Chapter 13

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}+x y-2 y^{2}}{2 x^{2}-x y-y^{2}}$$

The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. $$g(x, y)=\left(x^{2}-x-2\right)\left(y^{2}+2 y\right).$$

The density of a thin circular plate of radius 2 is given by \(\rho(x, y)=4+x y .\) The edge of the plate is described by the parametric equations \(x=2 \cos t, y=2 \sin t\), for \(0 \leq t \leq 2 \pi\). a. Find the rate of change of the density with respect to \(t\) on the edge of the plate. b. At what point(s) on the edge of the plate is the density a maximum?

Gradients in three dimensions Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\) a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=\frac{x-z}{y-z} ; P(3,2,-1) ;\left\langle\frac{1}{3}, \frac{2}{3},-\frac{1}{3}\right\rangle$$

Two electrical resistors When two electrical resistors with resistance \(R_{1}>0\) and \(R_{2}>0\) are wired in parallel in a circuit (see figure), the combined resistance \(R,\) measured in ohms \((\Omega),\) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) a. Estimate the change in \(R\) if \(R_{1}\) increases from \(2 \Omega\) to \(2.05 \Omega\) and \(R_{2}\) decreases from \(3 \Omega\) to \(2.95 \Omega\) b. Is it true that if \(R_{1}=R_{2}\) and \(R_{1}\) increases by the same small amount as \(R_{2}\) decreases, then \(R\) is approximately unchanged? Explain. c. Is it true that if \(R_{1}\) and \(R_{2}\) increase, then \(R\) increases? Explain. d. Suppose \(R_{1}>R_{2}\) and \(R_{1}\) increases by the same small amount as \(R_{2}\) decreases. Does \(R\) increase or decrease?

Compute the first partial derivatives of the following functions. $$f(x, y)=1-\cos (2(x+y))+\cos ^{2}(x+y)$$

What point on the plane \(x-y+z=2\) is closest to the point (1,1,1)\(?\)

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(1,0)} \frac{y \ln y}{x}$$

The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. $$h(x, y)=1-e^{-\left(x^{2}+y^{2}-2 x\right)}.$$

Suppose you follow the spiral path \(C: x=\cos t, y=\sin t, z=t,\) for \(t \geq 0,\) through the domain of the function \(w=f(x, y, z)=(x y z) /\left(z^{2}+1\right)\). a. Find \(w^{\prime}(t)\) along \(C\). b. Estimate the point \((x, y, z)\) on \(C\) at which \(w\) has its maximum value.

Determine whether the following statements are true and give an explanation or counterexample. a. If \(f(x, y)=x^{2}+y^{2}-10,\) then \(\nabla f(x, y)=2 x+2 y\) b. Because the gradient gives the direction of maximum increase of a function, the gradient is always positive. c. The gradient of \(f(x, y, z)=1+x y z\) has four components. d. If \(f(x, y, z)=4,\) then \(\nabla f=\mathbf{0}\)

Three electrical resistors Extending Exercise \(62,\) when three electrical resistors with resistance \(R_{1}>0, R_{2}>0\) and \(R_{3}>0\) are wired in parallel in a circuit (see figure), the combined resistance \(R\), measured in ohms \((\Omega),\) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}} .\) Estimate the change in \(R\) if \(R_{1}\) increases from \(2 \Omega\) to \(2.05 \Omega, R_{2}\) decreases from \(3 \Omega\) to \(2.95 \Omega,\) and \(R_{3}\) increases from \(1.5 \Omega\) to \(1.55 \Omega\).

Compute the first partial derivatives of the following functions. $$h(x, y, z)=(1+x+2 y)^{z}$$

Let \(R\) be a closed bounded set in \(\mathbb{R}^{2}\) and let \(f(x, y)=a x+b y+c,\) where \(a, b,\) and \(c\) are real numbers, with \(a\) and \(b\) not both zero. Give a geometrical argument explaining why the absolute maximum and minimum values of \(f\) over \(R\) occur on the boundaries of \(R\)

Note that Cartesian and polar coordinates are related through the transformation equations \(\left\\{\begin{array}{l}x=r \cos \theta \\ y=r \sin \theta\end{array} \quad \text { or } \quad\left\\{\begin{array}{l}r^{2}=x^{2}+y^{2} \\ \tan \theta=y / x\end{array}\right.\right.\). a. Evaluate the partial derivatives \(x_{r}, y_{r}, x_{\theta},\) and \(y_{\theta}\). b. Evaluate the partial derivatives \(r_{x}, r_{y}, \theta_{x},\) and \(\theta_{y}\). c. For a function \(z=f(x, y),\) find \(z_{r}\) and \(z_{\theta},\) where \(x\) and \(y\) are expressed in terms of \(r\) and \(\theta\). d. For a function \(z=g(r, \theta),\) find \(z_{x}\) and \(z_{y},\) where \(r\) and \(\theta\) are expressed in terms of \(x\) and \(y\). e. Show that \(\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}\).

Consider the function \(F(x, y, z)=e^{x y z}\) a. Write \(F\) as a composite function \(f^{\circ} g,\) where \(f\) is a function of one variable and \(g\) is a function of three variables. b. Relate \(\nabla F\) to \(\nabla g\)

Power functions and percent change Suppose that \(z=f(x, y)=x^{a} y^{b},\) where \(a\) and \(b\) are real numbers. Let \(d x / x, d y / y,\) and \(d z / z\) be the approximate relative (percent) changes in \(x, y,\) and \(z,\) respectively. Show that \(d z / z=a(d x) / x+b(d y) / y ;\) that is, the relative changes are additive when weighted by the exponents \(a\) and \(b\)

Compute the first partial derivatives of the following functions. $$g(x, y, z)=\frac{4 x-2 y-2 z}{3 y-6 x-3 z}$$

Let \(x, y,\) and \(z\) be nonnegative numbers with \(x+y+z=200\). a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\). b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\). c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\). d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\).

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{|x-y|}{|x+y|}$$

An important derivative operation in many applications is called the Laplacian; in Cartesian coordinates, for \(z=f(x, y),\) the Laplacian is \(z_{x x}+z_{y y} .\) Determine the Laplacian in polar coordinates using the following steps. a. Begin with \(z=g(r, \theta)\) and write \(z_{x}\) and \(z_{y}\) in terms of polar coordinates (see Exercise 64). b. Use the Chain Rule to find \(z_{x x}=\frac{\partial}{\partial x}\left(z_{x}\right) .\) There should be two major terms, which, when expanded and simplified, result in five terms. c. Use the Chain Rule to find \(z_{y y}=\frac{\partial}{\partial y}\left(z_{y}\right) .\) There should be two major terms, which, when expanded and simplified, result in five terms. d. Combine parts (b) and (c) to show that $$ z_{x x}+z_{y y}=z_{r r}+\frac{1}{r} z_{r}+\frac{1}{r^{2}} z_{\theta \theta}. $$

Find the directions in the xy-plane in which the following functions have zero change at the given point. Express the directions in terms of unit vectors. $$f(x, y)=12-4 x^{2}-y^{2} ; P(1,2,4)$$

Prove that the level curves of the plane \(a x+b y+c z=d\) are parallel lines in the \(x y\) -plane, provided \(a^{2}+b^{2} \neq 0\) and \(c \neq 0.\)

Consider the function \(z=x / y^{2}.\) a. Compute \(z_{x}\) and \(z_{y}\) b. Sketch the level curves for \(z=1,2,3,\) and 4 c. Move along the horizontal line \(y=1\) in the \(x y\) -plane and describe how the corresponding \(z\) -values change. Explain how this observation is consistent with \(z_{x}\) as computed in part (a). d. Move along the vertical line \(x=1\) in the \(x y\) -plane and describe how the corresponding \(z\) -values change. Explain how this observation is consistent with \(z_{y}\) as computed in part (a).

Logarithmic differentials Let \(f\) be a differentiable function of one or more variables that is positive on its domain. a. Show that \(d(\ln f)=\frac{d f}{f}\) b. Use part (a) to explain the statement that the absolute change in \(\ln f\) is approximately equal to the relative change in \(f\) c. Let \(f(x, y)=x y,\) note that \(\ln f=\ln x+\ln y,\) and show that relative changes add; that is, \(d f / f=d x / x+d y / y\) d. Let \(f(x, y)=x / y,\) note that \(\ln f=\ln x-\ln y,\) and show that relative changes subtract; that is \(d f / f=d x / x-d y / y\) e. Show that in a product of \(n\) numbers, \(f=x_{1} x_{2} \cdots x_{n},\) the relative change in \(f\) is approximately equal to the sum of the relative changes in the variables.

Assume that \(x+y+z=1\) with \(x \geq 0\), \(y \geq 0,\) and \(z \geq 0\). a. Find the maximum and minimum values of \(\left(1+x^{2}\right)\left(1+y^{2}\right)\left(1+z^{2}\right)\) b. Find the maximum and minimum values of \((1+\sqrt{x})(1+\sqrt{y})(1+\sqrt{z})\)

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(-1,0)} \frac{x y e^{-y}}{x^{2}+y^{2}}$$

Suppose \(x\) and \(y\) are related by the equation \(F(x, y)=0 .\) Interpret the solution of this equation as the set of points \((x, y)\) that lie on the intersection of the surface \(z=F(x, y)\) with the \(x y\) -plane \((z=0)\). a. Make a sketch of a surface and its intersection with the \(x y\) -plane. Give a geometric interpretation of the result that \(\frac{d y}{d x}=-\frac{F_{x}}{F_{y}}\). b. Explain geometrically what happens at points where $$F_{y}=0.$$

Find an equation for the family of level surfaces pe corresponding to \(f .\) Describe the level surfaces.$$f(x, y, z)=\frac{1}{x^{2}+y^{2}+z^{2}}.$$

Find the directions in the xy-plane in which the following functions have zero change at the given point. Express the directions in terms of unit vectors. $$f(x, y)=x^{2}-4 y^{2}-8 ; P(4,1,4)$$

The volume of the cap of a sphere of radius \(r\) and thickness \(h\) is \(V=\frac{\pi}{3} h^{2}(3 r-h),\) for \(0 \leq h \leq r.\) a. Compute the partial derivatives \(V_{h}\) and \(V_{r}\) b. For a sphere of any radius, is the rate of change of volume with respect to \(r\) greater when \(h=0.2 r\) or when \(h=0.8 r ?\) c. For a sphere of any radius, for what value of \(h\) is the rate of change of volume with respect to \(r\) equal to \(1 ?\) d. For a fixed radius \(r,\) for what value of \(h(0 \leq h \leq r)\) is the rate of change of volume with respect to \(h\) the greatest?

Distance from a plane to an ellipsoid (Adapted from 1938 Putnam Exam) Consider the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) and the plane \(P\) given by \(A x+B y+C z+1=0 .\) Let \(h=\left(A^{2}+B^{2}+C^{2}\right)^{-1 / 2}\) and \(m=\left(a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}\right)^{1 / 2}\) a. Find the equation of the plane tangent to the ellipsoid at the point \((p, q, r)\) b. Find the two points on the ellipsoid at which the tangent plane is parallel to \(P\) and find equations of the tangent planes. c. Show that the distance between the origin and the plane \(P\) is \(h\) d. Show that the distance between the origin and the tangent planes is \(h m\) e. Find a condition that guarantees that the plane \(P\) does not intersect the ellipsoid.

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 75 about Steiner's problem.)

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}}$$

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$-x^{2}+\frac{y^{2}}{4}-\frac{z^{2}}{9}=1$$

In the implicit relationship \(F(x, y, z)=0,\) any two of the variables may be considered independent, which then determines the third variable. To avoid confusion, we use a subscript to indicate which variable is held fixed in a derivative calculation; for example \(\left(\frac{\partial z}{\partial x}\right)_{y}\) means that \(y\) is held fixed in taking the partial derivative of \(z\) with respect to \(x\). (In this context, the subscript does not mean a derivative.) a. Differentiate \(F(x, y, z)=0\) with respect to \(x\) holding \(y\) fixed to show that \(\left(\frac{\partial z}{\partial x}\right)_{y}=-\frac{F_{x}}{F_{z}}\). b. As in part (a), find \(\left(\frac{\partial y}{\partial z}\right)_{x}\) and \(\left(\frac{\partial x}{\partial y}\right)_{z}\). c. Show that \(\left(\frac{\partial z}{\partial x}\right)_{y}\left(\frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial x}{\partial y}\right)_{z}=-1\). d. Find the relationship analogous to part (c) for the case \(F(w, x, y, z)=0\).

Find an equation for the family of level surfaces pe corresponding to \(f .\) Describe the level surfaces.$$f(x, y, z)=x^{2}+y^{2}-z.$$

All triangles satisfy the Law of Cosines \(c^{2}=a^{2}+b^{2}-2 a b \cos \theta\) (see figure). Notice that when \(\theta=\pi / 2,\) the Law of cosines becomes the Pythagorean Theorem. Consider all triangles with a fixed angle \(\theta=\pi / 3,\) in which case, \(c\) is a function of \(a\) and \(b,\) where \(a>0\) and \(b>0\) a. Compute \(\frac{\partial c}{\partial a}\) and \(\frac{\partial c}{\partial b}\) by solving for \(c\) and differentiating. b. Compute \(\frac{\partial c}{\partial a}\) and \(\frac{\partial c}{\partial b}\) by implicit differentiation. Check for agreement with part (a). c. What relationship between \(a\) and \(b\) makes \(c\) an increasing function of \(a\) (for constant \(b\) )?

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{\sqrt{x^{2}+y^{2}}}$$

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$1-4 x^{2}+y^{2}+\frac{z^{2}}{2}=0$$

Let \(f(x, y)=0\) define \(y\) as a twice differentiable function of \(x\). a. Show that \(y^{\prime \prime}(x)=\frac{f_{x x} f_{y}^{2}-2 f_{x} f_{y} f_{x y}+f_{y y} f_{x}^{2}}{f_{y}^{3}}\). b. Verify part (a) using the function \(f(x, y)=x y-1\).

Find the directions in the xy-plane in which the following functions have zero change at the given point. Express the directions in terms of unit vectors. $$f(x, y)=e^{1-x y} ; P(1,0, e)$$

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{x^{2}+y^{2}}$$

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$-\frac{x^{2}}{3}+3 y^{2}-\frac{z^{2}}{12}=1$$

Let \(w=f(x, y, z)=2 x+3 y+4 z\), which is defined for all \((x, y, z)\) in \(\mathbb{R}^{3}\). Suppose that we are interested in the partial derivative \(w_{x}\) on a subset of \(\mathbb{R}^{3}\), such as the plane \(P\) given by \(z=4 x-2 y .\) The point to be made is that the result is not unique unless we specify which variables are considered independent. a. We could proceed as follows. On the plane \(P\), consider \(x\) and \(y\) as the independent variables, which means \(z\) depends on \(x\) and \(y,\) so we write \(w=f(x, y, z(x, y)) .\) Differentiate with respect to \(x\) holding \(y\) fixed to show that \(\left(\frac{\partial w}{\partial x}\right)_{y}=18,\) where the subscript \(y\) indicates that \(y\) is held fixed. b. Alternatively, on the plane \(P,\) we could consider \(x\) and \(z\) as the independent variables, which means \(y\) depends on \(x\) and \(z,\) so we write \(w=f(x, y(x, z), z)\) and differentiate with respect to \(x\) holding \(z\) fixed. Show that \(\left(\frac{\partial w}{\partial x}\right)_{z}=8,\) where the subscript \(z\) indicates that \(z\) is held fixed. c. Make a sketch of the plane \(z=4 x-2 y\) and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find \(\left(\frac{\partial w}{\partial y}\right)_{x}\), \(\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},\) and \(\left(\frac{\partial w}{\partial z}\right)_{y}\).

Find an equation for the family of level surfaces corresponding to \(f .\) Describe the level surfaces.$$f(x, y, z)=\sqrt{x^{2}+2 z^{2}}.$$

Suppose a long sloping hillside is described by the plane \(z=a x+b y+c,\) where \(a, b,\) and \(c\) are constants. Find the path in the \(x y\) -plane, beginning at \(\left(x_{0}, y_{0}\right),\) that corresponds to the path of steepest ascent on the hillside.

The electric potential in the \(x y\) -plane associated with two positive charges, one at (0,1) with twice the magnitude as the charge at \((0,-1),\) is $$\varphi(x, y)=\frac{2}{\sqrt{x^{2}+(y-1)^{2}}}+\frac{1}{\sqrt{x^{2}+(y+1)^{2}}}$$ a. Compute \(\varphi_{x}\) and \(\varphi_{y}.\) b. Describe how \(\varphi_{x}\) and \(\varphi_{y}\) behave as \(x, y \rightarrow \pm \infty.\) c. Evaluate \(\varphi_{x}(0, y),\) for all \(y \neq \pm 1 .\) Interpret this result. d. Evaluate \(\varphi_{y}(x, 0),\) for all \(x .\) Interpret this result.

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{x^{2}+x y+y^{2}}$$

Level curves of a savings account Suppose you make a one-time deposit of \(P\) dollars into a savings account that earns interest at an annual rate of \(p \%\) compounded continuously. The balance in the account after \(t\) years is \(B(P, r, t)=P e^{r t},\) where \(r=p / 100\) (for example, if the annual interest rate is \(4 \%,\) then \(r=0.04\) ). Let the interest rate be fixed at \(r=0.04.\) a. With a target balance of \(\$ 2000,\) find the set of all points \((P, t)\) that satisfy \(B=2000 .\) This curve gives all deposits \(P\) and times \(t\) that result in a balance of \(\$ 2000\). b. Repeat part (a) with \(B=\$ 500, \$ 1000, \$ 1500,\) and \(\$ 2500\) and draw the resulting level curves of the balance function. c. In general, on one level curve, if \(t\) increases, does \(P\) increase or decrease?