BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621
Chapter 14.3, Problem 76E
Textbook Problem
1 views

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

(b) u = x 2 y 2

(c) u = x 3 + 3 x y 2

(d) u = ln x 2 + y 2

(e) u = sin x cosh y + cos x sinh y

(f) u = e x cos y e y cos x

To determine

Whether the given function is the solution of Laplace’s equation  uxx+uyy=0

Explanation of Solution

Concept:

To find the partial derivatives uxx and uyy and substitute uxx and uyy in Laplace’s equation and check if it is satisfied or not.

Given:

u=x2+y2

Calculation:

Given function is,

u=x2+y2

Differentiate with respect to x

ux=2x

Again differentiate with respect to x

uxx=2

Now differentiate given function with respect to y

uy=2y

Again differentiate with respect to y

uyy=2

So, substitute the values of  uxx and uyy in left side of Laplace’s equation and simplify

 uxx+uyy=2+2=40

That is, the function u=x2+y2 is not the solution of Laplace’s equation uxx+uyy=0

Conclusion:

The function u=x2+y2 is not the solution of Laplace’s equation uxx+uyy=0

To determine:

Whether the given function is the solution of Laplace’s equation uxx+uyy=0

Solution:

The function u=x2-y2 is the solution of Laplace’s equation uxx+uyy=0

Concept:

To find the partial derivatives uxx and uyy and substitute uxx and uyy in Laplace’s equation and check satisfied or not

Given:

u=x2-y2

Calculation:

 Given function is,

u=x2-y2

Differentiate with respect to x

ux=2x

Again differentiate with respect to x

uxx=2

Now differentiate given function with respect to y

uy=-2y

Again differentiate with respect to y

uyy=-2

So, substitute the values of uxx and uyy in left side of Laplace’s equation and simplify

uxx+uyy=2+-2=0

That is, the function u=x2+y2 is the solution of Laplace’s equation uxx+uyy=0

Conclusion:

The function u=x2+y2 is the solution of Laplace’s equation uxx+uyy=0

To determine:

Whether the given function is the solution of Laplace’s equation uxx+uyy=0 or not?

Solution:

The function u=x3+3xy2 is not the solution of Laplace’s equation uxx+uyy=0

Concept:

To find the partial derivatives uxx and uyy and substitute uxx and uyy in Laplace’s equation and check satisfied or not

Given:

u=x3+3xy2

Calculation:

Given function is u=x3+3xy2

Differentiate with respect to x

ux=3x2+3y2

Again differentiate with respect to x

uxx=6x

Now differentiate given function with respect to y

uy=0+6xy

Again differentiate with respect to y

uyy=6x

So, substitute the values of uxx and uyy in left side of Laplace’s equation and simplify

 uxx+uyy=6x+6x=12x0

Thus, the function u=x3+3xy2 is not the solution of Laplace’s equation uxx+uyy=0.

Conclusion:

The function u=x3+3xy2 is not the solution of Laplace’s equation

uxx+uyy=0.

To determine:

Whether the given function is the solution of Laplace’s equation uxx+uyy=0

Solution:

The function u=ln x2+y2 is the solution of Laplace’s equation uxx+uyy=0

Concept:

To find the partial derivatives uxx and uyy and substitute uxx and uyy in Laplace’s equation and check if it is satisfied

Given:

u=ln x2+y2

Calculation:

 Given function is u=ln x2+y2

Differentiate with respect to x

ux=1x2+y212x2+y212-12x

ux=xx2+y2

Again differentiate with respect to x

uxx=x2+y2-x2xx2+y22

uxx=y2-x2x2+y22

Now differentiate given function with respect to y

uy=1x2+y212x2+y212-12y

uy=yx2+y2

Again differentiate with respect to y

uyy=x2+y2-y2yx2+y22

uyy=x2-y2x2+y22

So, substitute the values of uxx and uyy in left side of Laplace’s equation and simplify

 uxx+uyy=y2-x2x2+y22+x2-y2x2+y22=0

That is, the function u=ln x2+y2 is the solution of Laplace’s equation uxx+uyy=0

Conclusion:

The function u=ln x2+y2 is the solution of Laplace’s equation

uxx+uyy=0

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 14 Solutions

Calculus (MindTap Course List)
Show all chapter solutions
add
Ch. 14.1 - Let f(x,y,z)=x+y+z+ln(4x2y2z2). aEvaluate...Ch. 14.1 - Let g(x,y,z)=x3y2z10xyz. aEvaluate g(1,2,3). bFind...Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Sketch the graph of the function. f(x,y)=yCh. 14.1 - Sketch the graph of the function. f(x,y)=x2Ch. 14.1 - Sketch the graph of the function. f(x,y)=104x5yCh. 14.1 - Sketch the graph of the function. f(x,y)=cosyCh. 14.1 - Sketch the graph of the function. f(x,y)=sinxCh. 14.1 - Sketch the graph of the function. f(x,y)=2x2y2Ch. 14.1 - Sketch the graph of the function. f(x,y)=x2+4y2+1Ch. 14.1 - Sketch the graph of the function. f(x,y)=4x2+y2Ch. 14.1 - Sketch the graph of the function. f(x,y)=44x2y2Ch. 14.1 - Match the function with its graph labeled IVI....Ch. 14.1 - A contour map for a function f is shown. Use it to...Ch. 14.1 - Shown is a contour map of atmospheric pressure in...Ch. 14.1 - Level curves isothermals are shown for the typical...Ch. 14.1 - Two contour maps are shown. One is for a function...Ch. 14.1 - Locate the points A and B on the map of Lonesome...Ch. 14.1 - Make a rough sketch of a contour map for the...Ch. 14.1 - The body mass index BMI of a person is defined by...Ch. 14.1 - The body mass index is defined in Exercise 39....Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Sketch both a contour map and a graph of the...Ch. 14.1 - Sketch both a contour map and a graph of the...Ch. 14.1 - A thin metal plate, located in the xy-plane, has...Ch. 14.1 - If V(x,y) is the electric potential at a point...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Graph the function using various domains and...Ch. 14.1 - Graph the function using various domains and...Ch. 14.1 - Investigate the family of functions...Ch. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Graph the functions f(x,y)=x2+y2 f(x,y)=ex2+y2...Ch. 14.1 - aShow that, by taking logarithms, the general...Ch. 14.2 - Suppose that lim(x,y)(3,1)f(x,y)=6. What can you...Ch. 14.2 - Explain why each function is continuous or...Ch. 14.2 - Use a table of numerical values of f(x,y) for...Ch. 14.2 - Use a table of numerical values of f(x,y) for...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Use a computer graph of the function to explain...Ch. 14.2 - Use a computer graph of the function to explain...Ch. 14.2 - Find h(x,y)=g(f(x,y)) and the set of points at...Ch. 14.2 - Find h(x,y)=g(f(x,y)) and the set of points at...Ch. 14.2 - Graph the function and observe where it is...Ch. 14.2 - Graph the function and observe where it is...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Use polar coordinates to find the limit. If (r,)...Ch. 14.2 - Use polar coordinates to find the limit. If (r,)...Ch. 14.2 - Use polar coordinates to find the limit. If (r,)...Ch. 14.2 - At the beginning of this section we considered the...Ch. 14.2 - Graph and discuss the continuity of the function...Ch. 14.2 - Let f(x,y)={0ify0oryx41if0yx4 a Show that f(x,y)0...Ch. 14.2 - Show that the function f given by f(x)=|x| is...Ch. 14.2 - If cVn, show that the function f given by f(x)=cx...Ch. 14.3 - The temperature T in C at a location in the...Ch. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - The wind-chill index W is the perceived...Ch. 14.3 - The wave heights h in the open sea depend on the...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - The following surfaces, labeled a, b, and c, are...Ch. 14.3 - A contour map is given for a function f. Use it to...Ch. 14.3 - If f(x,y)=164x2y2, find fx(1,2) and fy(1,2) and...Ch. 14.3 - If f(x,y)=4x24y2, find fx(1,0) and fy(1,0) and...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Use the definition of partial derivatives as...Ch. 14.3 - Use the definition of partial derivatives as...Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Find z/y and z/y. a z=f(x)+g(y) b z=f(x+y)Ch. 14.3 - Find z/y and z/y. a z=f(x)g(y) b z=f(xy) c...Ch. 14.3 - Find all the second partial derivatives....Ch. 14.3 - Find all the second partial derivatives....Ch. 14.3 - Find all the second partial derivatives. z=y2x+3yCh. 14.3 - Find all the second partial derivatives. T=e2rcosCh. 14.3 - Find all the second partial derivatives....Ch. 14.3 - Find all the second partial derivatives. w=1+uv2Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives. W=u+v2;...Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives. w=xy+2z;...Ch. 14.3 - Find the indicated partial derivatives. u=xaybzc;...Ch. 14.3 - If f(x,y,z)=xy2z3+arcsin(xz), find fxzy. Hint:...Ch. 14.3 - If g(x,y,z)=1+xz+1xy, find gxzy. Hint: Use a...Ch. 14.3 - Use the table of values of f(x,y) to estimate the...Ch. 14.3 - Level curves are shown for a function f. Determine...Ch. 14.3 - Verify that the function u=e2k2tsinkx is a...Ch. 14.3 - Determine whether each of the following functions...Ch. 14.3 - Verify that the function u=1/x2+y2+z2 is a...Ch. 14.3 - Show that each of the following functions is a...Ch. 14.3 - If f and g are twice differentiable functions of a...Ch. 14.3 - If u=ea1x1+a2x2+...+anxn, where a12+a22+...+an2=1,...Ch. 14.3 - The diffusion equation ct=D2cx2 where D is a...Ch. 14.3 - The temperature at a point x, y on a flat metal...Ch. 14.3 - The total resistance R produced by three...Ch. 14.3 - Show that the Cobb-Douglas production function...Ch. 14.3 - Show that the Cobb-Douglas production function...Ch. 14.3 - Cobb and Douglas used the equation...Ch. 14.3 - The van der Waals equation for n moles of a gas is...Ch. 14.3 - The gas law for a fixed mass m of an ideal gas at...Ch. 14.3 - For the ideal gas of Exercise 88, show that...Ch. 14.3 - The wind-chill index is modeled by the function...Ch. 14.3 - A model for the surface area of a human body is...Ch. 14.3 - One of Poiseuilles laws states that the resistance...Ch. 14.3 - In the project on page 271 we expressed the power...Ch. 14.3 - The average energy E in kcal needed for a lizard...Ch. 14.3 - The kinetic energy of a body with mass m and...Ch. 14.3 - If a, b, c are the sides of a triangle and A, B, C...Ch. 14.3 - You are told that there is a function f whose...Ch. 14.3 - The paraboloid z=6xx22y2 intersects the plane x=1...Ch. 14.3 - The ellipsoid 4x2+2y2+z2=16 intersects the plane...Ch. 14.3 - In a study of frost penetration it was found that...Ch. 14.3 - Use Clairauts Theorem to show that if the...Ch. 14.3 - a How many nth-order partial derivatives does a...Ch. 14.3 - If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0). Hint:...Ch. 14.3 - If f(x,y)=x3+y33, find fx(0,0).Ch. 14.3 - Let f(x,y)={x3yxy3x2+y2if(x,y)(0,0)0if(x,y)=(0,0)...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Graph the surface and the tangent plane at the...Ch. 14.4 - Graph the surface and the tangent plane at the...Ch. 14.4 - Draw the graph of f and its tangent plane at the...Ch. 14.4 - Draw the graph of f and its tangent plane at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Verify the linear approximation at (0,0)....Ch. 14.4 - Verify the linear approximation at (0,0)....Ch. 14.4 - Given that f is a differentiable function with...Ch. 14.4 - Find the linear approximation of the function...Ch. 14.4 - Find the linear approximation of the function...Ch. 14.4 - The wave heights h in the open sea depend on the...Ch. 14.4 - Use the table in Example 3 to find a linear...Ch. 14.4 - The wind-chill index W is the perceived...Ch. 14.4 - Find the differential of the function. z=e2xcos2tCh. 14.4 - Find the differential of the function. u=x2+3y2Ch. 14.4 - Find the differential of the function. m=p5q3Ch. 14.4 - Find the differential of the function. T=v1+uvwCh. 14.4 - Find the differential of the function. R=2cosCh. 14.4 - Find the differential of the function. L=xzey2z2Ch. 14.4 - If z=5x2+y2 and (x,y) changes from (1,2) to...Ch. 14.4 - If z=x2xy+3y2 and (x,y) changes from (3,1) to...Ch. 14.4 - The length and width of a rectangle are measured...Ch. 14.4 - Use differentials to estimate the amount of metal...Ch. 14.4 - Use differentials to estimate the amount of tin in...Ch. 14.4 - The wind-chill index is modeled by the function...Ch. 14.4 - The tension T in the string of the yo-yo in the...Ch. 14.4 - The pressure, volume, and temperature of a mole of...Ch. 14.4 - If R is the total resistance of three resistors,...Ch. 14.4 - A model for the surface area of a human body is...Ch. 14.4 - In Exercise 14.1.39 and Example 14.3.3, the body...Ch. 14.4 - Suppose you need to know an equation of the...Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Prove that if f is a function of two variables...Ch. 14.4 - a The function...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Let p(t)=f(x,y), where f is differentiable,...Ch. 14.5 - Let R(s,t)=G(u(s,t),v(s,t)), where G, u and v are...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use Equation 6 to find dy/dx. ycosx=x2+y2Ch. 14.5 - Use Equation 6 to find dy/dx. cos(xy)=1+sinyCh. 14.5 - Use Equation 6 to find dy/dx. tan1(x2y)=x+xy2Ch. 14.5 - Use Equation 6 to find dy/dx. eysinx=x+xyCh. 14.5 - Use Equations 7 to find z/x and z/y. x2+2y2+3z2=1Ch. 14.5 - Use Equations 7 to find z/x and z/y. x2y2+z22z=4Ch. 14.5 - Use Equations 7 to find z/x and z/y. ez=xyzCh. 14.5 - Use Equations 7 to find z/x and z/y. yz+xlny=z2Ch. 14.5 - The temperature at a point x, y is Tx, y, measured...Ch. 14.5 - Wheat production W in a given year depends on the...Ch. 14.5 - The speed of sound traveling through ocean water...Ch. 14.5 - The radius of a right circular cone is increasing...Ch. 14.5 - The length l, width w, and height h of a box...Ch. 14.5 - The voltage V in a simple electrical circuit is...Ch. 14.5 - The pressure of 1 mole of an ideal gas is...Ch. 14.5 - A manufacturer has modeled its yearly production...Ch. 14.5 - One side of a triangle is increasing at a rate of...Ch. 14.5 - A sound with frequency fs is produced by a source...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - A function f is called homogeneous of degree n if...Ch. 14.5 - If f is homogeneous of degree n, show that...Ch. 14.5 - If f is homogeneous of degree n, show that...Ch. 14.5 - Suppose that the equation F(x,y,z)=0 implicitly...Ch. 14.5 - Equation 6 is a formula for the derivative dy/dx...Ch. 14.6 - Level curves for barometric pressure in millibars...Ch. 14.6 - The contour map shows the average maximum...Ch. 14.6 - A table of values for the wind-chill index...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Use the figure to estimate Duf(2,2).Ch. 14.6 - Find the directional derivative of f(x,y)=xy at...Ch. 14.6 - Find the directional derivative of f(x,y,z)=xy2z3...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - a Show that a differentiable function f decreases...Ch. 14.6 - Find the directions in which the directional...Ch. 14.6 - Find all points at which the direction of fastest...Ch. 14.6 - Near a buoy, the depth of a lake at the point with...Ch. 14.6 - The temperature T in a metal ball is inversely...Ch. 14.6 - The temperature at a point (x,y,z) is given by...Ch. 14.6 - Suppose that over a certain region of space the...Ch. 14.6 - Suppose you are climbing a hill whose shape is...Ch. 14.6 - Let f be a function of two variables that has...Ch. 14.6 - Shown is a topographic map of Blue River Pine...Ch. 14.6 - Show that the operation of taking the gradient of...Ch. 14.6 - Sketch the gradient vector f(4,6) for the function...Ch. 14.6 - The second directional derivative of f(x,y) is...Ch. 14.6 - a If u=a,b is a unit vector and f has continuous...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Use a computer to graph the surface, the tangent...Ch. 14.6 - Use a computer to graph the surface, the tangent...Ch. 14.6 - If f(x,y)=xy, find the gradient vector f(3,2) and...Ch. 14.6 - If g(x,y)=x2+y24x, find the gradient vector g(1,2)...Ch. 14.6 - Show that the equation of the tangent plane to the...Ch. 14.6 - Find the equation of the tangent plane to the...Ch. 14.6 - Show that the equation of the tangent plane to the...Ch. 14.6 - At what point on the ellipsoid x2+y2+2z2=1 is the...Ch. 14.6 - Are there any points on the hyperboloid x2y2z2=1...Ch. 14.6 - Show that the ellipsoid 3x2+2y2+z2=9 and the...Ch. 14.6 - Show that every plane that is tangent to the cone...Ch. 14.6 - Show that every normal line to the sphere...Ch. 14.6 - Where does the normal line to the paraboloid...Ch. 14.6 - At what points does the normal line through the...Ch. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Show that the pyramids cut off from the first...Ch. 14.6 - Find parametric equations for the tangent line to...Ch. 14.6 - a The plane y+z=3 intersects the cylinder x2+y2=5...Ch. 14.6 - Where does the helix r(t)=cost,sint,t intersect...Ch. 14.6 - The helix r(t)=cos(t/2),sin(t/2),t intersects the...Ch. 14.6 - a Two surfaces are called orthogonal at a point of...Ch. 14.6 - a Show that the function f(x,y)=xy3 is continuous...Ch. 14.6 - Suppose that the directional derivatives of f(x,y)...Ch. 14.6 - Show that if z=f(x,y) is differentiable at...Ch. 14.7 - Suppose 1, 1 is a critical point of a function f...Ch. 14.7 - Suppose 0, 2 is a critical point of a function g...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Show that f(x,y)=x2+4y24xy+2 has an infinite...Ch. 14.7 - Show that f(x,y)=x2yex2y2 has minimum values at...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - For functions of one variable it is impossible for...Ch. 14.7 - If a function of one variable is continuous on an...Ch. 14.7 - Find the shortest distance from the point (2,0,3)...Ch. 14.7 - Find the point on the plane x2y+3z=6 that is...Ch. 14.7 - Find the points on the cone z2=x2+y2 that are...Ch. 14.7 - Find the points on the surface y2=9+xz that are...Ch. 14.7 - Find three positive numbers whose sum is 100 and...Ch. 14.7 - Find three positive numbers whose sum is 12 and...Ch. 14.7 - Find the maximum volume of a rectangular box that...Ch. 14.7 - Find the dimensions of the box with volume 1000...Ch. 14.7 - Find the volume of the largest rectangular box in...Ch. 14.7 - Find the dimensions of the rectangular box with...Ch. 14.7 - Find the dimensions of a rectangular box of...Ch. 14.7 - The base of an aquarium with given volume V is...Ch. 14.7 - A cardboard box without a lid is to have a volume...Ch. 14.7 - A rectangular building is being designed to...Ch. 14.7 - If the length of the diagonal of a rectangular box...Ch. 14.7 - A model for the yield Y of an agricultural crop as...Ch. 14.7 - The Shannon index sometimes called the...Ch. 14.7 - Three alleles alternative versions of a gene A, B,...Ch. 14.7 - Suppose that a scientist has reason to believe...Ch. 14.7 - Find an equation of the plane that passes through...Ch. 14.8 - Pictured are a contour map of f and a curve with...Ch. 14.8 - a Use a graphing calculator or computer to graph...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - The method of Lagrange multipliers assumes that...Ch. 14.8 - Find the minimum value of f(x,y,z)=x2+2y2+3z2...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Consider the problem of maximizing the function...Ch. 14.8 - Consider the problem of minimizing the function...Ch. 14.8 - a If your computer algebra system plots implicitly...Ch. 14.8 - The total production P of a certain product...Ch. 14.8 - Referring to Exercise 27, we now suppose that the...Ch. 14.8 - Use Lagrange multipliers to prove that the...Ch. 14.8 - Use Lagrange multipliers to prove that the...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Find the maximum and minimum volumes of a...Ch. 14.8 - The plane x+y+2z=2 intersects the paraboloid...Ch. 14.8 - The plane 4x3y+8z=5 intersects the cone z2=x2+y2...Ch. 14.8 - Find the maximum and minimum values of f subject...Ch. 14.8 - Find the maximum and minimum values of f subject...Ch. 14.8 - a Find the maximum value of...Ch. 14.8 - a Maximize i=1nxiyi subject to the constraints...Ch. 14.R - aWhat is a function of two variables? bDescribe...Ch. 14.R - What is a function of three variables? How can you...Ch. 14.R - What does lim(x,y)(a,b)f(x,y)=L mean? How can you...Ch. 14.R - a What does it mean to say that f is continuous at...Ch. 14.R - a Write expressions for the partial derivatives...Ch. 14.R - What does Clairauts Theorem say?Ch. 14.R - How do you find a tangent plane to each of the...Ch. 14.R - Define the linearization of f at (a,b). What is...Ch. 14.R - aWhat does it mean to say that f is differentiable...Ch. 14.R - If z=f(x,y), what are the differentials dx, dy,...Ch. 14.R - State the Chain Rule for the case where z=f(x,y)...Ch. 14.R - If z is defined implicitly as a function of x and...Ch. 14.R - aWrite an expression as a limit for the...Ch. 14.R - aDefine the gradient vector f for a function f of...Ch. 14.R - What do the following statements mean? af has a...Ch. 14.R - aIf f has a local maximum at a, b, what can you...Ch. 14.R - State the Second Derivatives Test.Ch. 14.R - aWhat is a closed set in 2? What is a bounded set?...Ch. 14.R - Explain how the method of Lagrange multipliers...Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Find and sketch the domain of the function....Ch. 14.R - Find and sketch the domain of the function....Ch. 14.R - Sketch the graph of the function. f(x,y)=1y2Ch. 14.R - Sketch the graph of the function. f(x,y)=x2+(y2)2Ch. 14.R - Sketch several level curves of the function....Ch. 14.R - Sketch several level curves of the function....Ch. 14.R - Make a rough sketch of a contour map for the...Ch. 14.R - The contour map of a function f is shown. a...Ch. 14.R - Evaluate the limit or show that it does not exist....Ch. 14.R - Evaluate the limit or show that it does not exist....Ch. 14.R - A metal plate is situated in the xy-plane and...Ch. 14.R - Find a linear approximation to the temperature...Ch. 14.R - Find the first partial derivatives....Ch. 14.R - Find the first partial derivatives....Ch. 14.R - Find the first partial derivatives. F(,)=2ln(2+2)Ch. 14.R - Find the first partial derivatives....Ch. 14.R - Find the first partial derivatives....Ch. 14.R - The speed of sound traveling through ocean water...Ch. 14.R - Find all second partial derivatives of f....Ch. 14.R - Find all second partial derivatives of f. z=xe2yCh. 14.R - Find all second partial derivatives of f....Ch. 14.R - Find all second partial derivatives of f....Ch. 14.R - If z=xy+xey/x, show that xzx+yzy=xy+z.Ch. 14.R - If z=sin(x+sint), show that zx2zxt=zt2zx2Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Use a computer to graph the surface z=x2+y4 and...Ch. 14.R - Find the points on the hyperboloid x2+4y2z2=4...Ch. 14.R - Find du if u=ln(1+se2t).Ch. 14.R - Find the linear approximation of the function...Ch. 14.R - The two legs of a right triangle are measured as 5...Ch. 14.R - If u=x2y3+z4, where x=p+3p2, y=pep, and z=psinp,...Ch. 14.R - If v=x2siny+yexy, where x=s+2t and y=st, use the...Ch. 14.R - Suppose z=f(x,y), where x=g(s,t), y=h(s,t),...Ch. 14.R - Use a tree diagram to write out the Chain Rule for...Ch. 14.R - If z=y+f(x2y2), where f is differentiable, show...Ch. 14.R - The length x of a side of a triangle is increasing...Ch. 14.R - If z=f(u,v), where u=xy,v=y/x, and f has...Ch. 14.R - If cos(xyz)=1+x2y2+z2, find zx and zy.Ch. 14.R - Find the gradient of the function f(x,y,z)=x2eyz2.Ch. 14.R - a When is the directional derivative of f a...Ch. 14.R - 4546 Find the directional derivative of f at the...Ch. 14.R - 4546 Find the directional derivative of f at the...Ch. 14.R - Find the maximum rate of change of f(x,y)=x2y+y at...Ch. 14.R - Find the direction in which f(x,y,z)=zexy...Ch. 14.R - The contour map shows wind speed in knots during...Ch. 14.R - Find parametric equations of the tangent line at...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5556 Find the absolute maximum and minimum values...Ch. 14.R - 5556 Find the absolute maximum and minimum values...Ch. 14.R - Use a graph or level curves or both to estimate...Ch. 14.R - Use a graphing calculator or computer or Newtons...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - Find the points on the surface xy2z3=2 that are...Ch. 14.R - A package in the shape of a rectangular box can be...Ch. 14.R - A pentagon is formed by placing an isosceles...Ch. 14.P - A rectangle with length L and width W is cut into...Ch. 14.P - Marine biologists have determined that when a...Ch. 14.P - A long piece of galvanized sheet metal with width...Ch. 14.P - For what values of the number r is the function...Ch. 14.P - Suppose f is a differentiable function of one...Ch. 14.P - a Newtons method for approximating a root of an...Ch. 14.P - If the ellipse x2/a2+y2/b2=1 is to enclose the...Ch. 14.P - Show that the maximum value of the function...

Additional Math Textbook Solutions

Find more solutions based on key concepts
Show solutions add
Explain in your own words the meaning of limxf(x)=L and limxf(x)=M.

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

37. Are and inverse matrices if and

Mathematical Applications for the Management, Life, and Social Sciences

True or False: dmdv stands for the derivative of the function m with respect to the variable v.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: i + j is a unit vector in the direction of 5i + 5j.

Study Guide for Stewart's Multivariable Calculus, 8th

Perform the operations and simplify. (z+3)(3z2+z1)

College Algebra (MindTap Course List)

Identify the threats to internal validity for pre-post designs.

Research Methods for the Behavioral Sciences (MindTap Course List)