Gas law calculations Consider the Ideal Gas Law PV = kT,

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. \(G(r, s, t)=\sqrt{r s+r t+s t}\)

Second partial derivatives Find the four second partial derivatives of the following functions. \(f(x, y)=2 x^{5} y^{2}+x^{2} y\)

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. h(x, y, z)=cos (x+y+z)

Gas law calculations Consider the Ideal Gas Law PV=kT, where k>0 is a constant. Solve this equation for V in terms of P and T. (a) Determine the rate of change of the volume with respect to the pressure at constant temperature. Interpret the result. (b) Determine the rate of change of the volume with respect to the temperature at constant pressure. Interpret the result. (c) Assuming k=1, draw several level curves of the volume function and interpret the results as in Example 5.

Mixed partial derivatives a. Consider the function w = f(x, y, z). List all possible second partial derivatives that could be computed. b. Let \(f(x, y, z)=x^{2} y+2 x z^{2}-3 y^{2} z\) and determine which second partial derivatives are equal. c. How many second partial derivatives does p = g(w, x, y, z) have?

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. \(F(u, v, w)=\frac{u}{v+w}\)

Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=3 x^{2} y+x y^{3}\).

Miscellaneous partial derivatives Compute the first partial derivatives of the following functions. \(f(x, y)=\ln \left(1+e^{-x y}\right)\)

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. \(g(x, y, z)=2 x^{2} y-3 x z^{4}+10 y^{2} z^{2}\)

The volume of a right circular cylinder with radius r and height h is \(V=\pi r^{2} h\). Is the volume an increasing or decreasing function of the radius at a fixed height (assume r>0 and h>0)?

Partial derivatives Find the first partial derivatives of the following functions. \(f(x, y)=3 x^{2} y+2\)

Volume of a box A box with a square base of length x and height h has a volume \(V=x^{2} h\). (a) Compute the partial derivatives \(V_{x}\) and \(V_{h}\). (b) For a box with h=1.5 m, use linear approximation to estimate the change in volume if x increases from x=0.5 m to x=0.51 m. (c) For a box with x=0.5 m, use linear approximation to estimate the change in volume if h decreases from h=1.5 m to h=1.49 m. (d) For a fixed height, does a 10% change in x always produce (approximately) a 10% change in V? Explain. (e) For a fixed base length, does a 10% change in h always produce (approximately) a 10% change in V? Explain.

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. \(F(w, x, y, z)=w \sqrt{x+2 y+3 z}\)

Suppose you are standing on the surface z=f(x, y) at the point (a, b, f(a, b)). Interpret the meaning of \(f_{x}(a, b)\) and \(f_{y}(a, b)\) in terms of slopes or rates of change.

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

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. Q(x, y, z)=tan xyz

Second partial derivatives Find the four second partial derivatives of the following functions. \(h(x, y)=x^{3}+x y^{2}+1\)

Heat equation The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}\) where u is a measure of the temperature at a location x on the bar at time! and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heat equation with k = 1. \(u(x, t)=A e^{-a^{2} t} \cos a x\) for any real number a and A

Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension (for example, a wave on a string) wave motion is governed by the one-dimensional wave equation \(\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}\) where u(x, t) is the height or displacement of the wave surface at position x and time t, and c is the constant speed of the wave. Show that the following functions are solutions of the wave equation. u(x, t) = A f(x + ct) + B g(x - ct), where A and B are constants, and f and g are twice differentiable functions of one variable.

Equality of mixed partial derivatives Verify that \(f_{x y}=f_{y x}\) for the following functions. \(f(x, y)=x e^{y}\)

Laplace's equation A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions. Laplace's equation is \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\) Show that the following functions are harmonic; that is, they satisfy Laplace's equation. \(u(x, y)=e^{-x} \sin y\)

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. g(w, x, y, z)=cos (w+x) sin (y-z)

Nondifferentiability? Consider the following functions f. (a) Is f continuous at (0,0)? (b) Is f differentiable at (0,0)? (c) If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\). (d) Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0). (e) Explain why Theorems 12.5 and 12.6 are consistent with the results in part (a)-(d). \(f(x, y)= \begin{cases}\frac{2 x y^{2}}{x^{2}+y^{4}}\) & if \((x, y) \neq(0,0)\) \\ 0 & if \((x, y)=(0,0)\end{cases}\)

Second partial derivatives Find the four second partial derivatives of the following functions. f(x, y)=cos xy

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. \(h(w, x, y, z)=\frac{w z}{x y}\)

Partial derivatives Find the first partial derivatives of the following functions. \(h(u, v)=\sqrt{\frac{u v}{u-v}}\)

Equality of mixed partial derivatives Verify that \(f_{x y}=f_{y x}\) for the following functions. \(f(x, y)=2 x^{3}+3 y^{2}+1\)

Partial derivatives Find the first partial derivatives of the following functions. \(F(p, q)=\sqrt{p^{2}+p q+q^{2}}\)

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

An identity Show that if \(f(x, y)=\frac{a x+b y}{c x+d y}\), where a, b, c, and d are real numbers with ad - bc = 0, then \(f_{x}=f_{y}=0\), for all x and y in the domain of f. Give an explanation.

Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension (for example, a wave on a string) wave motion is governed by the one-dimensional wave equation \(\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}\) where u(x, t) is the height or displacement of the wave surface at position x and time t, and c is the constant speed of the wave. Show that the following functions are solutions of the wave equation. u(x, t)=5 cos (2(x+c t))+3 sin (x-ct)

Heat equation The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}\) where u is a measure of the temperature at a location x on the bar at time! and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heat equation with k = 1. \(u(x, t)=e^{-t}(2 \sin x+3 \cos x)\)

Equality of mixed partial derivatives Verify that \(f_{x y}=f_{y x}\) for the following functions. \(f(x, y)=\sqrt{x y}\)

Nondifferentiability? Consider the following functions f. (a) Is f continuous at (0,0)? (b) Is f differentiable at (0,0)? (c) If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\). (d) Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0). (e) Explain why Theorems 12.5 and 12.6 are consistent with the results in part (a)-(d). \(f(x, y)= \begin{cases}-\frac{x y}{x^{2}+y^{2}}\) if \((x, y) \neq(0,0)\) \\ 0 & if \((x, y)=(0,0)\end{cases}\)

Second partial derivatives Find the four second partial derivatives of the following functions. \(f(x, y)=y^{3} \sin 4 x\)

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

Heat equation The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}\) where u is a measure of the temperature at a location x on the bar at time! and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heat equation with k = 1. \(u(x, t)=10 e^{-t} \sin x\)

Nondifferentiability? Consider the following functions f. (a) Is f continuous at (0,0)? (b) Is f differentiable at (0,0)? (c) If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\). (d) Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0). (e) Explain why Theorems 12.5 and 12.6 are consistent with the results in part (a)-(d). f(x, y)=1-|x y|

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. \(f(w, x, y, z)=w^{2} x y^{2}+x y^{3} z^{2}\)

Heat equation The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}\) where u is a measure of the temperature at a location x on the bar at time! and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heat equation with k = 1. \(u(x, t)=4 e^{-4 t} \cos 2 x\)

Partial derivatives Find the first partial derivatives of the following functions. \(f(x, y)=y^{8}+2 x^{6}+2 x y\)

Partial derivatives Find the first partial derivatives of the following functions. \(G(s, t)=\frac{\sqrt{s t}}{s+t\)

Second partial derivatives Find the four second partial derivatives of the following functions. \(H(x, y)=\sqrt{4+x^{2}+y^{2}}\)

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

Partial derivatives with more than two variables Find the first partial derivatives of the following functions. f(x, y, z)=xy+xz+yz

Partial derivatives Find the first partial derivatives of the following functions. \(g(x, z)=x \ln \left(z^{2}+x^{2}\right)\)

Partial derivatives Find the first partial derivatives of the following functions. \(s(y, z)=z^{2} \tan y z\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. (a) \(\frac{\partial}{\partial x}\left(y^{10}\right)=10 y^{9}\) (b) \(\frac{\partial^{2}}{\partial x \partial y}(\sqrt{x y})=\frac{1}{\sqrt{x y}}\) (c) If f has continuous partial derivatives of all orders, then \(f_{x x y}=f_{y x x}\)

Find \(f_{x}\) and \(f_{y}\) when f(x, y)=x cos (xy).

Find the four second partial derivatives of \((x, y)=3 x^{2} y+x y^{3}\).

Explain how you would evaluate \(f_{z}\), for the differentiable function w=f(x, y, z).

Partial derivatives Find the first partial derivatives of the following functions. g(x, y)=cos 2xy

Partial derivatives Find the first partial derivatives of the following functions. \(h(x, y)=\left(y^{2}+1\right) e^{x}\)

Partial derivatives Find the first partial derivatives of the following functions. \(f(w, z)=\frac{w}{w^{2}+z^{2}}\)

Second partial derivatives Find the four second partial derivatives of the following functions. \(p(u, v)=\ln \left(u^{2}+v^{2}+4\right)\)

Second partial derivatives Find the four second partial derivatives of the following functions. Q(r, s)=r/s

Second partial derivatives Find the four second partial derivatives of the following functions. \(F(r, s)=r e^{s}\)

Equality of mixed partial derivatives Verify that \(f_{x y}=f_{y x}\) for the following functions. f(x, y)=cos xy

Equality of mixed partial derivatives Verify that \(f_{x y}=f_{y x}\) for the following functions. \(f(x, y)=3 x^{2} y^{-1}-2 x^{-1} y^{2}\)

Equality of mixed partial derivatives Verify that \(f_{x y}=f_{y x}\) for the following functions. \(f(x, y)=e^{x+y}\)

Nondifferentiability? Consider the following functions f. (a) Is f continuous at (0,0)? (b) Is f differentiable at (0,0)? (c) If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\). (d) Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0). (e) Explain why Theorems 12.5 and 12.6 are consistent with the results in part (a)-(d). \(f(x, y)=\sqrt{|x y|}\)

Partial derivatives and level curves 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 xy-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 xy-plane and describe how the corresponding z-values change. Explain how this observation is consistent with \(z_{y}\) as computed in part (a).

Spherical caps 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 \((0 \leq h \leq r)\) is the rate of change of volume with respect to h the greatest?

Law of Cosines 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)?

Body mass index The body mass index (BMI) for an adult human is given by the function \(B=w / h^{2}\), where w is the weight measured in kilograms and h is the height measured in meters. (The BMI for units of pounds and inches is \(B=703 w / h^{2}\).) (a) Find the rate of change of the BMI with respect to weight at a constant height. (b) For fixed h, is the BMI an increasing or decreasing function of w? Explain. (c) Find the rate of change of the BMI with respect to height at a constant weight. (d) For fixed w, is the BMI an increasing or decreasing function of h? Explain.

Electric potential function The electric potential in the xy-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.

Cobb-Douglas production function The output Q of an economic system subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b}\). Suppose \(a=\frac{1}{3}\), \(b=\frac{2}{3}\), and c=1. (a) Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\). (b) If L=10 is fixed and K increases from K=20 to K=20.5, use linear approximation to estimate the change in Q. (c) If K=20 is fixed and L decreases from L=10 to L=9.5, use linear approximation to estimate the change in Q. (d) Graph the level curves of the production function in the first quadrant of the LK-plane for Q=1, 2, 3. (e) If you move along the vertical line L=2 in the positive K-direction, how does Q change? Is this consistent with \(Q_{K}\) computed in part (a)? (f) If you move along the horizontal line K=2 in the positive L-direction, how does Q change? Is this consistent with \(Q_{L}\) computed in part (a)?

Resistors in parallel Two resistors in an electrical circuit with resistance \(R_{1}\) and \(R_{2}\) wired in parallel with a constant voltage give an effective resistance of R, where \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). (a) Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by solving for R and differentiating. (b) Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by differentiating implicitly. (c) Describe how an increase in \(R_{1}\) with \(R_{2}\) constant affects R. (d) Describe how a decrease in \(R_{2}\) with \(R_{1}\) constant affects R.

Wave on a string Imagine a string that is fixed at both ends (for example, a guitar string). When plucked, the string forms a standing wave. The displacement u of the string varies with position x and with time t. Suppose it is given by \(u=f(x, t)=2 \sin (\pi x) \sin (\pi t / 2)\) for \(0 \leq x \leq 1\) and \(t \geq 0\) (see figure). At a fixed point in time, the string forms a wave on [0,1]. Alternatively, if you focus on a point on the string (fix a value of x), that point oscillates up and down in time. (a) What is the period of the motion in time? (b) Find the rate of change of the displacement with respect to time at a constant position (which is the vertical velocity of a point on the string). (c) At a fixed time, what point on the string is moving fastest? (d) At a fixed position on the string, when is the string moving fastest? (e) Find the rate of change of the displacement with respect to position at a constant time (which is the slope of the string). (f) At a fixed time, where is the slope of the string greatest?

Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension (for example, a wave on a string) wave motion is governed by the one-dimensional wave equation \(\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}\) where u(x, t) is the height or displacement of the wave surface at position x and time t, and c is the constant speed of the wave. Show that the following functions are solutions of the wave equation. u(x, t)=cos (2(x+ct))

Laplace's equation A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace's equation is \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\) Show that the following functions are harmonic; that is, they satisfy Laplace’s equation \(u(x, y)=x\left(x^{2}-3 y^{2}\right)\)

Laplace's equation A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace's equation is \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\) Show that the following functions are harmonic; that is, they satisfy Laplace’s equation \(u(x, y)=e^{a x} \cos a y\) for any real number a

Laplace's equation A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace's equation is \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\) Show that the following functions are harmonic; that is, they satisfy Laplace’s equation \(u(x, y)=\tan ^{-1}\left(\frac{y}{x-1}\right)-\tan ^{-1}\left(\frac{y}{x+1}\right)\)

Differentiability Use the definition of differentiability to prove that the following functions are differentiable at (0,0). You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\), with the required properties. f(x,y) = x + y

Differentiability Use the definition of differentiability to prove that the following functions are differentiable at (0,0). You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\), with the required properties. f(x,y) = xy

Derivatives of an integral Let h be continuous for all real numbers. a. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{x}^{y} h(s) d s\) b. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{1}^{x y} h(s) d s\)

Cauchy-Riemann equations In the advanced subject of complex variables, a function typically has the form f(x, y) = u(x, y) + i v(x, y), where u and v are real-valued functions and \(i=\sqrt{-1}\) is the imaginary unit. A function f = u + iv is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: \(u_{x}=v_{y} \text { and } u_{y}=-v_{x}\) a. Show that \(f(x, y)=\left(x^{2}-y^{2}\right)+i(2 x y)\) is analytic. b. Show that \(f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)\) is analytic. c. Show that if f = u + iv is analytic, then \(u_{x x}+u_{y y}=0\) and \(v_{x x}+v_{y y}=0\)