Chapter 14

Near a buoy, the depth of a lake at the point with coordinates \((x, y)\) is \(z=200+0.02 x^{2}-0.001 y^{3},\) where \(x, y,\) and \(z\) are measured in meters. A fisherman in a small boat starts at the point \((80,60)\) and moves toward the buoy, which is located at \((0,0) .\) Is the water under the boat getting deeper or shallower when he departs? Explain.

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) $$\begin{array}{l}{f(x, y)=x^{2}+y^{2}+x^{2} y+4} \\ {D=\\{(x, y)| | x|\leqslant 1,| y | \leqslant 1\\}}\end{array}$$

\(29-38\) Determine the set of points at which the function is continuous. $$F(x, y)=\arctan (x+\sqrt{y})$$

If \(z=5 x^{2}+y^{2}\) and \((x, y)\) changes from \((1,2)\) to \((1.05,2.1)\) compare the values of \(\Delta z\) and \(d z\)

Find the first partial derivatives of the function. $$w=\ln (x+2 y+3 z)$$

The temperature \(T\) in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point \((1,2,2)\) is \(120^{\circ}\) . (a) Find the rate of change of \(T\) at \((1,2,2)\) in the direction toward the point \((2,1,3) .\) (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin.

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) $$\begin{array}{l}{f(x, y)=4 x+6 y-x^{2}-y^{2}} \\ {D=\\{(x, y) | 0 \leqslant x \leqslant 4,0 \leqslant y \leqslant 5\\}}\end{array}$$

\(29-38\) Determine the set of points at which the function is continuous. $$F(x, y)=e^{x^{2} y}+\sqrt{x+y^{2}}$$

If \(z=x^{2}-x y+3 y^{2}\) and \((x, y)\) changes from \((3,-1)\) to \((2.96,-0.95),\) compare the values of \(\Delta z\) and \(d z\)

Find the first partial derivatives of the function. $$w=z e^{x y z}$$

The temperature at a point \((x, y, z)\) is given by $$T(x, y, z)=200 e^{-x^{2}-3 y^{2}-9 z^{2}}$$ where \(T\) is measured in \(^{\circ} \mathrm{C}\) and \(x, y, z\) in meters. (a) Find the rate of change of temperature at the point \(P(2,-1,2)\) in the direction toward the point \((3,-3,3)\) . (b) In which direction does the temperature increase fastest at \(P ?\) (c) Find the maximum rate of increase at \(P .\)

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) $$\begin{array}{l}{f(x, y)=x^{4}+y^{4}-4 x y+2} \\ {D=\\{(x, y) | 0 \leqslant x \leqslant 3,0 \leqslant y \leqslant 2\\}}\end{array}$$

\(29-38\) Determine the set of points at which the function is continuous. $$G(x, y)=\ln \left(x^{2}+y^{2}-4\right)$$

The length and width of a rectangle are measured as 30 \(\mathrm{cm}\) and \(24 \mathrm{cm},\) respectively, with an error in measurement of at most 0.1 \(\mathrm{cm}\) in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

Find the first partial derivatives of the function. $$u=x y \sin ^{-1}(y z)$$

Suppose that over a certain region of space the electrical potential \(V\) is given by \(V(x, y, z)=5 x^{2}-3 x y+x y z\) . (a) Find the rate of change of the potential at \(P(3,4,5)\) in the direction of the vector \(\mathbf{v}=\mathbf{i}+\mathbf{j}-\mathbf{k} .\) (b) In which direction does \(V\) change most rapidly at \(P ?\) (c) What is the maximum rate of change at \(P ?\)

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) $$f(x, y)=x y^{2}, \quad D=\\{(x, y) | x \geqslant 0, y \geqslant 0, x^{2}+y^{2} \leqslant 3\\}$$

\(29-38\) Determine the set of points at which the function is continuous. $$G(x, y)=\tan ^{-1}\left((x+y)^{-2}\right)$$

The dimensions of a closed rectangular box are measured as \(80 \mathrm{cm}, 60 \mathrm{cm},\) and \(50 \mathrm{cm},\) respectively, with a possible error of 0.2 \(\mathrm{cm}\) in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box.

Find the first partial derivatives of the function. $$u=X^{y / z}$$

Suppose you are climbing a hill whose shape is given by the equation \(z=1000-0.005 x^{2}-0.01 y^{2},\) where \(x, y,\) and \(z\) are measured in meters, and you are standing at a point with coordinates \((60,40,966) .\) The positive \(x\) -axis points east and the positive \(y\) -axis points north. (a) If you walk due south, will you start to ascend or descend? At what rate? (b) If you walk northwest, will you start to ascend or descend? At what rate? (c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) $$f(x, y)=2 x^{3}+y^{4}, \quad D=\\{(x, y) | x^{2}+y^{2} \leqslant 1\\}$$

The temperature at a point \((x, y)\) is \(T(x, y),\) measured in degrees Celsius. A bug crawls so that its position after \(t\) seconds is given by \(X=\sqrt{1+t}, y=2+\frac{1}{3} t,\) where \(x\) and \(y\) are measured in centimeters. The temperature function satisfies \(T_{x}(2,3)=4\) and \(T_{y}(2,3)=3 .\) How fast is the temperature rising on the bug's path after 3 seconds?

\(29-38\) Determine the set of points at which the function is continuous. $$f(x, y, z)=\frac{\sqrt{y}}{x^{2}-y^{2}+z^{2}}$$

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 \(\mathrm{cm}\) and height 12 \(\mathrm{cm}\) if the tin 0.04 \(\mathrm{cm}\) thick.

Find the first partial derivatives of the function. $$f(x, y, z, t)=x y z^{2} \tan (y t)$$

Let \(f\) be a function of two variables that has continuous partial derivatives and consider the points \(A(1,3), B(3,3),\) \(C(1,7),\) and \(D(6,15) .\) The directional derivative of \(f\) at \(A\) in the direction of the vector \(\vec{A B}\) is 3 and the directional derivative at \(A\) in the direction of \(\vec{A C}\) is \(26 .\) Find the directional derivative of \(f\) at \(A\) in the direction of the vector \(\vec{A D}\)

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) $$\begin{array}{l}{f(x, y)=x^{3}-3 x-y^{3}+12 y, \quad D \text { is the quadrilateral }} \\ {\text { whose vertices are }(-2,3),(2,3),(2,2), \text { and }(-2,-2)}\end{array}$$

Wheat production \(W\) in a given year depends on the average temperature \(T\) and the annual rainfall \(R\) . Scientists estimate that the average temperature is rising at a rate of \(0.15^{\circ} \mathrm{C} /\) year and rainfall is decreasing at a rate of 0.1 \(\mathrm{cm} / \mathrm{year.They}\) also estimate that, at current production levels, \(\partial W / \partial T=-2\) and \(\partial W / \partial R=8\) (a) What is the significance of the signs of these partial derivatives? (b) Estimate the current rate of change of wheat production, \(\quad d W / d t .\)

\(29-38\) Determine the set of points at which the function is continuous. $$f(x, y, z)=\sqrt{x+y+z}$$

Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 \(\mathrm{cm}\) high and 4 \(\mathrm{cm}\) in diameter if the metal in the top and bottom is 0.1 \(\mathrm{cm}\) thick and the metal in the sides is 0.05 \(\mathrm{cm}\) thick.

Find the first partial derivatives of the function. $$f(x, y, z, t)=\frac{x y^{2}}{t+2 z}$$

For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the function $$f(x, y)=-\left(x^{2}-1\right)^{2}-\left(x^{2} y-x-1\right)^{2}$$ has only two critical points, but has local maxima at both of them. Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible.

\(29-38\) Determine the set of points at which the function is continuous. $$f(x, y)=\left\\{\begin{array}{ll}{\frac{x^{2} y^{3}}{2 x^{2}+y^{2}}} & {\text { if }(x, y) \neq(0,0)} \\ {1} & {\text { if }(x, y)=(0,0)}\end{array}\right.$$

A boundary stripe 3 in. wide is painted around a rectangle whose dimensions are 100 \(\mathrm{ft}\) by 200 \(\mathrm{ft.}\) Use differentials to approximate the number of square feet of paint in the stripe.

Find the first partial derivatives of the function. $$u=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}$$

Show that the operation of taking the gradient of a function has the given property. Assume that \(u\) and \(v\) are differentiable functions of \(x\) and \(y\) and that \(a, b\) are constants. $$\begin{array}{ll}{\text { (a) } \nabla(a u+b v)=a \nabla u+b \nabla v} & {\text { (b) } \nabla(u v)=u \nabla v+v \nabla u} \\ {\text { (c) } \nabla\left(\frac{u}{v}\right)=\frac{v \nabla u-u \nabla v}{v^{2}}} & {\text { (d) } \nabla u^{n}=n u^{a-1} \nabla u}\end{array}$$

If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an absolute maximum. But this is not true for functions of two variables. Show that the function $$f(x, y)=3 x e^{y}-x^{3}-e^{3 y}$$ has exactly one critical point, and that \(f\) has a local maximum there that is not an absolute maximum. Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible.

The radius of a right circular cone is increasing at a rate of 1.8 in/s while its height is decreasing at a rate of 2.5 in \(/\) s. At what rate is the volume of the cone changing when the radius is 120 in. and the height is 140 in.

\(29-38\) Determine the set of points at which the function is continuous. $$f(x, y)=\left\\{\begin{array}{ll}{\frac{x y}{x^{2}+x y+y^{2}}} & {\text { if }(x, y) \neq(0,0)} \\ {0} & {\text { if }(x, y)=(0,0)}\end{array}\right.$$

The pressure, volume, and temperature of a mole of an ideal gas are related by the equation \(P V=8.31 T,\) where \(P\) is measured in kilo pascals, \(V\) in liters, and \(T\) in kelvins. Use differentials to find the approximate change in the pressure if the volume increases from 12 L to 12.3 \(\mathrm{L}\) and the temperature decreases from 310 \(\mathrm{K}\) to 305 \(\mathrm{K}\) .

Find the first partial derivatives of the function. $$u=\sin \left(x_{1}+2 x_{2}+\cdots+n x_{n}\right)$$

Find the shortest distance from the point \((2,1,-1)\) to the plane \(x+y-z=1 .\)

The length \(\ell,\) width \(w,\) and height \(h\) of a box change with time. At a certain instant the dimensions are \(\ell=1 \mathrm{m}\) and \(w=h=2 \mathrm{m},\) and \(\ell\) and \(w\) are increasing at a rate of 2 \(\mathrm{m} / \mathrm{s}\) while \(h\) is decreasing at a rate of 3 \(\mathrm{m} / \mathrm{s}\) . At that instant find the rates at which the following quantities are changing. (a) The volume (b) The surface area (c) The length of a diagonal

If \(R\) is the total resistance of three resistors, connected in parallel, with resistances \(R_{1}, R_{2}, R_{3},\) then $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}$$ If the resistances are measured in ohms as \(R_{1}=25 \Omega\) , \(R_{2}=40 \Omega,\) and \(R_{3}=50 \Omega,\) with a possible error of 0.5\(\%\) in each case, estimate the maximum error in the calculated value of \(R .\)

Find the indicated partial derivatives. $$f(x, y)=\ln \left(x+\sqrt{x^{2}+y^{2}}\right) ; \quad f_{x}(3,4)$$

\(39-44\) Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. $$2(x-2)^{2}+(y-1)^{2}+(z-3)^{2}=10, \quad(3,3,5)$$

Draw a contour map of the function showing several level curves. $$f(x, y)=(y-2 x)^{2}$$

Find the point on the plane \(x-y+z=4\) that is closest to the point \((1,2,3) .\)

Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 \(\mathrm{cm}^{2}\) and whose total edge length is 200 \(\mathrm{cm}.\)