?(a) In Example 3 we found that for the function . We interpreted this result

At the beginning of this section we discussed the function I = f(T, H), where I is the heat index, T is the actual temperature, and H is the relative humidity. Use Table 1 to estimate fT(92, 60) and fH(92, 60). What are the practical interpretations of these values?

The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h = f(v, t) are recorded in feet in the following table. (a) What are the meanings of the partial derivatives ?h/?v and ?h/?t? (b) Estimate the values of fv(40, 15) and ft(40, 15). What are the practical interpretations of these values? (c) What appears to be the value of the following limit?

The temperature T (in °C) at a location in the Northern Hemisphere depends on the longitude x, latitude y, and time t, so we can write T = f(x, y, t). Let’s measure time in hours from the beginning of January. (a) What are the meanings of the partial derivatives ?T/?x, ?T/?y, and ?T/?t? (b) Honolulu has longitude 158°W and latitude 21°N. Suppose that at 9:00 am on January 1 the wind is blowing hot air to the northeast, so the air to the west and south is warm and the air to the north and east is cooler. Would you expect fx(158, 21, 9), fy(158, 21, 9), and ft(158, 21, 9) to be positive or negative? Explain.

Determine the signs of the partial derivatives for the function f whose graph is shown. (a) fx(1, 2) (b) fy(1, 2)

Determine the signs of the partial derivatives for the function f whose graph is shown. (a) fx(?1, 2) (b) fy(?1, 2)

A contour map is given for a function f. Use it to estimate fx(2, 1) and fy(2, 1).

If f(x, y) = 16 ? 4x2 ? y2, find fx(1, 2) and fy(1, 2) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.

If \(f(x, y)=\sqrt{4-x^{2}-4 y^{2}}\) , find \(f_{x}(1,0)\) and \(f_{y}(1,0)\) and interpret these numbers as slopes. Illustrate with either hand drawn sketches or computer plots.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function. \(w=\frac{u}{v^{2}}\)

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function. \(f(x, y)=x^y\)

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

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

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the first partial derivatives of the function. \(\phi(x, y, z, t)=\frac{\alpha x+\beta y^{2}}{\gamma z+\delta t^{2}}\) Equation Transcription: Text Transcription: phi(x,y,z,t)=alpha x+ beta y^2/gamma z+ delta t^2

Find the first partial derivatives of the function.

Find the first partial derivatives of the function.

Find the indicated partial derivative.

Find the indicated partial derivative. \(f(x, \ y)=y \sin ^{-1}(x y)\); \(f_{y}\left(1, \ \frac{1}{2}\right)\)

Find the indicated partial derivative.

Find the indicated partial derivative. \(f(x, \ y, \ z)=x^{y z}; \quad f_{z}(e, \ 1, \ 0)\)

Use implicit differentiation to find and .

Use implicit differentiation to find and .

Use implicit differentiation to find and .

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\). \(y z+x \ln \ y=z^{2}\)

Find \(\partial z / \partial x\) and \(\partial z / \partial y\). (a) \(z=f(x)+g(y)\) (b) \(z=f(x+y)\)

Find and . (a) (b) (c)

Find all the second partial derivatives.

Find all the second partial derivatives.

Find all the second partial derivatives.

Find all the second partial derivatives.

Find all the second partial derivatives.

Find all the second partial derivatives.

Verify that the conclusion of Clairaut’s Theorem holds, that is, .

Verify that the conclusion of Clairaut’s Theorem holds, that is, .

Verify that the conclusion of Clairaut’s Theorem holds, that is, \(u_{x y}=u_{y x}\) \(u=\cos \left(x^{2} y\right)\)

Verify that the conclusion of Clairaut’s Theorem holds, that is, .

Find the indicated partial derivative(s).

Find the indicated partial derivative(s).

Find the indicated partial derivative(s).

Find the indicated partial derivative(s).

Find the indicated partial derivative(s). \(W=\sqrt{u=v^2} ; \frac{\partial^3 W}{\partial u^2 \partial v}\)

Find the indicated partial derivative(s). \(V=\ln \left(r+s^{2}+t^{3}\right) ; \frac{\partial^{3} V}{\partial r \partial s \partial t}\) Equation Transcription: Text Transcription: V=ln(r+s^2+t^3); partial^3V/ partial r partial s partial t

Find the indicated partial derivative(s). \(w=\frac{x}{y+2 z}, \frac{\partial^{3} w}{\partial z \partial y \partial x}, \frac{\partial^{3} w}{\partial x^{2} \partial y}\) Equation Transcription: Text Transcription: w=x/y + 2z; partial^3 w/ partial z partial y partial x, partial^3w/ partial x^2 partial y

Find the indicated partial derivative(s).

Use Definition 4 to find and .

Use Definition 4 to find and .

If , find . [Hint: Which order of differentiation is easiest?]

If , find . [Hint: Use a different order of differentiation for each term.]

The following surfaces, labeled a, b, and c, are graphs of a function f and its partial derivatives fx and fy. Identify each surface and give reasons for your choices.

Find \(f_{x}\) and \(f_{y}\) and graph \(f\), \(f_{x}\), and \(f_{y}\) with domains and viewpoints that enable you to see the relationships between them. \(f(x, y)=\frac{y}{1+x^{2} y^{2}}\)

Find \(f_{x}\) and \(f_{y}\) and graph \(f, \ f_{x}\), and \(f_{y}\) with domains and viewpoints that enable you to see the relationships between them. \(f(x, \ y)=x^{2} y^{3}\)

Determine the signs of the partial derivatives for the function f whose graph is shown in Exercises 4–5. (a) (b) (c) (d)

Use the table of values of \(f(x, \ y)\) to estimate the values of \(f_{x}(3, \ 2), \ f_{x}(3, \ 2.2)\), and \(f_{x y}(3, \ 2)\).

Level curves are shown for a function f . Determine whether the following partial derivatives are positive or negative at the point P. (a) (b) (c) (d) (e)

(a) In Example 3 we found that for the function . We interpreted this result geometrically as the slope of the tangent line to the curve at the point P(1, 1, 1), where is the trace of the graph of in the plane y = 1 . (See the figure.) Verify this interpretation by finding a vector equation for , computing the tangent vector to at P, and then finding the slope of the tangent line to at P in the plane y = 1 . (b) Use a similar method to verify that .

If , where , show that

Show that the function u = u(x, t) is a solution of the wave equation . (a) (b) (c) (d)

Determine whether each of the following functions is a solution of Laplace’s equation . (a) (b) (c) (d) (e) (f)

Verify that the function is a solution of the three-dimensional Laplace equation .

Verify that the function is a solution of the heat conduction equation .

The diffusion equation where D is a positive constant, describes the diffusion of heat through a solid, or the concentration of a pollutant at time t at a distance x from the source of the pollution, or the invasion of alien species into a new habitat. Verify that the function is a solution of the diffusion equation.

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 R1, R2, R3 connected in a parallel electrical circuit is given by the formula Find .

The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. (a) Show that . (b) Show that .

Van der Waals Equation The Van der Waals equation for n moles of a gas is \(\left(P+\frac{n^{2} a}{V^{2}}\right)(V-n b)=n R T\) where \(P\) is the pressure, \(V\) is the volume, and \(T\) is the temperature of the gas. The constant \(R\) is the universal gas constant and \(a\) and \(b\) are positive constants that are characteristic of a particular gas. Calculate \(\partial T / \partial P\) and \(\partial P / \partial V\).

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 \(\mathrm{T}\) is the temperature \(\left({ }^{\circ} \mathrm{C}\right )\). and \(\mathrm{v}\) is the wind speed (in \(\mathrm{km} / \mathrm{h}\) ). When \(\mathrm{T}=-15^{\circ} \mathrm{C}\) and \(\mathrm{v}=30 \mathrm{~km} / \mathrm{h}\), by how much would you expect the apparent temperature \(\mathrm{W}\) to drop if the actual temperature decreases by \(1^{\circ} \mathrm{C}\) ? What if the wind speed increases by \(1 \mathrm{~km} / \mathrm{h}\) ?

A model for the surface area of a human body is given by the function S = f(w, h) = 0.1091w0.425h0.725 where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. Calculate and interpret the partial derivatives.

One of Poiseuille’s laws states that the resistance of blood flowing through an artery is where L and r are the length and radius of the artery and C is a positive constant determined by the viscosity of the blood. Calculate ?R/?L and ?R/?r and interpret them.

In the project following Section 4.7 we expressed the power needed by a bird during its flapping mode as where A and B are constants specific to a species of bird, v is the velocity of the bird, m is the mass of the bird, and x is the fraction of the flying time spent in flapping mode. Calculate ?P/?v, ?P/?x, and ?P/?m and interpret them.

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 where ???? = 2????/365 and ???? is a positive constant. (a) Find ?T/?x. What is its physical significance? (b) Find ?T/?t. What is its physical significance? (c) Show that T satisfies the heat equation Tt ? kTxx for a certain constant k. (d) Graph T(x, t) for ???? = 0.2, T0 = 0, and T1 = 10. (e) What is the physical significance of the term 2x in the expression sin(????t ? ????x)?

The kinetic energy of a body with mass m and velocity v is mv2. Show that

The average energy E (in kcal) needed for a lizard to walk or run a distance of 1 km has been modeled by the equation \(E(m, v)=2.65 m^{0.66}+\frac{3.5 m^{0.75}}{v}\) where m is the body mass of the lizard (in grams) and v is its speed (in km/h). Calculate \(\mathrm{E}_{\mathrm{m}}(400,8)\) and \(\mathrm{E}_{\mathrm{v}}(400,8)\) and interpret your answers.

The ellipsoid 4x2 + 2y2 + z2 = 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).

The paraboloid z = 6 ? x ? x2 ? 2y2 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.

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

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

Use Clairaut’s Theorem to show that if the third-order partial derivatives of f are continuous, then fxyy = fyxy = fyyx

(a) How many nth-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.

If find fx(1, 0). [Hint: Instead of finding fx(x, y) first, note that it’s easier to use Equation 1 or Equation 2.]

If f(x, y) = , find fx(0, 0).

Let (a) Graph f. (b) Find fx(x, y) and fy(x, y) when (x, y) (0, 0). (c) Find fx(0, 0) and fy(0, 0) using Equations 2 and 3. (d) Show that fxy(0, 0) = ?1 and fyx(0, 0) = 1. (e) Does the result of part (d) contradict Clairaut’s Theorem? Use graphs of fxy and fyx to illustrate your answer.