Chapter 4: Partial Differentiation

$\mathit{z}\mathbf{=}\mathit{x}{\mathit{e}}^{\mathbf{-}\mathbf{y}}$,$\mathit{x}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathit{t}\mathbf{ }$,$\mathit{y}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{t}$, find$\frac{\mathbf{d}\mathbf{z}}{\mathbf{d}\mathbf{t}}$.

If $\mathit{y}\mathbf{=}\mathit{y}\mathit{z}$and $\mathit{y}\mathbf{=}\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\left(y+z\right)$, find$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{y}}\mathit{a}\mathit{n}\mathit{d}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{\mathbf{d}{\mathbf{y}}^{\mathbf{2}}}$ .

Use differentials to show that, for very large n , $\frac{\mathbf{1}}{{\left(n+1\right)}^{\mathbf{3}}}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{3}}}\mathbf{\equiv }\frac{\mathbf{3}}{{\mathbf{n}}^{\mathbf{4}}}$.

To find the familiar "second derivative test "for a maximum or minimum point. That is show that ${\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{=}\mathbf{0}$, then${\mathit{f}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{>}\mathbf{0}$implies a minimum point at$\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathit{a}$and${\mathit{f}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{<}\mathbf{0}$implies a maximum point at $\mathit{x}\mathbf{=}\mathit{a}$.

Consider a function $\mathit{f}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}$which can be expanded in a two-variable power series, (2.3) or (2.7). Let $\mathit{x}\mathbf{-}\mathit{a}\mathbf{=}\mathit{h}\mathbf{=}\mathbf{∆}\mathit{x}\mathbf{,}\mathit{y}\mathbf{-}\mathit{b}\mathbf{=}\mathit{k}\mathbf{=}\mathbf{∆}\mathit{y}$; then localid="1664363195022" $\mathit{x}\mathbf{=}\mathit{a}\mathbf{+}\mathbf{∆}\mathit{x}\mathbf{,}\mathit{y}\mathbf{=}\mathit{b}\mathbf{+}\mathbf{∆}\mathit{y}$, so that $\mathit{f}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}$becomes $\mathit{f}\mathbf{(}\mathit{a}\mathbf{+}\mathbf{∆}\mathit{x}\mathbf{,}\mathit{b}\mathbf{+}\mathbf{∆}\mathit{y}\mathbf{)}$. The change $\mathbf{∆}\mathit{z}$in $\mathit{z}\mathbf{=}\mathit{f}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}$when x changes from a to $\mathit{a}\mathbf{+}\mathbf{∆}\mathit{x}$and y changes from b to $\mathit{b}\mathbf{+}\mathbf{∆}\mathit{y}$is then

$\mathbf{∆}\mathit{z}\mathbf{=}\mathit{f}\mathbf{(}\mathit{a}\mathbf{+}\mathbf{∆}\mathit{x}\mathbf{,}\mathit{b}\mathbf{+}\mathbf{∆}\mathit{y}\mathbf{)}\mathbf{-}\mathit{f}\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}$.

Use the series (2.7) to obtain (3.11) and to see explicitly what ${\mathbf{\in }}_{\mathbf{1}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}{\mathbf{\in }}_{\mathbf{2}}$are and that they approach zero as $\mathbf{∆}\mathit{x}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathbf{∆}\mathit{y}\mathbf{\to }\mathbf{0}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathbf{∆}\mathit{y}\mathbf{\to }\mathbf{0}$.

If$\mathit{p}{\mathit{v}}^{\mathbf{a}}\mathbf{=}\mathit{c}$ (where $\mathit{a}$and$\mathit{c}$ are constants) find$\frac{\mathbf{dv}}{\mathbf{dp}}$ and$\frac{{\mathbf{d}}^{\mathbf{2}\mathbf{}}\mathbf{v}}{{\mathbf{d}}^{\mathbf{2}}\mathbf{p}}$ .

Find the two-variable Maclaurin series for the following functions.

cos x sinh y

Find the largest and smallest values of $\mathit{y}\mathbf{=}\mathbf{4}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{9}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{12}\mathit{x}\mathbf{+}\mathbf{3}$ if $\mathit{x}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$.

If $\mathit{u}\mathbf{+}\mathit{v}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{-}{\mathit{y}}^{\mathbf{3}}\mathbf{+}\mathbf{4}\mathbf{,}{\mathit{u}}^{\mathbf{2}}\mathbf{-}{\mathit{v}}^{\mathbf{2}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$, find ${\left(\frac{\partial u}{\partial x}\right)}_{\mathbf{y}}\mathbf{,}{\left(\frac{\partial u}{\partial x}\right)}_{\mathbf{v}}\mathbf{,}{\left(\frac{\partial x}{\partial u}\right)}_{\mathbf{y}}\mathbf{,}{\left(\frac{\partial x}{\partial u}\right)}_{\mathbf{v}}$at$\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{u}\mathbf{,}\mathbf{v}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{)}$.

If,$\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{,}\mathit{y}\mathbf{=}\mathit{r}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$, find the following partial derivatives.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }\mathbf{x}\mathbf{\partial }\mathbf{\theta }}$.