terms of $x, y, dx, dy$. Then write your answer as a matrix product.
- Compute the determinant of $\begin{bmatrix}2&3\\1&-2\end{bmatrix}$.
- Compute the determinant of $\begin{bmatrix}0&2&-1\\4&5&-2\\1&3&1\end{bmatrix}$.
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