Explanation 1) Concept: Use the definition of partial derivative of f with respect to x at a , b to calculate the partial derivative ∂ R ∂ L , ∂ R ∂ r , and use the given information to interpret them. 2) Definition: Partial derivative of f with respect to x : f x a , b = g ' a where g x = f x , b 3) Given: One of Poiseuille’s laws states that the resistance of blood flowing through an artery is R = C L r 4 , where L and r are the length and radius of the artery and C is a positive constant determined by the viscosity of the blood. 4) Calculation: R = C L r 4 … 1 By definition of partial derivative of f with respect to x Differentiate R partially with respect to L ∂ R ∂ L = ∂ ∂ L C L r 4 = C r 4 ∂ ∂ L L = C r 4 Therefore, ∂ R ∂ L = C r 4 ∂ R ∂ L = C r 4 ∂ R ∂ L = C r 4 is the rate at which the resistance of the flowing blood through an artery ( R ) is increase with respect to the length of the artery L when the radius of the artery r stays constant

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Chapter 14.3, Problem 92E

To determine

To calculate and interpret:

(a) ∂R∂L

(b) ∂R∂r

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Chapter 14.3, Problem 91E

Chapter 14.3, Problem 93E

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A graph of the function f is given below: Study the graph of ƒ at the value given below. Select each of the following that applies for the value a = 1 Of is defined at a. If is not defined at x = a. Of is continuous at x = a. If is discontinuous at x = a. Of is smooth at x = a. Of is not smooth at = a. If has a horizontal tangent line at = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. If has no tangent line at x = a. f(a + h) - f(a) lim is finite. h→0 h f(a + h) - f(a) lim h->0+ and lim h h->0- f(a + h) - f(a) h are infinite. lim does not exist. h→0 f(a+h) - f(a) h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

The graph below is the function f(z) 4 3 -2 -1 -1 1 2 3 -3 Consider the function f whose graph is given above. (A) Find the following. If a function value is undefined, enter "undefined". If a limit does not exist, enter "DNE". If a limit can be represented by -∞o or ∞o, then do so. lim f(z) +3 lim f(z) 1-1 lim f(z) f(1) = 2 = -4 = undefined lim f(z) 1 2-1 lim f(z): 2-1+ lim f(x) 2+1 -00 = -2 = DNE f(-1) = -2 lim f(z) = -2 1-4 lim f(z) 2-4° 00 f'(0) f'(2) = = (B) List the value(s) of x for which f(x) is discontinuous. Then list the value(s) of x for which f(x) is left- continuous or right-continuous. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5). If there are none, enter "none". Discontinuous at z = Left-continuous at x = Invalid use of a comma.syntax incomplete. Right-continuous at z = Invalid use of a comma.syntax incomplete. (C) List the value(s) of x for which f(x) is non-differentiable. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5).…

A graph of the function f is given below: Study the graph of f at the value given below. Select each of the following that applies for the value a = -4. f is defined at = a. f is not defined at 2 = a. If is continuous at x = a. Of is discontinuous at x = a. Of is smooth at x = a. f is not smooth at x = a. If has a horizontal tangent line at x = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. Of has no tangent line at x = a. f(a + h) − f(a) h lim is finite. h→0 f(a + h) - f(a) lim is infinite. h→0 h f(a + h) - f(a) lim does not exist. h→0 h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

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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