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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 14.3, Problem 76E

Textbook Problem

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Determine whether each of the following functions is a solution of Laplace’s equation

(a)

(b)

(c)

(d)

(e)

(f)

To determine

Whether the given function is the solution of Laplace’s equation

**Concept:**

To find the partial derivatives and

**Given:**

**Calculation:**

Given function is,

Differentiate with respect to

Again differentiate with respect to

Now differentiate given function with respect to

Again differentiate with respect to

So, substitute the values of and

That is, the function is not the solution of Laplace’s equation

**Conclusion:**

The function is not the solution of Laplace’s equation

**To determine:**

Whether the given function is the solution of Laplace’s equation

**Solution:**

The function is the solution of Laplace’s equation

**Concept:**

To find the partial derivatives and

**Given:**

**Calculation:**

Given function is,

Differentiate with respect to

Again differentiate with respect to

Now differentiate given function with respect to

Again differentiate with respect to

So, substitute the values of and

That is, the function is the solution of Laplace’s equation

**Conclusion:**

The function is the solution of Laplace’s equation

**To determine:**

Whether the given function is the solution of Laplace’s equation or not?

**Solution:**

The function is not the solution of Laplace’s equation

**Concept:**

To find the partial derivatives and

**Given:**

**Calculation:**

Given function is

Differentiate with respect to

Again differentiate with respect to

Now differentiate given function with respect to

Again differentiate with respect to

So, substitute the values of and

Thus, the function is not the solution of Laplace’s equation

**Conclusion:**

The function is not the solution of Laplace’s equation

.

**To determine:**

Whether the given function is the solution of Laplace’s equation

**Solution:**

The function is the solution of Laplace’s equation

**Concept:**

To find the partial derivatives and

**Given:**

**Calculation:**

Given function is

Differentiate with respect to

Again differentiate with respect to

Now differentiate given function with respect to

Again differentiate with respect to

So, substitute the values of and

That is, the function is the solution of Laplace’s equation

**Conclusion:**

The function is the solution of Laplace’s equation

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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...

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