Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term
8th Edition
ISBN: 9781305779198
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 14.8, Problem 30E
Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.
Hint: Use Heron’s formula for the area:
where s = p/2 and x, y, z are the lengths of the sides.
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Chapter 14 Solutions
Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term
Ch. 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 - Prob. 3ECh. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - Prob. 5ECh. 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 - Prob. 11ECh. 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 - Prob. 19ECh. 14.1 - Prob. 20ECh. 14.1 - Prob. 21ECh. 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 - Prob. 24ECh. 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 - Prob. 28ECh. 14.1 - Prob. 29ECh. 14.1 - Sketch the graph of the function. 30....Ch. 14.1 - Prob. 31ECh. 14.1 - Match the function with its graph (labeled IVI)....Ch. 14.1 - Prob. 33ECh. 14.1 - Shown is a contour map of atmospheric pressure in...Ch. 14.1 - Prob. 35ECh. 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 - Prob. 40ECh. 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 - Prob. 43ECh. 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 - Prob. 48ECh. 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 - Prob. 52ECh. 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 - Prob. 56ECh. 14.1 - Prob. 57ECh. 14.1 - Prob. 58ECh. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Prob. 60ECh. 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 - Prob. 63ECh. 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 - Prob. 66ECh. 14.1 - Prob. 67ECh. 14.1 - Describe the level surfaces of the function. 68....Ch. 14.1 - Describe the level surfaces of the function. 69....Ch. 14.1 - Prob. 70ECh. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Prob. 72ECh. 14.1 - Prob. 73ECh. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Prob. 75ECh. 14.1 - Prob. 76ECh. 14.1 - Prob. 77ECh. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Prob. 80ECh. 14.1 - Prob. 81ECh. 14.2 - Prob. 1ECh. 14.2 - Explain why each function is continuous or...Ch. 14.2 - Prob. 3ECh. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Prob. 6ECh. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Prob. 12ECh. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Prob. 17ECh. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Prob. 20ECh. 14.2 - Prob. 21ECh. 14.2 - Prob. 22ECh. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 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 - Prob. 28ECh. 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 - Prob. 32ECh. 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 - Prob. 35ECh. 14.2 - Prob. 36ECh. 14.2 - Prob. 37ECh. 14.2 - Prob. 38ECh. 14.2 - Prob. 39ECh. 14.2 - Prob. 40ECh. 14.2 - Prob. 41ECh. 14.2 - Prob. 42ECh. 14.2 - Graph and discuss the continuity of the function...Ch. 14.2 - Prob. 44ECh. 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 - Prob. 1ECh. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - Prob. 3ECh. 14.3 - Prob. 4ECh. 14.3 - Prob. 5ECh. 14.3 - Prob. 6ECh. 14.3 - Prob. 7ECh. 14.3 - Prob. 8ECh. 14.3 - Prob. 9ECh. 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 - Prob. 15ECh. 14.3 - Prob. 16ECh. 14.3 - Prob. 17ECh. 14.3 - Prob. 18ECh. 14.3 - Prob. 19ECh. 14.3 - Prob. 20ECh. 14.3 - Prob. 21ECh. 14.3 - Prob. 22ECh. 14.3 - Prob. 23ECh. 14.3 - Prob. 24ECh. 14.3 - Prob. 25ECh. 14.3 - Prob. 26ECh. 14.3 - Prob. 27ECh. 14.3 - Prob. 28ECh. 14.3 - Prob. 29ECh. 14.3 - Prob. 30ECh. 14.3 - Prob. 31ECh. 14.3 - Prob. 32ECh. 14.3 - Prob. 33ECh. 14.3 - Prob. 34ECh. 14.3 - Prob. 35ECh. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - Prob. 38ECh. 14.3 - Prob. 39ECh. 14.3 - Prob. 40ECh. 14.3 - Prob. 41ECh. 14.3 - Find the indicated partial derivative. 42....Ch. 14.3 - Prob. 43ECh. 14.3 - Find the indicated partial derivative. 44. f(x, y,...Ch. 14.3 - Prob. 45ECh. 14.3 - Prob. 46ECh. 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 - Prob. 50ECh. 14.3 - Prob. 51ECh. 14.3 - Find z/x and z/y. 52. (a) z = f(x)g(y) (b) z =...Ch. 14.3 - Prob. 53ECh. 14.3 - Find all the second partial derivatives. 54. f(x,...Ch. 14.3 - Find all the second partial derivatives. 55....Ch. 14.3 - Prob. 56ECh. 14.3 - Prob. 57ECh. 14.3 - Prob. 58ECh. 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 - Prob. 62ECh. 14.3 - Find the indicated partial derivative(s). 63. f(x,...Ch. 14.3 - Prob. 64ECh. 14.3 - Prob. 65ECh. 14.3 - Prob. 66ECh. 14.3 - Find the indicated partial derivative(s). 67....Ch. 14.3 - Prob. 68ECh. 14.3 - Find the indicated partial derivative(s). 69....Ch. 14.3 - Prob. 70ECh. 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 - Prob. 74ECh. 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 - Prob. 80ECh. 14.3 - The diffusion equation ct=D2cx2 where D is a...Ch. 14.3 - Prob. 82ECh. 14.3 - Prob. 83ECh. 14.3 - Prob. 84ECh. 14.3 - Prob. 85ECh. 14.3 - Cobb and Douglas used the equation P(L, K) =...Ch. 14.3 - Prob. 87ECh. 14.3 - Prob. 88ECh. 14.3 - Prob. 89ECh. 14.3 - The wind-chill index is modeled by the function W...Ch. 14.3 - Prob. 91ECh. 14.3 - One of Poiseuilles laws states that the resistance...Ch. 14.3 - Prob. 93ECh. 14.3 - The average energy E (in kcal) needed for a lizard...Ch. 14.3 - Prob. 95ECh. 14.3 - If a, b, c are the sides of a triangle and A, B, C...Ch. 14.3 - Prob. 97ECh. 14.3 - Prob. 98ECh. 14.3 - Prob. 99ECh. 14.3 - In a study of frost penetration it was found that...Ch. 14.3 - Prob. 101ECh. 14.3 - Prob. 102ECh. 14.3 - Prob. 103ECh. 14.3 - Prob. 104ECh. 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 - Prob. 7ECh. 14.4 - Prob. 8ECh. 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 - Prob. 23ECh. 14.4 - Prob. 24ECh. 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 - Prob. 30ECh. 14.4 - Prob. 31ECh. 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 - Prob. 36ECh. 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 - Prob. 40ECh. 14.4 - Prob. 41ECh. 14.4 - Suppose you need to know an equation of the...Ch. 14.4 - Prob. 43ECh. 14.4 - Prob. 44ECh. 14.4 - Prob. 45ECh. 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 - Prob. 10ECh. 14.5 - Use the Chain Rule to find z/s and z/t. 11. z = er...Ch. 14.5 - Prob. 12ECh. 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 - Prob. 17ECh. 14.5 - Prob. 18ECh. 14.5 - Prob. 19ECh. 14.5 - Prob. 20ECh. 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 - Prob. 28ECh. 14.5 - Prob. 29ECh. 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 - Prob. 32ECh. 14.5 - Prob. 33ECh. 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 - Prob. 36ECh. 14.5 - The speed of sound traveling through ocean water...Ch. 14.5 - Prob. 38ECh. 14.5 - Prob. 39ECh. 14.5 - The voltage V in a simple electrical circuit is...Ch. 14.5 - Prob. 41ECh. 14.5 - Prob. 42ECh. 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 - Prob. 47ECh. 14.5 - Prob. 48ECh. 14.5 - Prob. 49ECh. 14.5 - Prob. 50ECh. 14.5 - Prob. 51ECh. 14.5 - Prob. 53ECh. 14.5 - Prob. 55ECh. 14.5 - If f is homogeneous of degree n, show that...Ch. 14.5 - Prob. 57ECh. 14.5 - Prob. 58ECh. 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 - Prob. 14ECh. 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 - Prob. 32ECh. 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 - Prob. 35ECh. 14.6 - Shown is a topographic map of Blue River Pine...Ch. 14.6 - Prob. 37ECh. 14.6 - Sketch the gradient vector f(4. 6) for the...Ch. 14.6 - Prob. 39ECh. 14.6 - Prob. 40ECh. 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 - Prob. 47ECh. 14.6 - Prob. 48ECh. 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 - Prob. 58ECh. 14.6 - Where does the normal line to the paraboloid z =...Ch. 14.6 - Prob. 60ECh. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Prob. 62ECh. 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 - Prob. 66ECh. 14.6 - (a) Two surfaces are called orthogonal at a point...Ch. 14.6 - Prob. 68ECh. 14.6 - Suppose that the directional derivatives of f(x,...Ch. 14.6 - Prob. 70ECh. 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 - Prob. 12ECh. 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 - Prob. 23ECh. 14.7 - Prob. 24ECh. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Prob. 26ECh. 14.7 - Prob. 27ECh. 14.7 - Prob. 28ECh. 14.7 - Use a graphing device as in Example 4 (or Newtons...Ch. 14.7 - Prob. 30ECh. 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 - Prob. 39ECh. 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 - Prob. 43ECh. 14.7 - Prob. 44ECh. 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 - Prob. 57ECh. 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 - Prob. 13ECh. 14.8 - Prob. 14ECh. 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 - Prob. 31ECh. 14.8 - Prob. 32ECh. 14.8 - Prob. 33ECh. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Prob. 35ECh. 14.8 - Prob. 36ECh. 14.8 - Prob. 37ECh. 14.8 - Prob. 38ECh. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Prob. 40ECh. 14.8 - Prob. 41ECh. 14.8 - Prob. 42ECh. 14.8 - Prob. 43ECh. 14.8 - Prob. 44ECh. 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 - Prob. 50ECh. 14 - (a) What is a function of two variables? (b)...Ch. 14 - What is a function of three variables? How can you...Ch. 14 - Prob. 3RCCCh. 14 - Prob. 4RCCCh. 14 - Prob. 5RCCCh. 14 - Prob. 6RCCCh. 14 - Prob. 7RCCCh. 14 - Prob. 8RCCCh. 14 - Prob. 9RCCCh. 14 - Prob. 10RCCCh. 14 - State the Chain Rule for the case where z = f(x,...Ch. 14 - Prob. 12RCCCh. 14 - Prob. 13RCCCh. 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 - Prob. 16RCCCh. 14 - Prob. 17RCCCh. 14 - Prob. 18RCCCh. 14 - Prob. 19RCCCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 2RQCh. 14 - Prob. 3RQCh. 14 - Prob. 4RQCh. 14 - Prob. 5RQCh. 14 - Prob. 6RQCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 8RQCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 10RQCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 12RQCh. 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 - Prob. 9RECh. 14 - Prob. 10RECh. 14 - Prob. 11RECh. 14 - Find a linear approximation to the temperature...Ch. 14 - Prob. 13RECh. 14 - Prob. 14RECh. 14 - Prob. 15RECh. 14 - Prob. 16RECh. 14 - Prob. 17RECh. 14 - Prob. 18RECh. 14 - Prob. 19RECh. 14 - Prob. 20RECh. 14 - Prob. 21RECh. 14 - Prob. 22RECh. 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 - Prob. 27RECh. 14 - Prob. 28RECh. 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 - Prob. 32RECh. 14 - Prob. 33RECh. 14 - The two legs of a right triangle are measured as 5...Ch. 14 - Prob. 35RECh. 14 - If v = x2sin y + yexy, where x = s + 2t and y =...Ch. 14 - Prob. 37RECh. 14 - Prob. 38RECh. 14 - Prob. 39RECh. 14 - Prob. 40RECh. 14 - Prob. 41RECh. 14 - Prob. 42RECh. 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 - Prob. 52RECh. 14 - Prob. 53RECh. 14 - Prob. 54RECh. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Prob. 57RECh. 14 - Prob. 58RECh. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Prob. 61RECh. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Prob. 63RECh. 14 - Prob. 64RECh. 14 - A pentagon is formed by placing an isosceles...Ch. 14 - Prob. 1PCh. 14 - Marine biologists have determined that when a...Ch. 14 - A long piece of galvanized sheet metal with width...Ch. 14 - Prob. 4PCh. 14 - Suppose f is a differentiable function of one...Ch. 14 - Prob. 6PCh. 14 - If the ellipse x2/a2 + y2/b2 = 1 is to enclose the...Ch. 14 - Show that the maximum value of the function...
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- (10) (16 points) Let R>0. Consider the truncated sphere S given as x² + y² + (z = √15R)² = R², z ≥0. where F(x, y, z) = −yi + xj . (a) (8 points) Consider the vector field V (x, y, z) = (▼ × F)(x, y, z) Think of S as a hot-air balloon where the vector field V is the velocity vector field measuring the hot gasses escaping through the porous surface S. The flux of V across S gives the volume flow rate of the gasses through S. Calculate this flux. Hint: Parametrize the boundary OS. Then use Stokes' Theorem. (b) (8 points) Calculate the surface area of the balloon. To calculate the surface area, do the following: Translate the balloon surface S by the vector (-15)k. The translated surface, call it S+ is part of the sphere x² + y²+z² = R². Why do S and S+ have the same area? ⚫ Calculate the area of S+. What is the natural spherical parametrization of S+?arrow_forward(1) (8 points) Let c(t) = (et, et sint, et cost). Reparametrize c as a unit speed curve starting from the point (1,0,1).arrow_forward(9) (16 points) Let F(x, y, z) = (x² + y − 4)i + 3xyj + (2x2 +z²)k = - = (x²+y4,3xy, 2x2 + 2²). (a) (4 points) Calculate the divergence and curl of F. (b) (6 points) Find the flux of V x F across the surface S given by x² + y²+2² = 16, z ≥ 0. (c) (6 points) Find the flux of F across the boundary of the unit cube E = [0,1] × [0,1] x [0,1].arrow_forward
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