Chapter 19

In Exercises 25 and 26 , find \(f_{x}(x, y)\) and \(f_{y}(x, y)\). $$ f(x, y)=\int_{x}^{y} e^{\cos t} d t $$

In Exercises 24 through 29 , determine if the indicated limit exists. \(\lim _{(x, y) \rightarrow(2,-2)} \frac{\sin (x+y)}{x+y}\)

In Exercises 21 through 26 , draw a sketch of a contour map of the function \(f\) showing the level curves of \(f\) at the given numbers. The function \(f\) for which \(f(x, y)=(x-3) /(y+2)\) at \(4,2,1, \frac{1}{2}, \frac{1}{4}, 0,-\frac{1}{4},-\frac{1}{2},-1,-2\), and \(-4\).

Suppose \(f\) is a differentiable function of \(x\) and \(y\) and \(u=f(x, y)\). Then if \(x=\cosh v \cos w\) and \(y=\sinh v \sin w\), express \(\partial u / \partial v\) and \(\partial u / \partial w\) in terms of \(\partial u / \partial x\) and \(\partial u / \partial y\).

The specific gravity \(s\) of an object is given by the formula $$ s=\frac{A}{A-W} $$ where \(A\) is the number of pounds in the weight of the object in air and \(W\) is the number of pounds in the weight of the object in water. If the weight of an object in air is read as \(20 \mathrm{lb}\) with a possible error of \(0.01 \mathrm{lb}\) and its weight in water is read as \(12 \mathrm{lb}\) with a possible error of \(0.02 \mathrm{lb}\), find approximately the largest possible error in calculating \(s\) from these measurements. Also find the largest possible relative error.

$$ \text { Given } u=\sin \frac{r}{t}+\ln \frac{t}{r} . \text { Verify } t \frac{\partial u}{\partial t}+r \frac{\partial u}{\partial r}=0 $$.

In Exercises 24 through 29 , determine if the indicated limit exists. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y}{x^{2}+y^{2}} $$

In Exercises 27 and 28 , a function \(f\) and a function \(g\) are defined. Find \(h(x, y)\) if \(h=f \circ g\), and also find the domain of \(h\). \(f(t)=\sin ^{-1} t ; g(x, y)=\sqrt{1-x^{2}-y^{2}}\)

Suppose \(f\) is a differentiable function of \(x, y\), and \(z\) and \(u=f(x, y, z) .\) Then if \(x=r \sin \phi \cos \theta, y=r \sin \phi \sin \theta\), and \(z=r \cos \phi\), express \(\partial u / \partial r, \partial u / \partial \phi\), and \(\partial u / \partial \theta\) in terms of \(\partial u / \partial x, \partial u / \partial y\), and \(\partial u / \partial z\).

A wooden box is to be made of lumber that is \(\frac{2}{3}\) in. thick. The inside length is to be \(6 \mathrm{ft}\), the inside width is to be \(3 \mathrm{ft}\), the inside depth is to be \(4 \mathrm{ft}\), and the box is to have no top. Use the total differential to find the approximate amount of lumber to be used in the box.

$$ \text { Given } w=x^{2} y+y^{2} z+z^{2} x \text {. Verify } \frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}+\frac{\partial w}{\partial z}=(x+y+z)^{2} . $$

In Exercises 24 through 29 , determine if the indicated limit exists. $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x y}{x^{2}+y^{2}}+y \sin \frac{1}{x} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right\\} \lim _{(x, y)-(0,0)} f(x, y) $$

In Exercises 27 and 28 , a function \(f\) and a function \(g\) are defined. Find \(h(x, y)\) if \(h=f \circ g\), and also find the domain of \(h\). \(f(t)=\tan ^{-1} t ; g(x, y)=\sqrt{x^{2}-y^{2}}\)

Given \(u=9 x^{2}+4 y^{2}, x=r \cos \theta, y=r \sin \theta\). Find \(\partial^{2} u / \partial r^{2}\) in three ways: (a) by first expressing \(u\) in terms of \(r\) and \(\theta\); (b) by using the formula of Example 5; (c) by using the chain rule.

At a given instant, the length of one leg of a right triangle is \(10 \mathrm{ft}\) and it is increasing at the rate of \(1 \mathrm{ft} / \mathrm{min}\) and the length of the other leg of the right triangle is \(12 \mathrm{ft}\) and it is decreasing at the rate of \(2 \mathrm{ft} / \mathrm{min}\). Find the rate of change of the measure of the acute angle opposite the leg of length \(12 \mathrm{ft}\) at the given instant.

A company has contracted to manufacture 10,000 closed wooden crates having dimensions \(3 \mathrm{ft}, 4 \mathrm{ft}\), and \(5 \mathrm{ft}\). The cost of the wood to be used is \(5 \notin\) per square foot. If the machines that are used to cut the pieces of wood have a possible error of \(0.05 \mathrm{ft}\) in each dimension, find approximately, by using the total differential, the greatest possible error in the estimate of the cost of the wood.

$$ \text { Given } f(x, y)= \begin{cases}\frac{x^{3}+y^{3}}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases} $$

In Exercises 24 through 29 , determine if the indicated limit exists. $$ f(x, y)=\left\\{\begin{array}{ll} x \sin \frac{1}{y}+y \sin \frac{1}{x} & \text { if } x \neq 0 \text { and } y \neq 0 \\ 0 & \text { if either } x=0 \text { or } y=0 \end{array}\right\\} \lim _{(x, y)-(0,0)} f(x, y) $$

Given \(f(x, y)=x-y, g(t)=\sqrt{t}, h(s)=s^{2} .\) Find (a) \((g \circ f)(5,1) ;\) (b) \(f(h(3), g(9)) ;\) (c) \(f(g(x), h(y)) ;(\) d) \(g((h \circ f)(x, y)) ;\) (e) \((g \circ h)(f(x, y))\).

In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem \(19.5 .5\) are sufficient but not necessary for differentiability. The function \(f\) in these exercises is defined by In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem \(19.5 .5\) are sufficient but not necessary for differentiability. The function \(f\) in these exercises is defined by $$ f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases} $$ $$ \text { Find } \Delta f(0,0) \text {. } $$

Given \(f(x, y)= \begin{cases}\frac{x^{2}-x y}{x+y} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}\) Find (a) \(f_{1}(0, y)\) if \(y \neq 0\); (b) \(f_{1}(0,0)\).

Given \(f(x, y)=x / y^{2}, g(x)=x^{2}, h(x)=\sqrt{x}\). Find (a) \((h \circ f)(2,1) ;\) (b) \(f(g(2), h(4)) ;\) (c) \(f\left(g(\sqrt{x}), h\left(x^{2}\right)\right) ;\) (d) \(h((g \circ f)(x, y)) ;\) (e) \((h \circ g)(f(x, y))\).

In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem \(19.5 .5\) are sufficient but not necessary for differentiability. The function \(f\) in these exercises is defined by $$ f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases} $$ $$ \text { Find } D_{1} f(x, y) \text { and } D_{2} f(x, y) \text {. } $$

The electric potential at a point \((x, y)\) of the \(x y\) plane is \(V\) volts and \(V=4 / \sqrt{9-x^{2}-y^{2}} .\) Draw the equipotential curves for \(V=16,12,8,4,1, \frac{1}{2}\), and \(\frac{1}{4}\).

Water is flowing into a tank in the form of a right-circular cylinder at the rate of \(\frac{4}{5} \pi \mathrm{ft}^{3} / \mathrm{min}\). The tank is stretching in such a way that even though it remains cylindrical, its radius is increasing at the rate of \(0.002 \mathrm{ft} / \mathrm{min}\). How fast is the surface of the water rising when the radius is \(2 \mathrm{ft}\) and the volume of water in the tank is \(20 \pi \mathrm{ft}^{3}\) ?

Find the slope of the tangent line to the curve of intersection of the surface \(36 x^{2}-9 y^{2}+4 z^{2}+36=0\) with the plane \(x=1\) at the point \((1, \sqrt{12},-3) .\) Interpret this slope as a partial derivative.

The temperature at a point \((x, y)\) of a flat metal plate is \(t\) degrees and \(t=4 x^{2}+2 y^{2} .\) Draw the isothermals for \(t=12\), \(8,4,1\), and \(0 .\)

The one-dimensional heat-conduction partial differential equation is $$ \frac{\partial u}{\partial t}=k^{2} \frac{\partial^{2} u}{\partial x^{2}} $$ Show that if \(f\) is a function of \(x\) satisfying the equation $$ \frac{d^{2} f}{d x^{2}}+\lambda^{2} f(x)=0 $$ and \(g\) is a function of \(t\) satisfying the equation \(d g / d t+k^{2} \lambda^{2} g(t)=0\), then if \(u=f(x) g(t)\), the partial differential equation is satisfied. \(k\) and \(\lambda\) are constants.

In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem \(19.5 .5\) are sufficient but not necessary for differentiability. The function \(f\) in these exercises is defined by $$ f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases} $$ Prove that \(D_{1} f\) and \(D_{2} f\) are not continuous at \((0,0)\).

For the production of a certain commodity, if \(x\) is the number of machines used and \(y\) is the number of man-hours, the number of units of the commodity produced is \(f(x, y)\) and \(f(x, y)=6 x y\). Such a function \(f\) is called a production function and the level curves of \(f\) are called constant product curves. Draw the constant product curves for this function \(f\) at \(30,24,18,12,6\), and 0 .

The partial differential equation for a vibrating string is $$ \frac{\partial^{2} u}{\partial t^{2}}=a^{2} \frac{\partial^{2} u}{\partial x^{2}} $$ Show that if \(f\) is a function of \(x\) satisfying the equation \(d^{2} f / d x^{2}+\lambda^{2} f(x)=0\) and \(g\) is a function of \(t\) satisfying the equation \(d^{2} g / d t^{2}+a^{2} \lambda^{2} g(t)=0\), then if \(u=f(x) g(t)\), the partial differential equation is satisfied. \(a\) and \(\lambda\) are constants.

Find equations of the tangent line to the curve of intersection of the surface \(x^{2}+y^{2}+z^{2}=9\) with the plane \(y=2\) at the point \((1,2,2)\).

The temperature at any point \((x, y)\) of a flat plate is \(T\) degrees and \(T=54-\frac{2}{3} x^{2}-4 y^{2} .\) If distance is measured in feet, find the rate of change of the temperature with respect to the distance moved along the plate in the directions of the positive \(x\) and \(y\) axes, respectively, at the point \((3,1)\).

Prove that if \(f\) is a function of two variables and all the partial derivatives of \(f\) up to the fourth order are continuous on some open disk, then $$ D_{1122} f=D_{2121} f $$

If \(V\) dollars is the present value of an ordinary annuity of equal payments of \(\$ 100\) per year for \(t\) years at an interest rate of \(100 i\) percent per year, then $$ V=100\left[\frac{1-(1+i)^{-t}}{i}\right] $$ (a) Find the instantaneous rate of change of \(V\) per unit change in \(i\) if \(t\) remains fixed at 8. (b) Use the result of part (a) to find the approximate change in the present value if the interest rate changes from \(6 \%\) to \(7 \%\) and the time remains fixed at 8 years. (c) Find the instantaneous rate of change of \(V\) per unit change in \(t\) if \(i\) remains fixed at \(0.06\). (d) Use the result of part (c) to find the approximate change in the present value if the time is decreased from 8 to 7 years and the interest rate remains fixed at \(6 \%\).