Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Question
Chapter 14.5, Problem 36E
To determine
(a)
To interpret:
What is the significance of the signs of the partial derivatives
To determine
(b)
To estimate:
The current rate of change of wheat production
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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. 3ECh. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - Prob. 5ECh. 14.1 - Prob. 6ECh. 14.1 - Prob. 7ECh. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Prob. 9ECh. 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. 15ECh. 14.1 - Prob. 16ECh. 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. 20ECh. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Prob. 22ECh. 14.1 - Prob. 23ECh. 14.1 - Prob. 24ECh. 14.1 - Prob. 25ECh. 14.1 - Prob. 26ECh. 14.1 - Sketch the graph of the function. f(x,y)=sinxCh. 14.1 - Prob. 28ECh. 14.1 - Prob. 29ECh. 14.1 - Sketch the graph of the function. f(x,y)=4x2+y2Ch. 14.1 - Sketch the graph of the function. f(x,y)=44x2y2Ch. 14.1 - Match the function with its graph labeled IVI....Ch. 14.1 - Prob. 33ECh. 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. 37ECh. 14.1 - Prob. 38ECh. 14.1 - The body mass index BMI of a person is defined by...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 - 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. 45ECh. 14.1 - Prob. 46ECh. 14.1 - Prob. 47ECh. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Prob. 49ECh. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Prob. 51ECh. 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. 54ECh. 14.1 - Prob. 55ECh. 14.1 - Prob. 56ECh. 14.1 - Prob. 57ECh. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Prob. 59ECh. 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 - 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. 67ECh. 14.1 - Prob. 68ECh. 14.1 - Prob. 69ECh. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Prob. 71ECh. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Prob. 73ECh. 14.1 - Prob. 74ECh. 14.1 - Graph the function using various domains and...Ch. 14.1 - Prob. 76ECh. 14.1 - Prob. 77ECh. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Prob. 79ECh. 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 - Prob. 27ECh. 14.2 - Prob. 28ECh. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Prob. 31ECh. 14.2 - Prob. 32ECh. 14.2 - Prob. 33ECh. 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 - Prob. 43ECh. 14.2 - Prob. 44ECh. 14.2 - Prob. 45ECh. 14.2 - Prob. 46ECh. 14.3 - Prob. 1ECh. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - Prob. 3ECh. 14.3 - The wave heights h in the open sea depend on the...Ch. 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)=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. 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 - Find the first partial derivatives of the...Ch. 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 - Find the first partial derivatives of the...Ch. 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 - Prob. 42ECh. 14.3 - Prob. 43ECh. 14.3 - Prob. 44ECh. 14.3 - Prob. 45ECh. 14.3 - Prob. 46ECh. 14.3 - Prob. 47ECh. 14.3 - Prob. 48ECh. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Prob. 50ECh. 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. 53ECh. 14.3 - Prob. 54ECh. 14.3 - Prob. 55ECh. 14.3 - Prob. 56ECh. 14.3 - Prob. 57ECh. 14.3 - Prob. 58ECh. 14.3 - Prob. 59ECh. 14.3 - Prob. 60ECh. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Prob. 62ECh. 14.3 - Prob. 63ECh. 14.3 - Prob. 64ECh. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Prob. 66ECh. 14.3 - Prob. 67ECh. 14.3 - Prob. 68ECh. 14.3 - Prob. 69ECh. 14.3 - Find the indicated partial derivatives. u=xaybzc;...Ch. 14.3 - Prob. 71ECh. 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. 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 - Prob. 78ECh. 14.3 - If f and g are twice differentiable functions of a...Ch. 14.3 - Prob. 80ECh. 14.3 - Prob. 81ECh. 14.3 - Prob. 82ECh. 14.3 - Prob. 83ECh. 14.3 - Prob. 84ECh. 14.3 - Prob. 85ECh. 14.3 - Prob. 86ECh. 14.3 - Prob. 87ECh. 14.3 - Prob. 88ECh. 14.3 - For the ideal gas of Exercise 88, show that...Ch. 14.3 - Prob. 90ECh. 14.3 - Prob. 91ECh. 14.3 - Prob. 92ECh. 14.3 - In the project on page 271 we expressed the power...Ch. 14.3 - Prob. 94ECh. 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 - 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. 101ECh. 14.3 - Prob. 102ECh. 14.3 - If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0). Hint:...Ch. 14.3 - Prob. 104ECh. 14.3 - Prob. 105ECh. 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 - Draw the graph of f and its tangent plane at the...Ch. 14.4 - Prob. 10ECh. 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. 13ECh. 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. 23ECh. 14.4 - Prob. 24ECh. 14.4 - Find the differential of the function. z=e2xcos2tCh. 14.4 - Find the differential of the function. u=x2+3y2Ch. 14.4 - Find the differential of the function. m=p5q3Ch. 14.4 - Find the differential of the function. T=v1+uvwCh. 14.4 - Prob. 29ECh. 14.4 - Find the differential of the function. L=xzey2z2Ch. 14.4 - Prob. 31ECh. 14.4 - Prob. 32ECh. 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 - Prob. 37ECh. 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. 42ECh. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Prob. 45ECh. 14.4 - Prob. 46ECh. 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. 12ECh. 14.5 - Prob. 13ECh. 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. 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 - Prob. 23ECh. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use Equation 6 to find dy/dx. ycosx=x2+y2Ch. 14.5 - Use Equation 6 to find dy/dx. cos(xy)=1+sinyCh. 14.5 - Prob. 29ECh. 14.5 - Prob. 30ECh. 14.5 - Use Equations 7 to find z/x and z/y. x2+2y2+3z2=1Ch. 14.5 - Prob. 32ECh. 14.5 - Prob. 33ECh. 14.5 - Prob. 34ECh. 14.5 - The temperature at a point x, y is Tx, y, measured...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 - Prob. 40ECh. 14.5 - The pressure of 1 mole of an ideal gas is...Ch. 14.5 - Prob. 42ECh. 14.5 - Prob. 43ECh. 14.5 - Prob. 44ECh. 14.5 - Assume that all the given functions are...Ch. 14.5 - Prob. 46ECh. 14.5 - Prob. 47ECh. 14.5 - Prob. 48ECh. 14.5 - Prob. 49ECh. 14.5 - Prob. 50ECh. 14.5 - Prob. 51ECh. 14.5 - Prob. 52ECh. 14.5 - Prob. 53ECh. 14.5 - Prob. 54ECh. 14.5 - Prob. 55ECh. 14.5 - Prob. 56ECh. 14.5 - Prob. 57ECh. 14.5 - Suppose that the equation F(x,y,z)=0 implicitly...Ch. 14.5 - Prob. 59ECh. 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. 5ECh. 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. 8ECh. 14.6 - Prob. 9ECh. 14.6 - Prob. 10ECh. 14.6 - Prob. 11ECh. 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. 15ECh. 14.6 - Prob. 16ECh. 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. 19ECh. 14.6 - Prob. 20ECh. 14.6 - Prob. 21ECh. 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. 24ECh. 14.6 - Prob. 25ECh. 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. 32ECh. 14.6 - Prob. 33ECh. 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 - 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. 42ECh. 14.6 - Prob. 43ECh. 14.6 - Prob. 44ECh. 14.6 - Prob. 45ECh. 14.6 - Prob. 46ECh. 14.6 - Prob. 47ECh. 14.6 - Prob. 48ECh. 14.6 - If f(x,y)=xy, find the gradient vector f(3,2) and...Ch. 14.6 - Prob. 50ECh. 14.6 - Prob. 51ECh. 14.6 - Prob. 52ECh. 14.6 - Prob. 53ECh. 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. 57ECh. 14.6 - Show that every normal line to the sphere...Ch. 14.6 - Prob. 59ECh. 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 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. 66ECh. 14.6 - Prob. 67ECh. 14.6 - Prob. 68ECh. 14.6 - Suppose that the directional derivatives of f(x,y)...Ch. 14.6 - Prob. 70ECh. 14.7 - Prob. 1ECh. 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. 6ECh. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Prob. 8ECh. 14.7 - Prob. 9ECh. 14.7 - Prob. 10ECh. 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. 14ECh. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Prob. 16ECh. 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. 19ECh. 14.7 - Prob. 20ECh. 14.7 - Prob. 21ECh. 14.7 - Show that f(x,y)=x2yex2y2 has minimum 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 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Prob. 29ECh. 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 - Prob. 34ECh. 14.7 - Prob. 35ECh. 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. 38ECh. 14.7 - Prob. 39ECh. 14.7 - Prob. 40ECh. 14.7 - Find the shortest distance from the point (2,0,3)...Ch. 14.7 - Prob. 42ECh. 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 - Prob. 49ECh. 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. 56ECh. 14.7 - The Shannon index sometimes called the...Ch. 14.7 - Prob. 58ECh. 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. 2ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 4ECh. 14.8 - Prob. 5ECh. 14.8 - Prob. 6ECh. 14.8 - Prob. 7ECh. 14.8 - Prob. 8ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 10ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 12ECh. 14.8 - Prob. 13ECh. 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. 17ECh. 14.8 - Prob. 18ECh. 14.8 - Prob. 19ECh. 14.8 - Prob. 20ECh. 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. 24ECh. 14.8 - Consider the problem of minimizing the function...Ch. 14.8 - Prob. 26ECh. 14.8 - The total production P of a certain product...Ch. 14.8 - Prob. 28ECh. 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. 32ECh. 14.8 - Prob. 33ECh. 14.8 - Prob. 34ECh. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Prob. 36ECh. 14.8 - Prob. 37ECh. 14.8 - Prob. 38ECh. 14.8 - Prob. 39ECh. 14.8 - Prob. 40ECh. 14.8 - Prob. 41ECh. 14.8 - Prob. 42ECh. 14.8 - Prob. 43ECh. 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. 47ECh. 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. 1CCCh. 14.R - Prob. 2CCCh. 14.R - Prob. 3CCCh. 14.R - Prob. 4CCCh. 14.R - Prob. 5CCCh. 14.R - What does Clairauts Theorem say?Ch. 14.R - Prob. 7CCCh. 14.R - Prob. 8CCCh. 14.R - Prob. 9CCCh. 14.R - Prob. 10CCCh. 14.R - Prob. 11CCCh. 14.R - Prob. 12CCCh. 14.R - Prob. 13CCCh. 14.R - aDefine the gradient vector f for a function f of...Ch. 14.R - Prob. 15CCCh. 14.R - Prob. 16CCCh. 14.R - Prob. 17CCCh. 14.R - Prob. 18CCCh. 14.R - Prob. 19CCCh. 14.R - Prob. 1TFQCh. 14.R - Prob. 2TFQCh. 14.R - Prob. 3TFQCh. 14.R - Prob. 4TFQCh. 14.R - Prob. 5TFQCh. 14.R - Prob. 6TFQCh. 14.R - Prob. 7TFQCh. 14.R - Prob. 8TFQCh. 14.R - Prob. 9TFQCh. 14.R - Prob. 10TFQCh. 14.R - Prob. 11TFQCh. 14.R - Prob. 12TFQCh. 14.R - Prob. 1ECh. 14.R - Prob. 2ECh. 14.R - Prob. 3ECh. 14.R - Prob. 4ECh. 14.R - Prob. 5ECh. 14.R - Prob. 6ECh. 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. 9ECh. 14.R - Prob. 10ECh. 14.R - Prob. 11ECh. 14.R - Prob. 12ECh. 14.R - Prob. 13ECh. 14.R - Prob. 14ECh. 14.R - Prob. 15ECh. 14.R - Prob. 16ECh. 14.R - Prob. 17ECh. 14.R - Prob. 18ECh. 14.R - Prob. 19ECh. 14.R - Prob. 20ECh. 14.R - Prob. 21ECh. 14.R - Prob. 22ECh. 14.R - Prob. 23ECh. 14.R - Prob. 24ECh. 14.R - Prob. 25ECh. 14.R - Prob. 26ECh. 14.R - Prob. 27ECh. 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. 30ECh. 14.R - Find the points on the hyperboloid x2+4y2z2=4...Ch. 14.R - Prob. 32ECh. 14.R - Prob. 33ECh. 14.R - The two legs of a right triangle are measured as 5...Ch. 14.R - Prob. 35ECh. 14.R - Prob. 36ECh. 14.R - Suppose z=f(x,y), where x=g(s,t), y=h(s,t),...Ch. 14.R - Prob. 38ECh. 14.R - Prob. 39ECh. 14.R - Prob. 40ECh. 14.R - Prob. 41ECh. 14.R - Prob. 42ECh. 14.R - Prob. 43ECh. 14.R - Prob. 44ECh. 14.R - Prob. 45ECh. 14.R - Prob. 46ECh. 14.R - Prob. 47ECh. 14.R - Prob. 48ECh. 14.R - The contour map shows wind speed in knots during...Ch. 14.R - Prob. 50ECh. 14.R - Prob. 51ECh. 14.R - Prob. 52ECh. 14.R - Prob. 53ECh. 14.R - Prob. 54ECh. 14.R - Prob. 55ECh. 14.R - Prob. 56ECh. 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. 59ECh. 14.R - Prob. 60ECh. 14.R - Prob. 61ECh. 14.R - Prob. 62ECh. 14.R - Prob. 63ECh. 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. 3PCh. 14.P - For what values of the number r is the function...Ch. 14.P - Prob. 5PCh. 14.P - Prob. 6PCh. 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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- Find the derivative of the economic functions. Show full solutions and properly name the derivative. TR (Q) = 18Q − 0.5Q2 C (Y ) = 1500 + 0.75Y AC(Q)=1.5Q+4+46/Qarrow_forwardFind the partial derivative. f (p, q) = - Sp е я afarrow_forwardExample 2: Approximate the derivative of y=x´ at x=1 by forming small changes Solution :- JADY) = 5(x+AX)² +3(x+AXJ +=5(+2XDメ+泌+38 4り25メー+10XDX+5DX-2 y(1) = (1)° = 1 y(1.01) = (1.01)² =1.0201 Ay = 1.0201–1=0.0201 y+y =(5x x3X)+ DX (10 dy Ay 0.0201 = 2.01 Dy z D X\OX+5XAX dx Ar 0.01arrow_forward
- Determine the second-order partial derivative for the indicated function f(c,y) = cos(2x² – 3y?) axdyla-3)arrow_forwardIf y = f(x1.X2,u,w) where x1 = g(u.w) and x2 = h(u,w) which of the following is the correct expression for the total- partial derivative Sy/Sw? %3D Select one: о ду дх1 ду дх2 , ду Me Ixe dy дx2 , ду me Ixe дх, ду ду дх2 , ду Ixe ^p o me 2xp me Ixp de ду dх, , ду дх, dw me 2xe, Txe me dx,' dw 3D Ixp ke dy dx2 dx2 dw dx, dw mp +arrow_forwardSuppose the number of crimes z committed in a city depends only upon the number x of unemployed people and the size y of the police force. This oversimplifies the problem, but assume Z= f(x,y) x = number of unemployed y = size of police force. Explain in a sentence what the partial derivative of f with respect to x represents in the context of this problem. Include units in your explanation. Do you expect this partial to be positive or negative? Explain. Explain in a sentence what the partial derivative of f with respect to y represents in the context of this problem. Include units in your explanation. Do you expect this partial to be positive or negative? Explainarrow_forward
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