Chapter 4: Partial Differentiation

In discussing the velocity distribution of molecules of an ideal gas, a function $\mathbf{F}\left(\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\right)\mathbf{=}\mathbf{f}\left(\mathbf{x}\right)\mathbf{f}\left(\mathbf{y}\right)\mathbf{f}\left(\mathbf{z}\right)$ is needed such that $\mathbf{d}\left(\mathbf{lnF}\right)\mathbf{=}\mathbf{0}$when Then by the Lagrange multiplier method$\mathbf{d}\left(\mathbf{lnF}\mathbf{+}\mathbf{\lambda f}\right)\mathbf{=}\mathbf{0}$ . Use this to show that $\mathbf{F}\left(\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\right)\mathbf{=}{\mathbf{Ae}}^{\mathbf{-}\left(\mathbf{\lambda }\mathbf{/}\mathbf{2}\right)\left({\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\right)}$.

The time dependent temperature at a point of a long bar is given by $\mathbf{T}\left(\mathbf{t}\right)\mathbf{=}\mathbf{100}\mathbf{°}\left(\mathbf{1}\mathbf{-}\frac{\mathbf{2}}{\sqrt{\mathbf{\pi }}}\underset{\mathbf{0}}{\overset{8/\sqrt{t}}{\int }}{\mathbf{e}}^{\mathbf{-}{\tau }^{\mathbf{2}}}\mathbf{d}\tau \right)$When $\mathbf{t}\mathbf{=}\mathbf{64}\mathbf{,}\mathbf{T}\mathbf{=}\mathbf{15}\mathbf{.}{\mathbf{73}}^{\mathbf{o}}$. Use differentials to estimate how long it will be until .

(a). Given the point $\left(2,1\right)$in the$\left(x,y\right)$ plane and the line $\mathbf{3}\mathit{x}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{4}$, find the distance from the point to the line by using the method of Chapter 3, Section 5.

(b). Solve part (a) by writing a formula for the distance from $\left(2,1\right)$to$\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$ and minimizing the distance (use Lagrange multipliers).

(c). Derive the formula

$\mathit{D}\mathbf{=}\left|\frac{a{x}_0+b{y}_0-c}{\sqrt{a^2+b^2}}\right|$

For the distance from $\left(x_0,y_0\right)$to $\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathit{y}\mathbf{=}\mathit{c}$by the methods suggested in parts (a) and (b).

If $\mathbf{s}\mathbf{=}{\mathbf{\int }}_{\mathbf{u}}^{\mathbf{v}}\frac{\mathbf{1}\mathbf{-}{\mathbf{e}}^{\mathbf{t}}}{\mathbf{t}}\mathbf{dt}\mathbf{,}$find$\frac{\mathbf{\partial }\mathbf{s}}{\mathbf{\partial }\mathbf{v}}\mathbf{}\mathbf{and}\frac{\mathbf{\partial }\mathbf{s}}{\mathbf{\partial }\mathbf{u}}$ and also their limits as $\mathit{u}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{v}$tend to zero.

Given$\mathit{w}\mathbf{=}\sqrt{{\mathbf{u}}^{\mathbf{2}}\mathbf{+}{\mathbf{v}}^{\mathbf{2}}}$,

$\mathbf{u}\mathbf{=}\mathbf{cos}\left[\mathrm{Intan}\left(p+\frac{1}{4}\pi \right)\right]\mathbf{v}\mathbf{=}\mathbf{sin}\left[\mathrm{In}\tan \left(p+\frac{1}{4}\pi \right)\right]$, find$\frac{\mathbf{d}\mathbf{w}}{\mathbf{d}\mathbf{p}}$.

If$\mathit{y}{\mathit{e}}^{\mathbf{x}\mathbf{y}}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$ find$\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$ and${\mathit{d}}^{\mathbf{2}}\mathit{y}\mathbf{/}\mathit{d}{\mathit{x}}^{\mathbf{2}}$ atrole="math" localid="1658830042567" $(0,0)$ .

To find the familiar "second derivative test "for a maximum or minimum point of the functions of two variables if${\mathit{f}}_{\mathbf{x}}\mathbf{=}{\mathit{f}}_{\mathbf{y}}\mathbf{=}\mathbf{0}$atlocalid="1664265078344" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}$then,

localid="1664265157617" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}$ Is maximum point if at $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{}\mathbf{,}\mathbf{}{\mathit{f}}_{\mathbf{x}\mathbf{x}}\mathbf{>}\mathbf{0}\mathbf{,}{\mathit{f}}_{\mathbf{y}\mathbf{y}}\mathbf{>}\mathbf{0}{\mathit{f}}_{\mathbf{x}\mathbf{x}}{\mathit{f}}_{\mathbf{y}\mathbf{y}}\mathbf{>}{\mathit{f}}_{\mathbf{x}\mathbf{y}}^{\mathbf{2}}$.

$\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}$ Is maximum point if at$\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{}\mathbf{,}\mathbf{}{\mathit{f}}_{\mathbf{x}\mathbf{x}}\mathbf{<}\mathbf{0}\mathbf{,}{\mathit{f}}_{\mathbf{y}\mathbf{y}}\mathbf{<}\mathbf{0}{\mathit{f}}_{\mathbf{x}\mathbf{x}}{\mathit{f}}_{\mathbf{y}\mathbf{y}}\mathbf{>}{\mathit{f}}_{\mathbf{x}\mathbf{y}}^{\mathbf{2}}$

$\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}$ Is neither a maximum nor minimum point if ${\mathit{f}}_{\mathbf{x}\mathbf{x}}{\mathit{f}}_{\mathbf{y}\mathbf{y}}\mathbf{<}{\mathit{f}}_{\mathbf{x}\mathbf{y}}^{\mathbf{2}}$.

Use differentials to show that, for large n and small a, $\sqrt{\mathbf{n}\mathbf{+}\mathbf{a}}\mathbf{-}\sqrt{\mathbf{n}}\mathbf{\cong }\frac{\mathbf{a}}{\mathbf{2}\sqrt{\mathbf{n}}}$.Find the approximate value of $\sqrt{{\mathbf{10}}^{\mathbf{26}}\mathbf{+}\mathbf{5}}\mathbf{-}\sqrt{{\mathbf{10}}^{\mathbf{26}}}$.

If $\mathit{P}\mathbf{=}\mathit{r}\mathit{c}\mathit{o}\mathit{s}\mathit{t}$ and role="math" localid="1664266535265" $\mathit{r}\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{-}\mathbf{2}\mathit{t}{\mathit{e}}^{\mathbf{r}}\mathbf{=}\mathbf{0}$, find $\frac{\mathbf{d}\mathbf{P}}{\mathbf{d}\mathbf{t}}$ .

What proportions will maximize the volume of a projectile in the form of a circular cylinder with one conical end and one flat end, if the surface area is given?