Chapter 4: Partial Differentiation

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

cos ( x + y )

Evaluate $\frac{{\mathbf{d}}^{\mathbf{2}}}{{\mathbf{dx}}^{\mathbf{2}}}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\int }}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\int }}\mathbf{f}\left(\mathbf{s}\mathbf{,}\mathbf{t}\right)\mathbf{dsdt}$ .

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

If $\mathit{z}\mathbf{=}\mathit{I}\mathit{n}\sqrt{{\mathbf{u}}^{\mathbf{2}}\mathbf{+}{\mathbf{v}}^{\mathbf{2}}\mathbf{+}{\mathbf{w}}^{\mathbf{2}}}$,find$\mathbf{\partial }\mathit{z}\mathbf{/}\mathbf{\partial }\mathit{u}\mathbf{,}\mathbf{\partial }\mathit{z}\mathbf{/}\mathbf{\partial }\mathit{v}\mathbf{,}\mathbf{\partial }\mathit{z}\mathbf{/}\mathit{w}$

Given $\mathit{r}\mathbf{=}{\mathit{e}}^{\mathbf{-}{\mathbf{p}}^{\mathbf{2}}\mathbf{-}{\mathbf{q}}^{\mathbf{2}}}\mathbf{,}\mathit{p}\mathbf{=}{\mathit{e}}^{\mathbf{5}}\mathbf{,}\mathit{q}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{5}}\mathbf{,}$find$\frac{\mathbf{d}\mathbf{r}}{\mathbf{d}\mathbf{s}}$.

If ${\mathit{x}}^{\mathbf{y}}\mathbf{=}{\mathit{y}}^{\mathbf{x}}$, find $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$ at (2,4).

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

$\frac{\mathbf{In}\left(1+x\right)}{\mathbf{1}\mathbf{+}\mathbf{y}}$

$\mathbf{If}\mathbf{}\mathbf{z}\mathbf{=}{\mathbf{\int }}_{\mathbf{sinx}}^{{}^{\mathbf{cosx}}}\mathbf{}\frac{\mathbf{sin}\mathbf{}\mathbf{t}}{\mathbf{dt}}\mathbf{,}\mathbf{find}\frac{\mathbf{}\mathbf{dz}}{\mathbf{dx}}\mathbf{.}$

The thin lens formula is $\frac{\mathbf{1}}{\mathbf{i}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{o}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{f}}$.

Where f is the focal length of the lens and oand iare the distances from the lens to the object and image. If $\mathbf{i}\mathbf{=}\mathbf{15}$ ,when $\mathit{o}\mathbf{=}\mathbf{10}$use differentials to find i whenlocalid="1659153107776" role="math" $\mathbf{o}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{1}$.

3. If $\mathit{z}\mathbf{=}\mathit{x}{\mathit{e}}^{\mathbf{-}\mathbf{y}}$andlocalid="1664273019902" $\mathit{y}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{s}$,find $\frac{\mathbf{\partial }\mathbf{z}}{\mathbf{\partial }\mathbf{s}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\frac{\mathbf{\partial }\mathbf{z}}{\mathbf{\partial }\mathbf{t}}$.