The Ideal Gas Law The pressure, temperature, and volume of

Fluid flow The x- and y-components of a fluid moving in two dimensions are given by the following functions u and v. The speed of the fluid at (x, y) is s(x, y) = \(\sqrt{u(x, y)^{2}+v(x, y)^{2}}\). Use the Chain Rule to find \(\partial s / \partial x\) and \(\partial x / \partial y\). u(x,y) = 2y and v(x,y) = -2x; \(x \geq 0 \text { and } y \geq 0\)

Change of coordinates continued 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 \(z_{x} \text { and } z_{y}\) in terms of polar coordinates (see Exercise 60). b. Use the Chain Rule to find \(z_{x x}=\frac{d}{\partial x}\left(z_{x}\right)\). There should be two major terms, which when expanded and simplified, result in five terms. c. Use the Chain Rule to find \(z_{yy}=\frac{d}{\partial x}\left(z_{y}\right)\). There should be two major terms, which when expanded and simplified, result in five terms. d. Combine part (a) and (b) to show that \(z_{x x}+z_{y y}=z_{r r}+\frac{1}{r} z_{r}+\frac{1}{r^{2}} z_{\theta \theta }\)

Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x \(y e^{x y}-2=0\)

If w is a function of x, y, and z, which are each functions of t, explain how to find dw/dt.

Implicit differentiation with three variables Use the result of Exercise 44 to evaluate \(\frac{\partial z}{\partial x} \text { and } \frac{\partial z}{\partial y^{2}}\) for the following relations. \(x^{2}+2 y^{2}-3 z^{2}=1\)

Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the curve C in the xy-plane given parametrically in the form x = g(t), y = h(t). a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of increasing t. Find the values of 1 for which you are walking uphill (that is, z is increasing). \(z=\sqrt{1-x^{2}-y^{2}}, C: x=e^{-t}, y=e^{-t} ;\) \(t \geq \frac{1}{2} \ln 2\)

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dQ/dt, where \(Q=\sqrt{x^{2}+y^{2}+z^{2}}), x = sin t, y= cos t, and z =cos t

Chain Rule with several independent variables Find the following derivatives. \(z_{s} \text { and } z_{t}\) where \(z=x y-x^{2}y\), x =s + t, and y = s- t

Chain Rule with several independent variables Find the following derivatives. \(w_{s} \text { and } w_{t}\), \(w=\frac{x-z}{y+z}\), x =s+t, y =st and z =s-1

If F(x, y) = 0 and y is a differentiable function of x, explain how to find dy/dx.

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dw/dt, where \(w=\cos 2 x \sin 3 y, x=t / 2, \text { and } y=t^{4})

Changing cylinder The volume of a right circular cylinder with radius r and height h is \(V=\pi r^{2} h) a. Assume that r and h are functions of t. Find V'(t). b. Suppose that \(r=e^{t} \text { and } h=e^{-2 t}\), for t \(\geq\) 0. Use part (a) to find V'(t). c. Does the volume of the cylinder in part (b) increase or decrease as t increases?

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dV/dy, where \(V=\frac{x-y}{y+z}\), x = t, y= 2t, and z=3t

If z = f(x, y), x = g(s, t), and y = h(s, t), explain how to find \(\partial z / \partial t\).

Implicit differentiation with three variables Use the result of Exercise 44 to evaluate \(\frac{\partial z}{\partial x} \text { and } \frac{\partial z}{\partial y^{2}}\) for the following relations. More than one way Let \(e^{x y z}=2 \). Find \(z_{x} \text { and } z_{y}\) in three ways(and check for agreement). a. Use the result of Exercise 44 b. Take logarithms of both sides and differentiate xyz = ln 2 c. Solve for z and differentiate z =ln 2/(xy).

The Ideal Gas Law The pressure, temperature, and volume of an ideal gas are related by PV = kT, where k > O is a constant. Any two of the variables may be considered independent, which determines the third variable. a. Use implicit differentiation to compute the partial derivatives \(\frac{\partial P}{\partial V}, \frac{\partial T}{\partial P}, \text { and } \frac{\partial V}{\partial T}\) b. Show that . Show that \(\frac{\partial P}{\partial V} \frac{\partial T}{\partial P} \frac{\partial V}{\partial T}=-1\). (See Exercise 63 for a generalization.)

Derivative practice Find the indicated derivative for the following functions. \(\frac{\partial z}{\partial x}\) where \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1)

Changing pyramid The volume of a pyramid with a square base x units on a side and a height of his \(V=\frac{1}{3} x^{2} h\). a. Assume that x and h are functions of t. Find V'(t). b. Suppose that x = 1/(t + 1) and h = 1/(t+1), for t \(\geq\) 0. Use part (a) to find V'(t). c. Does the volume of the pyramid in part (b) increase or decrease as t increases?

Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the curve C in the xy-plane given parametrically in the form x = g(t), y = h(t). a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of increasing t. Find the values of 1 for which you are walking uphill (that is, z is increasing). \(z=x^{2}+4 y^{2}+1, C: x=\cos t, y=\sin t ; 0 \leq t \leq 2 \pi\)

Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x \(\sqrt{x^{2}+2 x y+y^{4}}=3\)

Utility functions in economics Economists use utility functions to describe consumers' relative preference for two or more commodities (for example, vanilla vs. chocolate ice cream or leisure time vs. material goods). The Cobb-Douglas family of utility functions has the form \(U(x, y)=x^{a} y^{1-a}\), where x and y are the amounts of two commodities and 0 < a < l is a parameter. Level curves on which the utility function is constant are called indifference curves; the preference is the same for all combinations of x and y along an indifference curve (see figure). a. The marginal utilities of the commodities x and y are defined to be au fax and au say, respectively. Compute the marginal utilities for the utility function \(U(x, y)=x^{a} y^{1-a}\) b. The marginal rate of substitution (MRS) is the slope of the indifference curve at the point (x, y). Use the Chain Rule to show that for \(U(x, y)=x^{a} y^{1-a}\). the MRS is \(-\frac{a}{1-a} \frac{y}{x}\) c. Find the MRS for the utility function \(U(x, y)=x^{0.4} y^{0.6}\) at (x, y) = (8, 12).

General three-variable relationship 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 may 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} \text { and }\left(\frac{\partial x}{\partial y}\right)) 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.

Body surface area One of several empirical formulas that relates the surface area of a human body to the height h and weight w of the body is the Mosteller formula \(S(h, w)=\frac{1}{60} \sqrt{h w}\), where h is measured in cm, w is measured in kg, and S is measured in \(m^{2}\). Suppose that h and w are functions of t. a. Find S'(t). b. Show that the condition that the surface area remains constant as h and w change is wh' (t) + hw' (t) = 0. c. Show that part (b) implies that for constant surface area, h and w must be inversely related; that is, h = C/w, where C is a constant

Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x \(x^{2}-2 y^{2}-1=0\)

Implicit differentiation with three variables Use the result of Exercise 44 to evaluate \(\frac{\partial z}{\partial x} \text { and } \frac{\partial z}{\partial y^{2}}\) for the following relations. xy+xz+yz=3

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dw/dt, where w = xy sin z, \(x=t^{2}, y=4 t^{3}, \text { and } z=t+1)

Given that w = F(x, y, z), and x, y, and z are functions of r and s, sketch a tree diagram with branches labeled with the appropriate derivatives.

Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x \(y \ln \left(x^{2}+y^{2}+4\right)=3\)

Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x \(x^{3}+3 x y^{2}-y^{5}=0\)

Making trees Use a tree diagram to write the required Chain Rule formula. w is a function of z, where z is a function of x and y, each of which is a function of t, Find dw/dt

Implicit differentiation with three variables Use the result of Exercise 44 to evaluate \(\frac{\partial z}{\partial x} \text { and } \frac{\partial z}{\partial y^{2}}\) for the following relations. x y z+x+y-z=0

Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the curve C in the xy-plane given parametrically in the form x = g(t), y = h(t). a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of increasing t. Find the values of 1 for which you are walking uphill (that is, z is increasing). \(z=4 x^{2}-y^{2}+1, C: x=\cos t, y=\sin t ; 0 \leq t \leq 2 \pi\)

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dz/dt, where z =x sin y, \(x=t^{2}, \text { and } y=4 t^{3}\)

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dU/dt, where \(U=\ln (x+y+z), x=t, y=t^{2}, \text { and } z=t^{3}\)

Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x 2 sin xy = 1

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dz/dt, where \(z=\sqrt{r^{2}+s^{2}}, r=\cos 2 t, \text { and } s=\sin 2 t)

Second derivative 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) = xy - I.

Chain Rule with several independent variables Find the following derivatives. \(w_{r}, w_{s} \text { and } w_{t}\), where \(w=\sqrt{x^{2}+y^{2}+z^{2}}\), x= st, y =rs, z =rt

Fluid flow The x- and y-components of a fluid moving in two dimensions are given by the following functions u and v. The speed of the fluid at (x, y) is s(x, y) = \(\sqrt{u(x, y)^{2}+v(x, y)^{2}}\). Use the Chain Rule to find \(\partial s / \partial x\) and \(\partial x / \partial y\). u(x,y) = x(1-x)(1-2y) and v(x,y) = y(y-1)(1-2x); \(0 \leq x \leq 1,0<y<1\)

Derivative practice Find the indicated derivative for the following functions. \(\partial w / \partial x\), where w = cos z - cos x cos y + sin x sin y, and z = x+y

Constant volume tori The volume of a solid torus (a bagel or doughnut) is given by \(V=\left(\pi^{2} / 4\right)(R+r)(R-r)^{2}\), where r and R are the inner and outer radii and R >r (see figure). a. If R and r increase at the same rate, does the volume of the torus increase, decrease, or remain constant? b. If R and r decrease at the same rate, does the volume of the torus increase, decrease, or remain constant?

Chain Rule with one independent variable Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. dz/dt, where \(z=x^{2} y-x y^{3}, x=t^{2}, \text { and } y=t^{-2})

Spiral through a domain Suppose you follow the spiral path C: x = cos 1, y = sint, z = t, for t \(\geq\) 0, through the domain of the function w = f(x, y, z) = (xyz)/(\(z^{2}\) + 1). a. Find w' (t) along C. b. Estimate the point (x, y, z) on C at which w has its maximum value.

Derivative practice Find the indicated derivative for the following functions. dw/dt, where w =xyz, \(x=2 t^{4}, y=3 t^{-1}, \text { and } z=4 t^{-3}\)

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

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume all partial derivatives exist. a. If z = (x + y) sin xy, where x and y are functions of s, then \(\frac{\partial z}{\partial s}=\frac{d z}{d x} \frac{d x}{d s}\) b. Given that w = f(x(s, t), y(s, t), z(s, t)), the rate of change of w with respect to t is dw/dt.

Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the curve C in the xy-plane given parametrically in the form x = g(t), y = h(t). a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of increasing t. Find the values of 1 for which you are walking uphill (that is, z is increasing). \(z=2 x^{2}+y^{2}+1, C: x=1+\cos t, y=\sin t ; 0 \leq t \leq 2 \pi\)

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

Derivative practice Find the indicated derivative for the following functions. \(\partial z / \partial x\), where xy-z =1

Change on a line Suppose w = f(x, y, z) and l is the line r(t) = (at, bt, ct), for \((-\infty<t<\infty\). a. Find w'(t) on l (in terms of a, b, c, \(w_{x}, w_{y}, \text { and } w_{z}\)). b. Apply part (a) to find w't) when f(x, y, z) = xyz. c. Apply part (a) to find w' (t) when f(x, y, z) =\(\sqrt{x^{2}+y^{2}+z^{2}}\)? d. For a general function w = f(x, y, z), find w"(t).

Implicit differentiation rule with three variables Assume that F(x, y, z(x, y) = 0 implicitly defines z as a differentiable function of x and y. Extend Theorem 12.9 to show that \(\frac{\partial z}{\partial x}=-\frac{F_{x}}{F_{z}} \text { and } \frac{\partial z}{\partial y}=-\frac{F_{y}}{F_{z}})

Suppose z = f(x, y), where x and y are functions of t. How many dependent, intermediate, and independent variables are there?

If z is a function of x and y, while x and y are functions of t, explain how to find \(\frac{d z}{d t}\)

Chain Rule with several independent variables Find the following derivatives. \(z_{s} \text { and } z_{t}\), where z =sin x cos 2 y, x =s + t, and y =s -t

Chain Rule with several independent variables Find the following derivatives. \(z_{s} \text { and } z_{t}\) where \(z=e^{x+y}\), x =st, and y =s+t

Chain Rule with several independent variables Find the following derivatives. \(z_{s} \text { and } z_{t}\), where z =xy - 2x +3y, x =cos s, and y =sin t

Making trees Use a tree diagram to write the required Chain Rule formula. w = f(x,y,z) where x = g(t), y = h(s,t), z = p(r,s,t). Find \(\partial w / \partial t .\)

Making trees Use a tree diagram to write the required Chain Rule formula. u = f(v), where v = g(w,x,y), w = h(z 0, x = p(t,z), y = q(t,z). Find \(\partial u / \partial z).

Making trees Use a tree diagram to write the required Chain Rule formula. U =f(v,w,x), where v =g(r,s,t), w =h(r,s,t), x = p(r,s,t), r =F(z). Find \(\partia; y / \partial z\)

Derivative practice two ways Find the indicated derivative in two ways: a. Replace x and y to write z as a function of t and differentiate. b. Use the Chain Rule. \(z^{\prime}(t), \text { where } z=\ln (x+y), x=t e^{t} \text {, and } y=e^{t}\)

Derivative practice two ways Find the indicated derivative in two ways: a. Replace x and y to write z as a function of t and differentiate. b. Use the Chain Rule. z’(t), where \(z=\frac{1}{x}+\frac{1}{y}\), \(x=t^{2}+2 t, \text { and } y=t^{3}-2\)

Derivative practice Find the indicated derivative for the following functions. \(\partial z / \partial p\), where z =x/y, x = p+q, and y = p-q

Conservation of energy A projectile is launched into the air on a parabolic trajectory. For t \(\geq\) 0, its horizontal and vertical coordinates are \(x(t)=u_{0} t \text { and } y(t)=-(1 / 2) g t^{2}+v_{0} t\), respectively, where \(u_{0}\) is the initial horizontal velocity, \(v_{0}\) is the initial vertical velocity, and g is the acceleration due to gravity. Recalling that u(t) = x'(t) and v(t) = y’(t) are the components of the velocity, the energy of the projectile (kinetic plus potential) is \(E(t)=\frac{1}{2} m\left(u^{2}+v^{2}\right)+m g y\) Use the Chain Rule to compute E'(t) and show that E' (t) = 0 for all t \(\geq\) 20. Interpret the result.

Geometry of implicit differentiation 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 xy-plane (z = 0). a. Make a sketch of a surface and its intersection with the xy-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.

Subtleties of the Chain Rule Let w = f(x, y, z) = 2x + 3y + 4z, which is defined for all (x, y, z) in \(R^{3}\). Suppose that we are interested in the partial derivative w, on a subset of \(R^{3}\), such as the plane P given by z = 4x - 2y. 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 \(\left(\frac{\partial w}{\partial x}\right)_{y}=8\), where the subscript z indicates that z is held fixed. c. Make a sketch of the plane z = 4x - 2y and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find \(\left(\frac{\partial w}{\partial y}\right)_{x},\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},\left(\frac{\partial w}{\partial z}\right)_{y}\)