8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Textbook Question

Chapter 14.R, Problem 14CC

(a) Define the gradient vector ∇ f for a function f of two or three variables.

(b) Express D u f in terms of ∇ f .

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Chapter 14.R, Problem 13CC

Chapter 14.R, Problem 15CC

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2. Calculate the gradient vector Vf of the function f (x, y) = x² – x + y - x²y - 2y2 at the point (2,1) and sketch it on the attached contour plot (you can save the picture, open in photo editor and use drawing tools). Explain in one paragraph (about 200-300 words) the meaning of the gradient vector Vf(2,1), negative gradient vector -Vf(2,1).

Suppose a function: R Rhas, at a e R the gradient vector V (a) = (-6, -2, -20, -7) Suppose a particle P moves with unit speed through a= (-13,3, 28, 17) with a velocity vector u that makes the angle 4 with Vf(a). Then what rate of change does P experience at that instant? Answer

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Chapter 14 Solutions

Calculus (MindTap Course List)

Ch. 14.1 - In Example 2 we considered the function W=f(T,v),...Ch. 14.1 - The temperature-humidity index I or humidex, for...Ch. 14.1 - Prob. 3E Ch. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Prob. 9E Ch. 14.1 - Let F(x,y)=1+4y2. aEvaluate F(3,1). bFind and...

Ch. 14.1 - Let f(x,y,z)=x+y+z+ln(4x2y2z2). aEvaluate...Ch. 14.1 - Let g(x,y,z)=x3y2z10xyz. aEvaluate g(1,2,3). bFind...Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Prob. 15E Ch. 14.1 - Prob. 16E Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Prob. 20E Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Prob. 22E Ch. 14.1 - Prob. 23E Ch. 14.1 - Prob. 24E Ch. 14.1 - Prob. 25E Ch. 14.1 - Prob. 26E Ch. 14.1 - Sketch the graph of the function. f(x,y)=sinx Ch. 14.1 - Prob. 28E Ch. 14.1 - Prob. 29E Ch. 14.1 - Sketch the graph of the function. f(x,y)=4x2+y2 Ch. 14.1 - Sketch the graph of the function. f(x,y)=44x2y2 Ch. 14.1 - Match the function with its graph labeled IVI....Ch. 14.1 - Prob. 33E Ch. 14.1 - Shown is a contour map of atmospheric pressure in...Ch. 14.1 - Level curves isothermals are shown for the typical...Ch. 14.1 - Two contour maps are shown. One is for a function...Ch. 14.1 - Prob. 37E Ch. 14.1 - Prob. 38E Ch. 14.1 - The body mass index BMI of a person is defined by...Ch. 14.1 - Prob. 40E 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 - Prob. 45E Ch. 14.1 - Prob. 46E Ch. 14.1 - Prob. 47E Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Prob. 49E Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Prob. 51E 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 - Prob. 54E Ch. 14.1 - Prob. 55E Ch. 14.1 - Prob. 56E Ch. 14.1 - Prob. 57E Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Prob. 59E Ch. 14.1 - Prob. 60E 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 - Prob. 67E Ch. 14.1 - Prob. 68E Ch. 14.1 - Prob. 69E Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Prob. 71E Ch. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Prob. 73E Ch. 14.1 - Prob. 74E Ch. 14.1 - Graph the function using various domains and...Ch. 14.1 - Prob. 76E Ch. 14.1 - Prob. 77E Ch. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Prob. 79E Ch. 14.1 - Prob. 80E Ch. 14.1 - Prob. 81E Ch. 14.2 - Prob. 1E Ch. 14.2 - Explain why each function is continuous or...Ch. 14.2 - Prob. 3E Ch. 14.2 - Prob. 4E Ch. 14.2 - Prob. 5E Ch. 14.2 - Prob. 6E Ch. 14.2 - Prob. 7E Ch. 14.2 - Prob. 8E Ch. 14.2 - Prob. 9E Ch. 14.2 - Prob. 10E Ch. 14.2 - Prob. 11E Ch. 14.2 - Prob. 12E Ch. 14.2 - Prob. 13E Ch. 14.2 - Prob. 14E Ch. 14.2 - Prob. 15E Ch. 14.2 - Prob. 16E Ch. 14.2 - Prob. 17E Ch. 14.2 - Prob. 18E Ch. 14.2 - Prob. 19E Ch. 14.2 - Prob. 20E Ch. 14.2 - Prob. 21E Ch. 14.2 - Prob. 22E Ch. 14.2 - Prob. 23E Ch. 14.2 - Prob. 24E Ch. 14.2 - Prob. 25E Ch. 14.2 - Find h(x,y)=g(f(x,y)) and the set of points at...Ch. 14.2 - Prob. 27E Ch. 14.2 - Prob. 28E Ch. 14.2 - Prob. 29E Ch. 14.2 - Prob. 30E Ch. 14.2 - Prob. 31E Ch. 14.2 - Prob. 32E Ch. 14.2 - Prob. 33E Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Prob. 35E Ch. 14.2 - Prob. 36E Ch. 14.2 - Prob. 37E Ch. 14.2 - Prob. 38E Ch. 14.2 - Prob. 39E Ch. 14.2 - Prob. 40E Ch. 14.2 - Prob. 41E Ch. 14.2 - Prob. 42E Ch. 14.2 - Prob. 43E Ch. 14.2 - Prob. 44E Ch. 14.2 - Prob. 45E Ch. 14.2 - Prob. 46E Ch. 14.3 - Prob. 1E Ch. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - Prob. 3E Ch. 14.3 - The wave heights h in the open sea depend on the...Ch. 14.3 - Prob. 5E Ch. 14.3 - Prob. 6E Ch. 14.3 - Prob. 7E Ch. 14.3 - Prob. 8E Ch. 14.3 - Prob. 9E Ch. 14.3 - A contour map is given for a function f. Use it to...Ch. 14.3 - If f(x,y)=164x2y2, find fx(1,2) and fy(1,2) and...Ch. 14.3 - If f(x,y)=4x24y2, find fx(1,0) and fy(1,0) and...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 14.3 - Prob. 15E Ch. 14.3 - Prob. 16E Ch. 14.3 - Prob. 17E Ch. 14.3 - Prob. 18E Ch. 14.3 - Prob. 19E Ch. 14.3 - Prob. 20E Ch. 14.3 - Prob. 21E Ch. 14.3 - Prob. 22E Ch. 14.3 - Prob. 23E Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Prob. 25E Ch. 14.3 - Prob. 26E Ch. 14.3 - Prob. 27E Ch. 14.3 - Prob. 28E Ch. 14.3 - Prob. 29E Ch. 14.3 - Prob. 30E Ch. 14.3 - Prob. 31E Ch. 14.3 - Prob. 32E Ch. 14.3 - Prob. 33E Ch. 14.3 - Prob. 34E Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Prob. 36E Ch. 14.3 - Prob. 37E Ch. 14.3 - Prob. 38E Ch. 14.3 - Prob. 39E Ch. 14.3 - Prob. 40E Ch. 14.3 - Prob. 41E Ch. 14.3 - Prob. 42E Ch. 14.3 - Prob. 43E Ch. 14.3 - Prob. 44E Ch. 14.3 - Prob. 45E Ch. 14.3 - Prob. 46E Ch. 14.3 - Prob. 47E Ch. 14.3 - Prob. 48E Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Prob. 50E Ch. 14.3 - Find z/y and z/y. a z=f(x)+g(y) b z=f(x+y)Ch. 14.3 - Find z/y and z/y. a z=f(x)g(y) b z=f(xy) c...Ch. 14.3 - Prob. 53E Ch. 14.3 - Prob. 54E Ch. 14.3 - Prob. 55E Ch. 14.3 - Prob. 56E Ch. 14.3 - Prob. 57E Ch. 14.3 - Prob. 58E Ch. 14.3 - Prob. 59E Ch. 14.3 - Prob. 60E Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Prob. 62E Ch. 14.3 - Prob. 63E Ch. 14.3 - Prob. 64E Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Prob. 66E Ch. 14.3 - Prob. 67E Ch. 14.3 - Prob. 68E Ch. 14.3 - Prob. 69E Ch. 14.3 - Find the indicated partial derivatives. u=xaybzc;...Ch. 14.3 - Prob. 71E Ch. 14.3 - If g(x,y,z)=1+xz+1xy, find gxzy. Hint: Use a...Ch. 14.3 - Use the table of values of f(x,y) to estimate the...Ch. 14.3 - Prob. 74E 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 - Prob. 78E Ch. 14.3 - If f and g are twice differentiable functions of a...Ch. 14.3 - Prob. 80E Ch. 14.3 - Prob. 81E Ch. 14.3 - Prob. 82E Ch. 14.3 - Prob. 83E Ch. 14.3 - Prob. 84E Ch. 14.3 - Prob. 85E Ch. 14.3 - Prob. 86E Ch. 14.3 - Prob. 87E Ch. 14.3 - Prob. 88E Ch. 14.3 - For the ideal gas of Exercise 88, show that...Ch. 14.3 - Prob. 90E Ch. 14.3 - Prob. 91E Ch. 14.3 - Prob. 92E Ch. 14.3 - In the project on page 271 we expressed the power...Ch. 14.3 - Prob. 94E Ch. 14.3 - Prob. 95E Ch. 14.3 - If a, b, c are the sides of a triangle and A, B, C...Ch. 14.3 - Prob. 97E Ch. 14.3 - The paraboloid z=6xx22y2 intersects the plane x=1...Ch. 14.3 - The ellipsoid 4x2+2y2+z2=16 intersects the plane...Ch. 14.3 - In a study of frost penetration it was found that...Ch. 14.3 - Prob. 101E Ch. 14.3 - Prob. 102E Ch. 14.3 - If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0). Hint:...Ch. 14.3 - Prob. 104E Ch. 14.3 - Prob. 105E 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 - Prob. 7E Ch. 14.4 - Prob. 8E Ch. 14.4 - Draw the graph of f and its tangent plane at the...Ch. 14.4 - Prob. 10E 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 - Prob. 13E Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Verify the linear approximation at (0,0)....Ch. 14.4 - Verify the linear approximation at (0,0)....Ch. 14.4 - Given that f is a differentiable function with...Ch. 14.4 - Find the linear approximation of the function...Ch. 14.4 - Find the linear approximation of the function...Ch. 14.4 - The wave heights h in the open sea depend on the...Ch. 14.4 - Prob. 23E Ch. 14.4 - Prob. 24E Ch. 14.4 - Find the differential of the function. z=e2xcos2t Ch. 14.4 - Find the differential of the function. u=x2+3y2 Ch. 14.4 - Find the differential of the function. m=p5q3 Ch. 14.4 - Find the differential of the function. T=v1+uvw Ch. 14.4 - Prob. 29E Ch. 14.4 - Find the differential of the function. L=xzey2z2 Ch. 14.4 - Prob. 31E Ch. 14.4 - Prob. 32E 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. 36E Ch. 14.4 - Prob. 37E 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 - Prob. 42E Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Prob. 45E Ch. 14.4 - Prob. 46E Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Prob. 12E Ch. 14.5 - Prob. 13E Ch. 14.5 - Let R(s,t)=G(u(s,t),v(s,t)), where G, u and v are...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Prob. 17E Ch. 14.5 - Prob. 18E Ch. 14.5 - Prob. 19E Ch. 14.5 - Prob. 20E 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 - Prob. 23E Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use Equation 6 to find dy/dx. ycosx=x2+y2 Ch. 14.5 - Use Equation 6 to find dy/dx. cos(xy)=1+siny Ch. 14.5 - Prob. 29E Ch. 14.5 - Prob. 30E Ch. 14.5 - Use Equations 7 to find z/x and z/y. x2+2y2+3z2=1 Ch. 14.5 - Prob. 32E Ch. 14.5 - Prob. 33E Ch. 14.5 - Prob. 34E Ch. 14.5 - The temperature at a point x, y is Tx, y, measured...Ch. 14.5 - Prob. 36E Ch. 14.5 - The speed of sound traveling through ocean water...Ch. 14.5 - Prob. 38E Ch. 14.5 - Prob. 39E Ch. 14.5 - Prob. 40E Ch. 14.5 - The pressure of 1 mole of an ideal gas is...Ch. 14.5 - Prob. 42E Ch. 14.5 - Prob. 43E Ch. 14.5 - Prob. 44E Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Prob. 46E Ch. 14.5 - Prob. 47E Ch. 14.5 - Prob. 48E Ch. 14.5 - Prob. 49E Ch. 14.5 - Prob. 50E Ch. 14.5 - Prob. 51E Ch. 14.5 - Prob. 52E Ch. 14.5 - Prob. 53E Ch. 14.5 - Prob. 54E Ch. 14.5 - Prob. 55E Ch. 14.5 - Prob. 56E Ch. 14.5 - Prob. 57E Ch. 14.5 - Suppose that the equation F(x,y,z)=0 implicitly...Ch. 14.5 - Prob. 59E Ch. 14.6 - Level curves for barometric pressure in millibars...Ch. 14.6 - The contour map shows the average maximum...Ch. 14.6 - A table of values for the wind-chill index...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Prob. 5E Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - Prob. 8E Ch. 14.6 - Prob. 9E Ch. 14.6 - Prob. 10E Ch. 14.6 - Prob. 11E 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. 15E Ch. 14.6 - Prob. 16E 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 - Prob. 19E Ch. 14.6 - Prob. 20E Ch. 14.6 - Prob. 21E 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 - Prob. 24E Ch. 14.6 - Prob. 25E 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 - Prob. 32E Ch. 14.6 - Prob. 33E Ch. 14.6 - Suppose you are climbing a hill whose shape is...Ch. 14.6 - Prob. 35E 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 - Prob. 42E Ch. 14.6 - Prob. 43E Ch. 14.6 - Prob. 44E Ch. 14.6 - Prob. 45E Ch. 14.6 - Prob. 46E Ch. 14.6 - Prob. 47E Ch. 14.6 - Prob. 48E Ch. 14.6 - If f(x,y)=xy, find the gradient vector f(3,2) and...Ch. 14.6 - Prob. 50E Ch. 14.6 - Prob. 51E Ch. 14.6 - Prob. 52E Ch. 14.6 - Prob. 53E 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 - Prob. 57E Ch. 14.6 - Show that every normal line to the sphere...Ch. 14.6 - Prob. 59E Ch. 14.6 - Prob. 60E Ch. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Prob. 62E 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 - Prob. 66E Ch. 14.6 - Prob. 67E Ch. 14.6 - Prob. 68E Ch. 14.6 - Suppose that the directional derivatives of f(x,y)...Ch. 14.6 - Prob. 70E Ch. 14.7 - Prob. 1E 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 - Prob. 6E Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Prob. 8E Ch. 14.7 - Prob. 9E Ch. 14.7 - Prob. 10E 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. 14E Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Prob. 16E 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. 19E Ch. 14.7 - Prob. 20E Ch. 14.7 - Prob. 21E Ch. 14.7 - Show that f(x,y)=x2yex2y2 has minimum values at...Ch. 14.7 - Prob. 23E Ch. 14.7 - Prob. 24E Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Prob. 26E Ch. 14.7 - Prob. 27E Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Prob. 29E Ch. 14.7 - Prob. 30E 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. 34E Ch. 14.7 - Prob. 35E 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. 38E Ch. 14.7 - Prob. 39E Ch. 14.7 - Prob. 40E Ch. 14.7 - Find the shortest distance from the point (2,0,3)...Ch. 14.7 - Prob. 42E Ch. 14.7 - Prob. 43E Ch. 14.7 - Prob. 44E 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 - Prob. 49E 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 - Prob. 56E Ch. 14.7 - The Shannon index sometimes called the...Ch. 14.7 - Prob. 58E 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 - Prob. 2E Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 4E Ch. 14.8 - Prob. 5E Ch. 14.8 - Prob. 6E Ch. 14.8 - Prob. 7E Ch. 14.8 - Prob. 8E Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 10E Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 12E Ch. 14.8 - Prob. 13E 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 - Prob. 17E Ch. 14.8 - Prob. 18E Ch. 14.8 - Prob. 19E Ch. 14.8 - Prob. 20E 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 - Prob. 24E Ch. 14.8 - Consider the problem of minimizing the function...Ch. 14.8 - Prob. 26E Ch. 14.8 - The total production P of a certain product...Ch. 14.8 - Prob. 28E 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 - Prob. 32E Ch. 14.8 - Prob. 33E Ch. 14.8 - Prob. 34E Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Prob. 36E Ch. 14.8 - Prob. 37E Ch. 14.8 - Prob. 38E Ch. 14.8 - Prob. 39E Ch. 14.8 - Prob. 40E Ch. 14.8 - Prob. 41E Ch. 14.8 - Prob. 42E Ch. 14.8 - Prob. 43E 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 - Prob. 47E 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 - Prob. 1CC Ch. 14.R - Prob. 2CC Ch. 14.R - Prob. 3CC Ch. 14.R - Prob. 4CC Ch. 14.R - Prob. 5CC Ch. 14.R - What does Clairauts Theorem say?Ch. 14.R - Prob. 7CC Ch. 14.R - Prob. 8CC Ch. 14.R - Prob. 9CC Ch. 14.R - Prob. 10CC Ch. 14.R - Prob. 11CC Ch. 14.R - Prob. 12CC Ch. 14.R - Prob. 13CC Ch. 14.R - aDefine the gradient vector f for a function f of...Ch. 14.R - Prob. 15CC Ch. 14.R - Prob. 16CC Ch. 14.R - Prob. 17CC Ch. 14.R - Prob. 18CC Ch. 14.R - Prob. 19CC Ch. 14.R - Prob. 1TFQ Ch. 14.R - Prob. 2TFQ Ch. 14.R - Prob. 3TFQ Ch. 14.R - Prob. 4TFQ Ch. 14.R - Prob. 5TFQ Ch. 14.R - Prob. 6TFQ Ch. 14.R - Prob. 7TFQ Ch. 14.R - Prob. 8TFQ Ch. 14.R - Prob. 9TFQ Ch. 14.R - Prob. 10TFQ Ch. 14.R - Prob. 11TFQ Ch. 14.R - Prob. 12TFQ Ch. 14.R - Prob. 1E Ch. 14.R - Prob. 2E Ch. 14.R - Prob. 3E Ch. 14.R - Prob. 4E Ch. 14.R - Prob. 5E Ch. 14.R - Prob. 6E 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 - Prob. 9E Ch. 14.R - Prob. 10E Ch. 14.R - Prob. 11E Ch. 14.R - Prob. 12E Ch. 14.R - Prob. 13E Ch. 14.R - Prob. 14E Ch. 14.R - Prob. 15E Ch. 14.R - Prob. 16E Ch. 14.R - Prob. 17E Ch. 14.R - Prob. 18E Ch. 14.R - Prob. 19E Ch. 14.R - Prob. 20E Ch. 14.R - Prob. 21E Ch. 14.R - Prob. 22E Ch. 14.R - Prob. 23E Ch. 14.R - Prob. 24E Ch. 14.R - Prob. 25E Ch. 14.R - Prob. 26E Ch. 14.R - Prob. 27E 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 - Prob. 30E Ch. 14.R - Find the points on the hyperboloid x2+4y2z2=4...Ch. 14.R - Prob. 32E Ch. 14.R - Prob. 33E Ch. 14.R - The two legs of a right triangle are measured as 5...Ch. 14.R - Prob. 35E Ch. 14.R - Prob. 36E Ch. 14.R - Suppose z=f(x,y), where x=g(s,t), y=h(s,t),...Ch. 14.R - Prob. 38E Ch. 14.R - Prob. 39E Ch. 14.R - Prob. 40E Ch. 14.R - Prob. 41E Ch. 14.R - Prob. 42E Ch. 14.R - Prob. 43E Ch. 14.R - Prob. 44E Ch. 14.R - Prob. 45E Ch. 14.R - Prob. 46E Ch. 14.R - Prob. 47E Ch. 14.R - Prob. 48E Ch. 14.R - The contour map shows wind speed in knots during...Ch. 14.R - Prob. 50E Ch. 14.R - Prob. 51E Ch. 14.R - Prob. 52E Ch. 14.R - Prob. 53E Ch. 14.R - Prob. 54E Ch. 14.R - Prob. 55E Ch. 14.R - Prob. 56E 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 - Prob. 59E Ch. 14.R - Prob. 60E Ch. 14.R - Prob. 61E Ch. 14.R - Prob. 62E Ch. 14.R - Prob. 63E 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 - Prob. 3P Ch. 14.P - For what values of the number r is the function...Ch. 14.P - Prob. 5P Ch. 14.P - Prob. 6P 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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(a) Define the gradient vector ∇ f for a function f of two or three variables. (b) Express D u f in terms of ∇ f . (c) Explain the geometric significance of the gradient.

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Chapter 14 Solutions