Chapter 4: Partial Differentiation

Find the largest rectangular parallelepiped (box) that can be shipped by parcel post (length plus girth$\mathbf{}\mathbf{=}\mathbf{}\mathbf{108}$in).

Find $\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{r}}{\mathbf{\partial }\mathbf{\theta }\mathbf{\partial }\mathbf{y}}$.

Do Problem 3 if $\mathit{i}\mathbf{=}\mathbf{12}$when $\mathit{o}\mathbf{=}\mathbf{18}$, to find i if$\mathit{o}\mathbf{=}\mathbf{17}\mathbf{.}\mathbf{5}$

Find the largest box (with faces parallel to the coordinate axes) that can be inscribed in$\frac{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{4}}\mathbf{+}\frac{{\mathbf{y}}^{\mathbf{2}}}{\mathbf{9}}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{2}}}{\mathbf{25}}\mathbf{=}\mathbf{1}\mathbf{.}$

If $\mathit{w}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{-}{\mathit{y}}^{\mathbf{3}}\mathbf{-}\mathbf{2}\mathit{x}\mathit{y}\mathbf{+}\mathbf{6}$,find${\mathbf{\partial }}^{\mathbf{2}}\mathit{w}\mathbf{/}\mathbf{\partial }{\mathit{x}}^{\mathbf{2}}$ and${\mathbf{\partial }}^{\mathbf{2}}\mathit{w}\mathbf{/}\mathbf{\partial }{\mathit{y}}^{\mathbf{2}}$at the points where$\mathbf{\partial }\mathit{w}\mathbf{/}\mathbf{\partial }\mathit{x}\mathbf{=}\mathbf{\partial }\mathit{w}\mathbf{/}\mathbf{\partial }\mathit{y}\mathbf{=}\mathbf{0}$.

To find the maximum and the minimum points of the given function.

x² - y²+ 2x - 4y + 10

Given $\mathbf{x}\mathbf{=}\mathbf{In}\left(u^2,v^2\right)\mathbf{,}\mathbf{u}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}$and $\mathbf{v}\mathbf{=}\mathbf{cos}\mathbf{}\mathbf{t}\mathbf{.}\mathbf{find}\frac{\mathbf{dx}}{\mathbf{dt}}$.

Use L’Hospital’s rule to evaluate $\underset{\mathbf{x}\mathbf{\to }\mathbf{2}}{\mathbf{lim}}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{2}}\mathbf{dt}{\mathbf{\int }}_{\mathbf{2}}^{\mathbf{x}}\frac{\mathbf{sint}}{\mathbf{t}}\mathbf{dt}$.

Find the two-variable Maclaurin series for the following functions.${\mathbf{e}}^{\mathbf{xy}}$

4. If$\mathit{w}\mathbf{=}{\mathit{e}}^{\mathbf{-}{\mathbf{r}}^{\mathbf{2}}\mathbf{-}{\mathbf{s}}^{\mathbf{2}}}$,$\mathit{r}\mathbf{=}\mathit{u}\mathit{v}$,$\mathit{s}\mathbf{=}\mathit{y}\mathbf{+}\mathbf{2}\mathit{v}$, find $\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{u}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{v}}$.