Chapter 14

Verify that the conclusion of Clairaut's Theorem holds, that is, \(u_{x y}=u_{y x}\) $$u=x y e^{y}$$

(a) The plane \(y+z=3\) intersects the cylinder \(x^{2}+y^{2}=5\) in an ellipse. Find parametric equations for the tangent line to this ellipse at the point \((1,2,1)\) . (b) Graph the cylinder, the plane, and the tangent line on the same screen.

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

Describe the level surfaces of the function. $$f(x, y, z)=x+3 y+5 z$$

(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations \(F(x, y, z)=0\) and \(G(x, y, z)=0\) are orthogonal at a point \(P\) where \(\nabla F \neq \mathbf{0}\) and \(\nabla G \neq \mathbf{0}\) if and only if $$F_{x} G_{x}+F_{y} G_{y}+F_{z} G_{z}=0 \quad\( at \)P$$ (b) Use part (a) to show that the surfaces \(z^{2}=x^{2}+y^{2}\) and \(x^{2}+y^{2}+z^{2}=r^{2}\) are orthogonal at every point of intersection. Can you see why this is true without using calculus?

Describe the level surfaces of the function. $$f(x, y, z)=x^{2}+3 y^{2}+5 z^{2}$$

(a) Show that the function \(f(x, y)=\sqrt[3]{x y}\) is continuous and the partial derivatives \(f_{x}\) and \(f_{y}\) exist at the origin but the directional derivatives in all other directions do not exist. (b) Graph \(f\) near the origin and comment on how the graph confirms part (a).

Find the indicated partial derivative. $$f(x, y, z)=\cos (4 x+3 y+2 z) ; \quad f_{x y z}, \quad f_{y z z}$$

Describe the level surfaces of the function. $$f(x, y, z)=x^{2}-y^{2}+z^{2}$$

Suppose that the directional derivatives of \(f(x, y)\) are known at a given point in two nonparallel directions given by unit vectors \(\mathbf{u}\) and \(\mathbf{v} .\) Is it possible to find \(\nabla f\) at this point? If so, how would you do it?

Find the indicated partial derivative. $$f(r, s, t)=r \ln \left(r s^{2} t^{3}\right) ; \quad f_{r s s t} \quad f_{r s t}$$

Describe the level surfaces of the function. $$f(x, y, z)=x^{2}-y^{2}$$

Find the indicated partial derivative. $$u=e^{r g} \sin \theta ; \quad \frac{\partial^{3} u}{\partial r^{2} \partial \theta}$$

Describe how the graph of \(g\) is obtained from the graph of \(f .\) $$g(x, y)=f(x, y)+2 \quad \text { (b) } g(x, y)=2 f(x, y)$$ $$g(x, y)=-f(x, y) \quad \text { (d) } g(x, y)=2-f(x, y)$$

Find the indicated partial derivative. $$z=u \sqrt{v-w} ; \frac{\partial^{3} z}{\partial u \partial v \partial w}$$

Describe how the graph of \(g\) is obtained from the graph of \(f .\) (a) \(g(x, y)=f(x-2, y) \quad\) (b) \(g(x, y)=f(x, y+2)\) (c)\(g(x, y)=f(x+3, y-4)\)

Find the indicated partial derivative. $$w=\frac{x}{y+2 z} ; \quad \frac{\partial^{3} w}{\partial z \partial y \partial x}, \quad \frac{\partial^{3} w}{\partial x^{2} \partial y}$$

Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"? $$f(x, y)=3 x-x^{4}-4 y^{2}-10 x y$$

Find the indicated partial derivative. $$u=x^{a} y^{b_{z}} c, \quad \frac{\partial^{6} u}{\partial x \partial y^{2} \partial z^{3}}$$

Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"?What about "local minimum points"? $$f(x, y)=x y e^{-x^{2}-y^{2}}$$

Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both \(x\) and \(y\) become large? What happens as \((x, y)\) approaches the origin? $$f(x, y)=\frac{x+y}{x^{2}+y^{2}}$$

Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both \(x\) and \(y\) become large? What happens as \((x, y)\) approaches the origin? $$f(x, y)=\frac{x y}{x^{2}+y^{2}}$$

Verify that the function \(u=e^{-u^{2} k^{2} t} \sin k x\) is a solution of the heat conduction equation \(u_{t}=\alpha^{2} u_{x x}\)

Use a computer to investigate the family of functions \(f(x, y)=e^{c x^{2}+y^{2}} .\) How does the shape of the graph depend on \(c ?\)

Determine whether each of the following functions is a solution of Laplace's equation \(u_{x x}+u_{y y}=0 .\) (a) \(u=x^{2}+y^{2} \quad\) (b) \(u=x^{2}-y^{2} \quad\) (c) \(u=x^{3}+3 x y^{2} \quad\) (d) \(u=\ln \sqrt{x^{2}+y^{2}} \quad\) (e) \(u=\sin x \cosh y+\cos x \sinh y\) (f) \(u=e^{-x} \cos y-e^{-y} \cos x\)

Use a computer to investigate the family of surfaces $$z=\left(a x^{2}+b y^{2}\right) e^{-x^{2}-y^{2}}$$ How does the shape of the graph depend on the numbers a and \(b ?\)

Use a computer to investigate the family of surfaces \(z=x^{2}+y^{2}+c x y .\) In particular, you should determine the transitional values of \(c\) for which the surface changes from one type of quadric surface to another.

Show that each of the following functions is a solution of the wave equation \(u_{t t}=a^{2} u_{x x} .\) (a) \(u=\sin (k x) \sin (a k t) \quad\) (b) \(u=t /\left(a^{2} t^{2}-x^{2}\right) \quad\)(c) \(u=(x-a t)^{6}+(x+a t)^{6}\quad\)(d) \(u=\sin (x-a t)+\ln (x+a t)\)

Verify that the function \(z=\ln \left(e^{x}+e^{y}\right)\) is a solution of the differential equations $$\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=1$$ and $$\frac{\partial^{2} z}{\partial x^{2}} \frac{\partial^{2} z}{\partial y^{2}}-\left(\frac{\partial^{2} z}{\partial x \partial y}\right)^{2}=0$$

Show that the Cobb-Douglas production function \(P=b L^{\alpha} K^{\beta}\) satisfies the equation $$L \frac{\partial P}{\partial L}+K \frac{\partial P}{\partial K}=(\alpha+\beta) P$$

The temperature at a point \((x, y)\) on a flat metal plate is given by \(T(x, y)=60 /\left(1+x^{2}+y^{2}\right),\) where \(T\) is measured in \(^{\circ} \mathrm{C}\) and \(x, y\) in meters. Find the rate of change of temperature with respect to distance at the point \((2,1)\) in (a) the \(x\) -direction and (b) the \(y\) -direction.

The total resistance \(R\) produced by three conductors with resistances \(R_{1}, R_{2}, R_{3}\) connected in a parallel electrical circuit is given by the formula $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}$$ Find \(\partial R / \partial R_{1}\)

The gas law for a fixed mass \(m\) of an ideal gas at absolute temperature \(T,\) pressure \(P,\) and volume \(V\) is \(P V=m R T,\) where \(R\) is the gas constant. Show that $$\frac{\partial P}{\partial V} \frac{\partial V}{\partial T} \frac{\partial T}{\partial P}=-1$$

The wind-chill index is modeled by the function $$W=13.12+0.6215 T-11.37 v^{0.16}+0.3965 T v^{0.16}$$ where \(T\) is the temperature \(\left(^{\circ} \mathrm{C}\right)\) and \(v\) is the wind speed \((\mathrm{km} / \mathrm{h}) .\) When \(T=-15^{\circ} \mathrm{C}\) and \(v=30 \mathrm{km} / \mathrm{h}\) , by how much would you expect the apparent temperature \(W\) to drop if the actual temperature decreases by \(1^{\circ} \mathrm{C} ?\) What if the wind speed increases by 1 \(\mathrm{km} / \mathrm{h}\) ?

The kinetic energy of a body with mass \(m\) and velocity \(v\) is \(K=\frac{1}{2} m v^{2} .\) Show that $$\frac{\partial K}{\partial m} \frac{\partial^{2} K}{\partial v^{2}}=K$$

If \(a, b, c\) are the sides of a triangle and \(A, B, C\) are the opposite angles, find \(\partial A / \partial a, \partial A / \partial b, \partial A / \partial c\) by implicit differentiation of the Law of Cosines.

You are told that there is a function \(f\) whose partial derivatives are \(f_{x}(x, y)=x+4 y\) and \(f_{y}(x, y)=3 x-y\) . Should you believe it?

The paraboloid \(z=6-x-x^{2}-2 y^{2}\) intersects the plane \(x=1\) in a parabola. Find parametric equations for the tangent line to this parabola at the point \((1,2,-4) .\) Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen.

The ellipsoid \(4 x^{2}+2 y^{2}+z^{2}=16\) intersects the plane \(y=2\) in an ellipse. Find parametric equations for the tangent line to this ellipse at the point \((1,2,2)\) .

In a study of frost penetration it was found that the temperature \(T\) at time \(t\) (measured in days) at a depth \(x\) (measured in feet) can be modeled by the function $$T(x, t)=T_{0}+T_{1} e^{-\lambda x} \sin (\omega t-\lambda x)$$ where \(\omega=2 \pi / 365\) and \(\lambda\) is a positive constant. (a) Find \(\partial T / \partial x\) . What is its physical significance? (b) Find \(\partial T / \partial t\) . What is its physical significance? (c) Show that \(T\) satisfies the heat equation \(T_{t}=k T_{x x}\) for a certain constant \(k .\) (d) If \(\lambda=0.2, T_{0}=0,\) and \(T_{1}=10\) , use a computer to graph \(T(x, t) .\) (e) What is the physical significance of the term \(-\lambda x\) in the \(\quad\) expression \(\sin (\omega t-\lambda x) ?\)

Use Clairaut's Theorem to show that if the third-order partial derivatives of \(f\) are continuous, then $$f_{x y y}=f_{y x y}=f_{y y x}$$

(a) How many \(n\) th-order partial derivatives does a function of two variables have? (b) If these partial derivatives are all continuous, how many of them can be distinct? (c) Answer the question in part (a) for a function of three variables.

$$f(x, y)=\sqrt[3]{x^{3}+y^{3}}, \text { find } f_{x}(0,0)$$

Let $$f(x, y)=\left\\{\begin{array}{ll}{\frac{x^{3} y-x y^{3}}{x^{2}+y^{2}}} & {\text { if }(x, y) \neq(0,0)} \\ {0} & {\text { if }(x, y)=(0,0)}\end{array}\right.$$ (a) Use a computer to graph \(f\) (b) Find \(f_{x}(x, y)\) and \(f_{y}(x, y)\) when \((x, y) \neq(0,0)\) (c) Find \(f_{x}(0,0)\) and \(f_{y}(0,0)\) using Equations 2 and 3 . (d) Show that \(f_{x y}(0,0)=-1\) and \(f_{y x}(0,0)=1\) (e) Does the result of part (d) contradict Clairaut's Theorem? Use graphs of \(f_{x y}\) and \(f_{y x}\) to illustrate your answer.