The plane 4 x – 3 y + 8 z = 5 intersects the cone z 2 = x 2 + y 2 in an ellipse. (a) Graph the cone and the plane, and observe the elliptical intersection. (b) Use Lagrange multipliers to find the highest and lowest points on the ellipse.
BuyFindarrow_forward
Multivariable Calculus
8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266643
BuyFindarrow_forward
Multivariable Calculus
8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266643
Solutions
Chapter
Section
Chapter 14.8, Problem 46E
Textbook Problem
The plane 4x – 3y + 8z = 5 intersects the cone z2 = x2 + y2 in an ellipse.
(a) Graph the cone and the plane, and observe the elliptical intersection.
(b) Use Lagrange multipliers to find the highest and lowest points on the ellipse.
check_circle
Expert Solution
Chapter 14 Solutions
Multivariable Calculus
Show all chapter solutions
addCh. 14.1 - In Example 2 we considered the function W = f(T,...Ch. 14.1 - The temperature-humidity index I (or humidex, for...Ch. 14.1 - A manufacturer has modeled its yearly production...Ch. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - A model for the surface area of a human body is...Ch. 14.1 - The wind-chill index W discussed in Example 2 has...Ch. 14.1 - The wave heights h in the open sea depend on the...Ch. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Let g(x, y) = cos(x + 2y). (a) Evaluate g(2, 1)....Ch. 14.1 - Let F(x,y)=1+4y2. (a) Evaluate F(3,1). (b) Find...
Ch. 14.1 - Let f(x,y,z)=x+y+z+ln(4x2y2z2). (a) Evaluate f(1,...Ch. 14.1 - Let g(x,y,z)=x3y2z10xyz. (a) Evaluate g(1, 2, 3)....Ch. 14.1 - Find and sketch the domain of the function. 13....Ch. 14.1 - Find and sketch the domain of the function. 14....Ch. 14.1 - Find and sketch the domain of the function. 15....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function. 17....Ch. 14.1 - Find and sketch the domain of the function. 18....Ch. 14.1 - Find and sketch the domain of the function. 19....Ch. 14.1 - Find and sketch the domain of the function. 20....Ch. 14.1 - Find and sketch the domain of the function. 21....Ch. 14.1 - Find and sketch the domain of the function. 22....Ch. 14.1 - Sketch the graph of the function. 23. f(x, y) = yCh. 14.1 - Sketch the graph of the function. 24. f(x, y) = x2Ch. 14.1 - Sketch the graph of the function. 25. f(x, y) = 10...Ch. 14.1 - Sketch the graph of the function. 26. f(x, y) =...Ch. 14.1 - Sketch the graph of the function. 27. f(x, y) =...Ch. 14.1 - Sketch the graph of the function. 28. f(x, y) = 2 ...Ch. 14.1 - Sketch the graph of the function. 29. f(x, y) = x2...Ch. 14.1 - Sketch the graph of the function. 30....Ch. 14.1 - Sketch the graph of the function. 31....Ch. 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...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...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. 67....Ch. 14.1 - Describe the level surfaces of the function. 68....Ch. 14.1 - Describe the level surfaces of the function. 69....Ch. 14.1 - Describe the level surfaces of the function. 70....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 - (a) Show that, by taking logarithms, the general...Ch. 14.2 - Suppose that lim(x, y) (3, 1) = 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 (x,...Ch. 14.2 - Use a table of numerical values of f(x, y) for (x,...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 - U se a computer graph of the function to explain...Ch. 14.2 - U se 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,...Ch. 14.2 - Show that the function f given by f(x) = |x| is...Ch. 14.2 - If c Vn, show that the function f given by f(x) =...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 in die 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) = 16 4x2 y2, find fx(1, 2) and fy(1,...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. 41. R(s, t)...Ch. 14.3 - Find the indicated partial derivative. 42....Ch. 14.3 - Find the indicated partial derivative. 43....Ch. 14.3 - Find the indicated partial derivative. 44. f(x, y,...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/x and z/y. 51. (a) z = f(x) + g(y) (b) z =...Ch. 14.3 - Find z/x and z/y. 52. (a) z = f(x)g(y) (b) z =...Ch. 14.3 - Find all the second partial derivatives. 53. f(x,...Ch. 14.3 - Find all the second partial derivatives. 54. f(x,...Ch. 14.3 - Find all the second partial derivatives. 55....Ch. 14.3 - Find all the second partial derivatives. 56. T =...Ch. 14.3 - Find all the second partial derivatives. 57. v =...Ch. 14.3 - Find all the second partial derivatives. 58....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 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Find the indicated partial derivative(s). 63. f(x,...Ch. 14.3 - Find the indicated partial derivative(s). 64. f(x,...Ch. 14.3 - Find the indicated partial derivative(s). 65....Ch. 14.3 - Find the indicated partial derivative(s). 66. g(r,...Ch. 14.3 - Find the indicated partial derivative(s). 67....Ch. 14.3 - Find the indicated partial derivative(s). 68. V =...Ch. 14.3 - Find the indicated partial derivative(s). 69....Ch. 14.3 - Find the indicated partial derivative(s). 70. u =...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 gxyz. [Hint: Which...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, show...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 P =...Ch. 14.3 - Show that the Cobb-Douglas production function...Ch. 14.3 - Cobb and Douglas used the equation P(L, K) =...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 W...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 344 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 = 6 x x22y2 intersects the...Ch. 14.3 - The ellipsoid 4x2 + 2y2 + z2 = 16 intersects the...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)....Ch. 14.3 - If f(x,y)=x3+y33 find fx(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 - 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). 17. ex...Ch. 14.4 - Verify the linear approximation at (0, 0). 18....Ch. 14.4 - Given that f is a differentiable function with...Ch. 14.4 - Find the linear approximation of the function f(x,...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. 25. z = e2x...Ch. 14.4 - Find the differential of the function. 26....Ch. 14.4 - Find the differential of the function. 27. m =...Ch. 14.4 - Find the differential of the function. 28....Ch. 14.4 - Find the differential of the function. 29. R = 2...Ch. 14.4 - Find the differential of the function. 30....Ch. 14.4 - If z = 5x2 + y2 and (x, y) changes from (1, 2) to...Ch. 14.4 - If z = x2 xy + 3y2 and (x, y) changes from (3, 1)...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 W...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. 1. z =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 2....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 3. z =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 4....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 5. w =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 6. w =...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 7. z = (x ...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 8. z =...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 9. z =...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 10. z =...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 11. z = er...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 12. z =...Ch. 14.5 - Let p(t) = f(x,y), where f is differentiable, x =...Ch. 14.5 - Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and...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. 27. y cos x = x2 +...Ch. 14.5 - Use Equation 6 to find dy/dx. 28. cos(xy) = 1 +...Ch. 14.5 - Use Equation 6 to find dy/dx. 29. tan1(x2y) = x +...Ch. 14.5 - Use Equation 6 to find dy/dx. 30. ey sin x = x +...Ch. 14.5 - Use Equations 7 to find z/x and z/y. 31. x2 + 2y2...Ch. 14.5 - Use Equations 7 to find z/x and z/y. 32. x2 y2 +...Ch. 14.5 - Use Equations 7 to find z/x and z/y. 33. ez = xyzCh. 14.5 - Use Equations 7 to find z/x and z/y. 34. yz + x ln...Ch. 14.5 - The temperature at a point (x, y) is T(x, y),...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 , width w, and height h of a box change...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 - 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 fx(tx,...Ch. 14.5 - Suppose that the equation F(x, y, z) = 0...Ch. 14.5 - Equation 6 is a formula for the derivative dy/dx...Ch. 14.6 - Level curves for barometric pressure (in...Ch. 14.6 - The contour map shows the average maximum...Ch. 14.6 - A table of values for the wind-chill index W =...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...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...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 Du, f(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) =...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...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...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...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...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)...Ch. 14.6 - If g(x, y) = x2 + y2 4x, find the gradient vector...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...Ch. 14.6 - Are there any points on the hyperboloid x2 y2 z2...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 x2 + y2...Ch. 14.6 - Where does the normal line to the paraboloid z =...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 tor the tangent line to...Ch. 14.6 - (a) The plane y + z = 3 intersects the cylinder x2...Ch. 14.6 - Where does the helix r(t) = cos t, sin t, t...Ch. 14.6 - The helix r(t) = cos(t/2), sin(t/2), t) intersects...Ch. 14.6 - (a) Two surfaces are called orthogonal at a point...Ch. 14.6 - (a) Show that the function f(x, y) = xy3 is...Ch. 14.6 - Suppose that the directional derivatives of f(x,...Ch. 14.6 - Show that if z = f(x, y) is differentiable at x0 =...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 y...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 + 4y2 4xy + 2 has an...Ch. 14.7 - Show that f(x,y)=x2yex2y2 has maximum 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,...Ch. 14.7 - Find the point on the plane x 2y + 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...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,...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 +...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 - 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 4x 3y + 8z = 5 intersects the cone z2 =...Ch. 14.8 - (a) Find the maximum value of...Ch. 14.8 - (a) Maximize i=1nxiyi subject to the constraints...Ch. 14 - (a) What is a function of two variables? (b)...Ch. 14 - What is a function of three variables? How can you...Ch. 14 - What does lim(x,y)(a,b)f(x,y)=L mean? How can you...Ch. 14 - (a) What does it mean to say that f is continuous...Ch. 14 - (a) Write expressions for the partial derivatives...Ch. 14 - What does Clairauts Theorem say?Ch. 14 - How do you find a tangent plane to each of the...Ch. 14 - Define the linearization of f at (a, b). What is...Ch. 14 - (a) What does it mean to say that f is...Ch. 14 - If z = f(x, y), what arc the differentials dx, dy,...Ch. 14 - State the Chain Rule for the case where z = f(x,...Ch. 14 - If z is defined implicitly as a function of x and...Ch. 14 - (a) Write an expression as a limit for the...Ch. 14 - (a) Define the gradient vector f for a function f...Ch. 14 - What do the following statements mean? (a) f has a...Ch. 14 - (a) If f has a local maximum at (a, b), what can...Ch. 14 - State the Second Derivatives Test.Ch. 14 - (a) What is a closed set in 2? What is a bounded...Ch. 14 - Explain how the method of Lagrange multipliers...Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Find and sketch the domain of the function. 1....Ch. 14 - Find and sketch the domain of the function. 2....Ch. 14 - Sketch the graph of the function. 3. f(x, y) = 1 ...Ch. 14 - Sketch the graph of the function. 4. f(x, y) = x2...Ch. 14 - Sketch several level curves of the function. 5....Ch. 14 - Sketch several level curves of the function. 6....Ch. 14 - Make a rough sketch of a contour map for the...Ch. 14 - The contour map of a function f is shown, (a)...Ch. 14 - Evaluate the limit or show that it does not exist....Ch. 14 - Evaluate the limit or show that it does not exist....Ch. 14 - A metal plate is situated in the xy-plane and...Ch. 14 - Find a linear approximation to the temperature...Ch. 14 - Find the first partial derivatives. 13. f(x, y) =...Ch. 14 - Find the first partial derivatives. 14....Ch. 14 - Find the first partial derivatives. 15. F(, ) = 2...Ch. 14 - Find the first partial derivatives. 16. G(x, y, z)...Ch. 14 - Find the first partial derivatives. 17....Ch. 14 - The speed of sound traveling through ocean water...Ch. 14 - Find all second partial derivatives of f. 19. f(x,...Ch. 14 - Find all second partial derivatives of f. 20. z =...Ch. 14 - Find all second partial derivatives of f. 21. f(x,...Ch. 14 - Find all second partial derivatives of f. 22. v =...Ch. 14 - If z = xy + xey/x show that xzx+yzy=xy+z.Ch. 14 - If z = sin(x + sin t), show that zx2zxt=zt2zx2Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Use a computer to graph the surface z = x2 + y4...Ch. 14 - Find the points on the hyperboloid x2 + 4y2 z2 =...Ch. 14 - Find du if u = ln(1 + se2t).Ch. 14 - Find the linear approximation of the function...Ch. 14 - The two legs of a right triangle are measured as 5...Ch. 14 - If u = x2y3 + z4, where x = p + 3p2, y = pep, and...Ch. 14 - If v = x2sin y + yexy, where x = s + 2t and y =...Ch. 14 - Suppose z = f(x, y), where x = g(s, t), y = h(s,...Ch. 14 - Use a tree diagram to write out the Chain Rule for...Ch. 14 - If z = y + f(x2 y2), where f is differentiable,...Ch. 14 - The length x of a side of a triangle is increasing...Ch. 14 - If z =f(u, v), where u = xy, v = y/x, and f has...Ch. 14 - If cos(xyz) = 1 + .x2y2 + z2, find zx and zy.Ch. 14 - Find the gradient of the function f(x,y,z)=x2eyz2.Ch. 14 - (a) When is the directional derivative of f a...Ch. 14 - Find the directional derivative of f at the given...Ch. 14 - Find the directional derivative of f at the given...Ch. 14 - Find the maximum rate of change of f(x,y)=x2y+y at...Ch. 14 - Find the direction in which f(x, y, z) =...Ch. 14 - The contour map shows wind speed in knots during...Ch. 14 - Find parametric equations of the tangent line at...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Use a graph or level curves or both to estimate...Ch. 14 - Use a graphing calculator or computer (or Newtons...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Find the points on the surface xy2z3= 2 that are...Ch. 14 - A package in the shape of a rectangular box can be...Ch. 14 - A pentagon is formed by placing an isosceles...Ch. 14 - A rectangle with length L and width W is cut into...Ch. 14 - Marine biologists have determined that when a...Ch. 14 - A long piece of galvanized sheet metal with width...Ch. 14 - For what values of the number r is the function...Ch. 14 - Suppose f is a differentiable function of one...Ch. 14 - (a) Newton's method for approximating a root of an...Ch. 14 - If the ellipse x2/a2 + y2/b2 = 1 is to enclose the...Ch. 14 - Show that the maximum value of the function...
Additional Math Textbook Solutions
Find more solutions based on key concepts
Show solutions addBirths and Deaths From 2008 to 2012, the total numbers of births B (in thousands) and deaths D (in thousands) i...
Calculus: An Applied Approach (MindTap Course List)
Solve the equations in Exercises 126. 2x3x12x+3x+1=0
Applied Calculus
Finding Values In Exercises 69-74, find f1 (a) for the function f and real number a. Function Real Number f(x)=...
Calculus: Early Transcendental Functions
Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate i...
Single Variable Calculus: Early Transcendentals, Volume I
A lamina occupies the part of the disk x2 + y2 a2 that lies in the first quadrant. (a) Find the centroid of th...
Calculus: Early Transcendentals
Solve the equations in Exercises 126. 2(x21)x2+1x4x2+1x2+1=0
Finite Mathematics and Applied Calculus (MindTap Course List)
Solving by Factoring Find all real solutions of the equation by factoring. 48. x2 + 8x + 12 = 0
Precalculus: Mathematics for Calculus (Standalone Book)
SW Compute the median and mean for both the pretest and posttest for the test scores first presented in problem...
Essentials Of Statistics
Find the mean for the scores in the following frequency distribution table: X f 6 1 5 4 4 2 3 2 2 1
Essentials of Statistics for The Behavioral Sciences (MindTap Course List)
Find an equation of the tangent plane to the given parametric surface at the specified point. r(u,v)=sinui+cosu...
Calculus (MindTap Course List)
A sample has a mean of M=25and a standard deviation of s=5. For this sample, find the X value corresponding to ...
Statistics for The Behavioral Sciences (MindTap Course List)
Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. 70.
Single Variable Calculus
In Exercises 13-20, sketch a set of coordinate axes and plot each point. 14. (1, 3)
Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach
Demand and Profit To generate a profit from its new oTunes service, OHaganBooks.com needs to know how the deman...
Finite Mathematics
Critical Thinking If two variables have a negative linear correlation, is the slope of the least-squares line p...
Understanding Basic Statistics
For each of the following assertions, state whether it is a legitimate statistical hypothesis and why: a. H: ...
Probability and Statistics for Engineering and the Sciences
26. Manufacturing Show that a box with a square base, an open top, and a fixed volume requires the least materi...
Mathematical Applications for the Management, Life, and Social Sciences
Use the probability demonstrator, in the left margin, to work Excursion. Explain why the greatest number of bal...
Mathematical Excursions (MindTap Course List)
Using Table 14-1 as needed, calculate the required information for the following mortgages. TABLE 14-11 Monthly...
Contemporary Mathematics for Business & Consumers
Classify each number as prime or not prime: 53
Elementary Technical Mathematics
Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. Objectives 2 x2+2xy...
Intermediate Algebra
In O, chord AB chord CD. Radius OE is perpendicular to AB and CD at points M and N, respectively. If OE=13, AB=...
Elementary Geometry For College Students, 7e
If , F(x)=2xf(t)dt, where f is the function whose graph is given, which of the following values is largest? (A)...
Single Variable Calculus: Early Transcendentals
Finding the Volume of a Solid In Exercises 33-36, find the volume of the solid generated by revolving the regio...
Calculus of a Single Variable
Let be in the field. Define the mapping by . Prove that is an automorphism.
Elements Of Modern Algebra
For Problems 21 through 24, use 3.1416 for unless your calculator has a key marked . Use a calculator to conve...
Trigonometry (MindTap Course List)
If we know that then the absolute value of the nth remainder of the Taylor series for y = f(x) about a for is...
Study Guide for Stewart's Multivariable Calculus, 8th
Comet Hale-Bopp The comet Hale-Bopp has an elliptical orbit with the sun a, one focus and has an eccentricity o...
Calculus: Early Transcendental Functions (MindTap Course List)
Sometimes, Always, or Never:
If {an} is increasing and bounded above, then {an} converges.
Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th
In Exercises 918, indicate whether the matrix is in row-reduced form. [0100|35]
Finite Mathematics for the Managerial, Life, and Social Sciences
Given: AOOBandOCbisectsACBinO Find: a mAB b mACB c mBC d mAOC
Elementary Geometry for College Students
Using Vector Operations In Exercise 27 and 28, And (a) 23u , (b) 3v, (c) vu (d) 2u+5v. u=3,8,v=8,7
Multivariable Calculus
Finding a Unit Vector In Exercises 15-18, find a unit vector that is orthogonal to both u and v. u=8,6,4v=10,12...
Calculus (MindTap Course List)
Give an example of a permutation and a similar example of combination.
Mathematics: A Practical Odyssey
Following is a portion of the computer output for a regression analysis relating y = maintenance expense (dolla...
STATISTICS F/BUSINESS+ECONOMICS-TEXT
Explain why the main effects in a factorial study may not provide an accurate description of the results?
Research Methods for the Behavioral Sciences (MindTap Course List)
A random sample of n = 347 students was selected, and each one was asked to complete several questionnaires, fr...
Introduction To Statistics And Data Analysis
The following formulas are used in the machine trades. Substitute the given values in each formula and solve fo...
Mathematics For Machine Technology
Use the Leading Coefficient Test to determine the end behavior of each polynomial. f(x)=5x7+10x32x
College Algebra (MindTap Course List)
entrepreneur magazine ranks franchises using performance measures such as growth rate, number of locations, sta...
Statistics for Business & Economics, Revised (MindTap Course List)
What can a researcher do if the treatment administered during the first B?
Research Methods for the Behavioral Sciences (MindTap Course List)
A random variable is normally distributed with a mean of and a standard deviation of .
Sketch a normal curve f...
Essentials Of Statistics For Business & Economics
Write an equation relating p(7,2) and (27) .
Discrete Mathematics With Applications
7. Are stock splits beneficial to stockholders? SNL Financial studied stock splits in the banking industry over...
Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)
In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not eva...
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
The Economic Order Quantity Model gives the optimal quantity Q of goods for a store to order at a time Q=2cNh. ...
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c...
Calculus Volume 1
