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Math Calculus Calculus: Early Transcendentals (3rd Edition)

Gradients in three dimensions Consider the following functions f, points P, and unit vectors u . a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 61. f ( x , y , z ) = ln ( 1 + x 2 + y 2 + z 2 ) ; P ( 1 , 1 , − 1 ) ; 〈 2 3 ′ 2 3 ′ − 1 3 〉

Gradients in three dimensions Consider the following functions f, points P, and unit vectors u . a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 61. f ( x , y , z ) = ln ( 1 + x 2 + y 2 + z 2 ) ; P ( 1 , 1 , − 1 ) ; 〈 2 3 ′ 2 3 ′ − 1 3 〉

Calculus: Early Transcendentals (3rd Edition)

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Textbook Question

Chapter 15.5, Problem 65E

Gradients in three dimensions Consider the following functions f, points P, and unit vectors u.

a. Compute the gradient of f and evaluate it at P
b. Find the unit vector in the direction of maximum increase of f at P.
c. Find the rate of change of the function in the direction of maximum increase at P.
d. Find the directional derivative at P in the direction of the given vector.

61. f ( x , y , z ) = ln ( 1 + x 2 + y 2 + z 2 ) ; P ( 1 , 1 , − 1 ) ; 〈 2 3 ′ 2 3 ′ − 1 3 〉

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Chapter 15.5, Problem 64E

Chapter 15.5, Problem 66E

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Chapter 15 Solutions

Calculus: Early Transcendentals (3rd Edition)

Ch. 15.1 - Find the domains of f(x, y) = sin xy and g(x, y) =...Ch. 15.1 - Does the graph of a hyperboloid of one sheet...Ch. 15.1 - Find a function whose graph is the lower half of...Ch. 15.1 - Can two level curves of a function intersect?...Ch. 15.1 - Prob. 5QC Ch. 15.1 - Prob. 6QC Ch. 15.1 - Prob. 7QC Ch. 15.1 - Prob. 8QC Ch. 15.1 - What is the domain of the function w = f(x, y, z)...Ch. 15.1 - What is domain of f(x, y) = x2y xy2?

Ch. 15.1 - What is the domain of g(x, y) = 1/(xy)?Ch. 15.1 - What is the domain of h(x,y)=xy?Ch. 15.1 - How many axes (or how many dimensions) are needed...Ch. 15.1 - Explain how to graph the level curves of a surface...Ch. 15.1 - Given the function f(x, y) = 10x+y, evaluate f(2,...Ch. 15.1 - Prob. 8E Ch. 15.1 - The function z = f(x, y) gives the elevation z (in...Ch. 15.1 - The function z = f(x, y) gives the elevation z (in...Ch. 15.1 - Describe in words the level curves of the...Ch. 15.1 - How many axes (or how many dimensions) are needed...Ch. 15.1 - The domain of Q = f(u, v, w, x, y, z) lies in n...Ch. 15.1 - Give two methods for graphically representing a...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Prob. 16E Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Domains Find the domain of the following...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Graphs of familiar functions Use what you learned...Ch. 15.1 - Matching level curves with surfaces Match surfaces...Ch. 15.1 - Matching surfaces Match functions ad with surfaces...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Level curves Graph several level curves of the...Ch. 15.1 - Earned run average A baseball pitchers earned run...Ch. 15.1 - Electric potential function The electric potential...Ch. 15.1 - Cobb-Douglas production function The output Q of...Ch. 15.1 - Resistors in parallel Two resistors wired in...Ch. 15.1 - Level curves of a savings account Suppose you make...Ch. 15.1 - Level curves of a savings plan Suppose you make...Ch. 15.1 - Domains of functions of three or more variables...Ch. 15.1 - Domains of functions of three or more variables...Ch. 15.1 - Domains of functions of three or more variables...Ch. 15.1 - Domains of functions of three or more variables...Ch. 15.1 - Domains of functions of three or more variables...Ch. 15.1 - Domains of functions of three or more variables...Ch. 15.1 - Prob. 56E Ch. 15.1 - Explain why or why not Determine whether the...Ch. 15.1 - Quarterback passer ratings One measurement of the...Ch. 15.1 - Ideal Gas Law Many gases can be modeled by the...Ch. 15.1 - Water waves A snapshot of a water wave moving...Ch. 15.1 - Approximate mountains Suppose the elevation of...Ch. 15.1 - Graphing functions a.Determine the domain and...Ch. 15.1 - Prob. 63E Ch. 15.1 - Prob. 64E Ch. 15.1 - Graphing functions a.Determine the domain and...Ch. 15.1 - Graphing functions a.Determine the domain and...Ch. 15.1 - Prob. 67E Ch. 15.1 - Prob. 68E Ch. 15.1 - Prob. 69E Ch. 15.1 - Prob. 70E Ch. 15.1 - Peaks and valleys The following functions have...Ch. 15.1 - Prob. 72E Ch. 15.1 - Prob. 73E Ch. 15.1 - Level surfaces Find an equation for the family of...Ch. 15.1 - Level surfaces Find an equation for the family of...Ch. 15.1 - Level surfaces Find an equation for the family of...Ch. 15.1 - Level surfaces Find an equation for the family of...Ch. 15.1 - Prob. 78E Ch. 15.1 - Challenge domains Find the domains of the...Ch. 15.1 - Prob. 80E Ch. 15.1 - Prob. 81E Ch. 15.1 - Prob. 82E Ch. 15.2 - Which of the following limits exist? Ch. 15.2 - Give an example of a set that contains none of its...Ch. 15.2 - Can the limit be evaluated by direct...Ch. 15.2 - What is the analog of the Two-Path Test for...Ch. 15.2 - Prob. 5QC Ch. 15.2 - Prob. 1E Ch. 15.2 - Explain why f(x, y) must approach a unique number...Ch. 15.2 - What does it mean to say that limits of...Ch. 15.2 - Suppose (a, b) is on the boundary of the domain of...Ch. 15.2 - Explain how examining limits along multiple paths...Ch. 15.2 - Explain why evaluating a limit along a finite...Ch. 15.2 - What three conditions must be met for a function f...Ch. 15.2 - Let R be the unit disk {(x, y): x2 + y2 1} with...Ch. 15.2 - At what points of 2 is a rational function of two...Ch. 15.2 - Prob. 10E Ch. 15.2 - Evaluate lim(x,y)(5,5)x2y2x+y Ch. 15.2 - Let f(x)=x22xy2+1x22xy2+1 Use the Two-Path Test to...Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Limits of functions Evaluate the following limits....Ch. 15.2 - Prob. 21E Ch. 15.2 - Limits at boundary points Evaluate the following...Ch. 15.2 - Limits at boundary points Evaluate the following...Ch. 15.2 - Limits at boundary points Evaluate the following...Ch. 15.2 - Limits at boundary points Evaluate the following...Ch. 15.2 - Prob. 26E Ch. 15.2 - Limits at boundary points Evaluate the following...Ch. 15.2 - Prob. 28E Ch. 15.2 - Prob. 29E Ch. 15.2 - Prob. 30E Ch. 15.2 - Nonexistence of limits Use the Two-Path Test to...Ch. 15.2 - Prob. 32E Ch. 15.2 - Nonexistence of limits Use the Two-Path Test to...Ch. 15.2 - Nonexistence of limits Use the Two-Path Test to...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity At what points of 2 are the following...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Continuity of composite functions At what points...Ch. 15.2 - Limits of functions of three variables Evaluate...Ch. 15.2 - Limits of functions of three variables Evaluate...Ch. 15.2 - Limits of functions of three variables Evaluate...Ch. 15.2 - Limits of functions of three variables Evaluate...Ch. 15.2 - Limits of functions of three variables Evaluate...Ch. 15.2 - Limits of functions of three variables Evaluate...Ch. 15.2 - Prob. 61E Ch. 15.2 - Miscellaneous limits Use the method of your choice...Ch. 15.2 - Miscellaneous limits Use the method of your choice...Ch. 15.2 - Miscellaneous limits Use the method of your choice...Ch. 15.2 - Prob. 65E Ch. 15.2 - Miscellaneous limits Use the method of your choice...Ch. 15.2 - Miscellaneous limits Use the method of your choice...Ch. 15.2 - Prob. 68E Ch. 15.2 - Miscellaneous limits Use the method of your choice...Ch. 15.2 - Prob. 70E Ch. 15.2 - Limits of composite functions Evaluate the...Ch. 15.2 - Limits of composite functions Evaluate the...Ch. 15.2 - Limits of composite functions Evaluate the...Ch. 15.2 - Prob. 74E Ch. 15.2 - Prob. 75E Ch. 15.2 - Prob. 76E Ch. 15.2 - Piecewise function Let...Ch. 15.2 - Prob. 78E Ch. 15.2 - Prob. 79E Ch. 15.2 - Prob. 80E Ch. 15.2 - Prob. 81E Ch. 15.2 - Prob. 82E Ch. 15.2 - Nonexistence of limits Show that...Ch. 15.2 - Prob. 84E Ch. 15.2 - Prob. 85E Ch. 15.2 - Limit proof Use the formal definition of a limit...Ch. 15.2 - Limit proof Use the formal definition of a limit...Ch. 15.2 - Proof of Limit Law 1 Use the formal definition of...Ch. 15.2 - Proof of Limit Law 3 Use the formal definition of...Ch. 15.3 - Compute fx and fy for f(x, y) = 2xy.Ch. 15.3 - Which of the following expressions are equivalent...Ch. 15.3 - Compute fxxx and f xxy for f(x, y) = x3y.Ch. 15.3 - Compute fxz and fzz for f(x, y, z) = xyz x2z +...Ch. 15.3 - Explain why, in Figure 15.33, the slopes of the...Ch. 15.3 - Suppose you are standing on the surface z = f(x,...Ch. 15.3 - Prob. 2E Ch. 15.3 - Prob. 3E Ch. 15.3 - Find fx and fy when f(x, y) = y8 + 2x6 + 2xy.Ch. 15.3 - Find fx and fy when f(x, y) = 3x2y + 2.Ch. 15.3 - Prob. 6E Ch. 15.3 - Verify that fxy = fyx. for f(x, y) = 2x3 + 3y2 +...Ch. 15.3 - Verify that fxy = fyx, for f(x, y) = xey.Ch. 15.3 - Find fx,, fy, and fz, for f(x, y, z) = xy + xz +...Ch. 15.3 - The volume of a right circular cylinder with...Ch. 15.3 - Prob. 11E Ch. 15.3 - Prob. 12E Ch. 15.3 - Prob. 13E Ch. 15.3 - Prob. 14E Ch. 15.3 - Prob. 15E Ch. 15.3 - Prob. 16E Ch. 15.3 - Prob. 17E Ch. 15.3 - Prob. 18E Ch. 15.3 - Prob. 19E Ch. 15.3 - Prob. 20E Ch. 15.3 - Prob. 21E Ch. 15.3 - Prob. 22E Ch. 15.3 - Prob. 23E Ch. 15.3 - Prob. 24E Ch. 15.3 - Prob. 25E Ch. 15.3 - Prob. 26E Ch. 15.3 - Prob. 27E Ch. 15.3 - Prob. 28E Ch. 15.3 - Prob. 29E Ch. 15.3 - Prob. 30E Ch. 15.3 - Partial derivatives Find the first partial...Ch. 15.3 - Prob. 32E Ch. 15.3 - Prob. 33E Ch. 15.3 - Prob. 34E Ch. 15.3 - Prob. 35E Ch. 15.3 - Miscellaneous partial derivatives Compute the...Ch. 15.3 - Prob. 37E Ch. 15.3 - Prob. 38E Ch. 15.3 - Prob. 39E Ch. 15.3 - Prob. 40E Ch. 15.3 - Prob. 41E Ch. 15.3 - Prob. 42E Ch. 15.3 - Prob. 43E Ch. 15.3 - Prob. 44E Ch. 15.3 - Prob. 45E Ch. 15.3 - Prob. 46E Ch. 15.3 - Prob. 47E Ch. 15.3 - Prob. 48E Ch. 15.3 - Prob. 49E Ch. 15.3 - Equality of mixed partial derivatives Verify that...Ch. 15.3 - Prob. 51E Ch. 15.3 - Equality of mixed partial derivatives Verify that...Ch. 15.3 - Prob. 53E Ch. 15.3 - Prob. 54E Ch. 15.3 - Prob. 55E Ch. 15.3 - Prob. 56E Ch. 15.3 - Prob. 57E Ch. 15.3 - Prob. 58E Ch. 15.3 - Prob. 59E Ch. 15.3 - Prob. 60E Ch. 15.3 - Prob. 61E Ch. 15.3 - Prob. 62E Ch. 15.3 - Prob. 63E Ch. 15.3 - Prob. 64E Ch. 15.3 - Prob. 65E Ch. 15.3 - Prob. 66E Ch. 15.3 - Prob. 67E Ch. 15.3 - Prob. 68E Ch. 15.3 - Gas law calculations Consider the Ideal Gas Law PV...Ch. 15.3 - Body mass index The body mass index (BMI) for an...Ch. 15.3 - Resistors in parallel Two resistors in an...Ch. 15.3 - Spherical caps The volume of the cap of a sphere...Ch. 15.3 - Heat equation The flow of hear along a thin...Ch. 15.3 - Heat equation The flow of hear along a thin...Ch. 15.3 - Heat equation The flow of hear along a thin...Ch. 15.3 - Prob. 76E Ch. 15.3 - Nondifferentiability? Consider the following...Ch. 15.3 - Nondifferentiability? Consider the following...Ch. 15.3 - Prob. 79E Ch. 15.3 - Prob. 80E Ch. 15.3 - Prob. 81E Ch. 15.3 - Prob. 82E Ch. 15.3 - Electric potential function The electric potential...Ch. 15.3 - Prob. 84E Ch. 15.3 - Prob. 85E Ch. 15.3 - Wave on a string Imagine a string that is fixed at...Ch. 15.3 - Wave equation Traveling waves (for example, water...Ch. 15.3 - Wave equation Traveling waves (for example, water...Ch. 15.3 - Wave equation Traveling waves (for example, water...Ch. 15.3 - Laplaces equation A classical equation of...Ch. 15.3 - Laplaces equation A classical equation of...Ch. 15.3 - Laplaces equation A classical equation of...Ch. 15.3 - Laplaces equation A classical equation of...Ch. 15.3 - Prob. 94E Ch. 15.3 - Differentiability Use the definition of...Ch. 15.3 - Nondifferentiability? Consider the following...Ch. 15.3 - Nondifferentiability? Consider the following...Ch. 15.3 - Prob. 98E Ch. 15.3 - Derivatives of an integral Let h be continuous for...Ch. 15.4 - Explain why Theorem 15.7 reduces to the Chain Rule...Ch. 15.4 - Suppose w = f(x, y, z), where x = g(s, t), y =...Ch. 15.4 - If Q is a function of w, x, y, and z, each of...Ch. 15.4 - Use the method of Example 5 to find dy/dx when...Ch. 15.4 - Suppose z = f(x, y), where x and y are functions...Ch. 15.4 - Let z be a function of x and y, while x and y are...Ch. 15.4 - Suppose w is a function of x, y and z, which are...Ch. 15.4 - Let z = f(x, y), x = g(s, t), and y = h(s, t)....Ch. 15.4 - Given that w = F(x, y, z), and x, y, and z are...Ch. 15.4 - Suppose F(x, y) = 0 and y is a differentiable...Ch. 15.4 - Evaluate dz/dt, where z = x2+y3, x = t2 and y = t,...Ch. 15.4 - Prob. 8E Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with one independent variable Use...Ch. 15.4 - Chain Rule with several independent variables Find...Ch. 15.4 - Chain Rule with several independent variables Find...Ch. 15.4 - Chain Rule with several independent variables Find...Ch. 15.4 - Chain Rule with several independent variables Find...Ch. 15.4 - Chain Rule with several independent variables Find...Ch. 15.4 - Prob. 24E Ch. 15.4 - Chain Rule with several independent variables Find...Ch. 15.4 - Prob. 26E Ch. 15.4 - Changing cylinder The volume of a right circular...Ch. 15.4 - Changing pyramid The volume of a pyramid with a...Ch. 15.4 - Derivative practice two ways Find the indicated...Ch. 15.4 - Derivative practice two ways Find the indicated...Ch. 15.4 - Making trees Use a tree diagram to write the...Ch. 15.4 - Prob. 32E Ch. 15.4 - Prob. 33E Ch. 15.4 - Prob. 34E Ch. 15.4 - Implicit differentiation Given the following...Ch. 15.4 - Implicit differentiation Given the following...Ch. 15.4 - Implicit differentiation Given the following...Ch. 15.4 - Implicit differentiation Given the following...Ch. 15.4 - Implicit differentiation Given the following...Ch. 15.4 - Implicit differentiation Given the following...Ch. 15.4 - Prob. 41E Ch. 15.4 - Prob. 42E Ch. 15.4 - Prob. 43E Ch. 15.4 - Prob. 44E Ch. 15.4 - Prob. 45E Ch. 15.4 - Prob. 46E Ch. 15.4 - Prob. 47E Ch. 15.4 - Prob. 48E Ch. 15.4 - Prob. 49E Ch. 15.4 - Derivative practice Find the indicated derivative...Ch. 15.4 - Derivative practice Find the indicated derivative...Ch. 15.4 - Derivative practice Find the indicated derivative...Ch. 15.4 - Derivative practice Find the indicated derivative...Ch. 15.4 - Prob. 54E Ch. 15.4 - Change on a line Suppose w=(x,y,z) and is the line...Ch. 15.4 - Prob. 56E Ch. 15.4 - Implicit differentiation with three variables Use...Ch. 15.4 - Prob. 58E Ch. 15.4 - Prob. 59E Ch. 15.4 - More than one way Let exyz = 2. Find zx and zy in...Ch. 15.4 - Walking on a surface Consider the following...Ch. 15.4 - Walking on a surface Consider the following...Ch. 15.4 - Walking on a surface Consider the following...Ch. 15.4 - Walking on a surface Consider the following...Ch. 15.4 - Conservation of energy A projectile with mass m is...Ch. 15.4 - Utility functions in economics Economists use...Ch. 15.4 - Constant volume tori The volume of a solid torus...Ch. 15.4 - Body surface area One of several empirical...Ch. 15.4 - The Ideal Gas Law The pressure, temperature, and...Ch. 15.4 - Prob. 70E Ch. 15.4 - Prob. 71E Ch. 15.4 - Change of coordinates Recall that Cartesian and...Ch. 15.4 - Change of coordinates continued An important...Ch. 15.4 - Prob. 75E Ch. 15.4 - Prob. 76E Ch. 15.4 - Prob. 77E Ch. 15.5 - Explain Why, when u = 1, 0 in the definition of...Ch. 15.5 - In the parametric description x = a + su1 and y =...Ch. 15.5 - In Example 1, evaluate Du f(3, 2) and Dv f(3, 2)....Ch. 15.5 - Draw a circle in the xy-plane centered at the...Ch. 15.5 - Prob. 5QC Ch. 15.5 - Prob. 6QC Ch. 15.5 - Prob. 1E Ch. 15.5 - How do you compute the gradient of the functions...Ch. 15.5 - Prob. 3E Ch. 15.5 - Prob. 4E Ch. 15.5 - Given a function f, explain the relationship...Ch. 15.5 - The level curves of the surface z=x2+y2 are...Ch. 15.5 - Suppose f is differentiable at (3, 4), f(3, 4) =...Ch. 15.5 - Suppose f is differentiable at (9, 9), f(9, 9) =...Ch. 15.5 - Suppose f is differentiable at (3, 4). Assume u,...Ch. 15.5 - Suppose f is differentiable at (1, 2) and ∇ f(1,...Ch. 15.5 - Directional derivatives Consider the function...Ch. 15.5 - Directional derivatives Consider the function...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing gradients Compute the gradient of the...Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Prob. 23E Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Prob. 26E Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Computing directional derivatives with the...Ch. 15.5 - Direction of steepest ascent and descent Consider...Ch. 15.5 - Direction of steepest ascent and descent Consider...Ch. 15.5 - Direction of steepest ascent and descent Consider...Ch. 15.5 - Direction of steepest ascent and descent Consider...Ch. 15.5 - Direction of steepest ascent and descent Consider...Ch. 15.5 - Direction of steepest ascent and descent Consider...Ch. 15.5 - Interpreting directional derivatives A function f...Ch. 15.5 - Interpreting directional derivatives A function f...Ch. 15.5 - Interpreting directional derivatives A function f...Ch. 15.5 - Interpreting directional derivatives A function f...Ch. 15.5 - Interpreting directional derivatives A function f...Ch. 15.5 - Interpreting directional derivatives A function f...Ch. 15.5 - Directions of change Consider the following...Ch. 15.5 - Prob. 44E Ch. 15.5 - Prob. 45E Ch. 15.5 - 43-46. Directions of change Consider the following...Ch. 15.5 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 15.5 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 15.5 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 15.5 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 15.5 - Level curves Consider the upper half of the...Ch. 15.5 - Level curves Consider the upper half of the...Ch. 15.5 - Level curves Consider the upper half of the...Ch. 15.5 - Prob. 54E Ch. 15.5 - Path of steepest descent Consider each of the...Ch. 15.5 - Path of steepest descent Consider each of the...Ch. 15.5 - Path of steepest descent Consider each of the...Ch. 15.5 - Path of steepest descent Consider each of the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Gradients in three dimensions Consider the...Ch. 15.5 - Explain why or why not Determine whether the...Ch. 15.5 - Gradient of a composite function Consider the...Ch. 15.5 - Directions of zero change Find the directions in...Ch. 15.5 - Prob. 70E Ch. 15.5 - Directions of zero change Find the directions in...Ch. 15.5 - Directions of zero change Find the directions in...Ch. 15.5 - Steepest ascent on a plane Suppose a long sloping...Ch. 15.5 - Gradient of a distance function Let (a, b) be a...Ch. 15.5 - Looking aheadtangent planes Consider the following...Ch. 15.5 - Prob. 76E Ch. 15.5 - Looking aheadtangent planes Consider the following...Ch. 15.5 - Prob. 78E Ch. 15.5 - Prob. 79E Ch. 15.5 - Prob. 80E Ch. 15.5 - Potential functions Potential functions arise...Ch. 15.5 - Potential functions Potential functions arise...Ch. 15.5 - Prob. 83E Ch. 15.5 - Prob. 84E Ch. 15.5 - Rules for gradients Use the definition of the...Ch. 15.5 - Prob. 86E Ch. 15.5 - Prob. 87E Ch. 15.5 - Prob. 88E Ch. 15.5 - Using gradient rules Use the gradient rules of...Ch. 15.5 - Using gradient rules Use the gradient rules of...Ch. 15.5 - Prob. 91E Ch. 15.6 - Write the function z = xy + x y in the form F(x,...Ch. 15.6 - Prob. 2QC Ch. 15.6 - Prob. 3QC Ch. 15.6 - Prob. 4QC Ch. 15.6 - Suppose n is a vector normal to the tangent plane...Ch. 15.6 - Write the explicit function z = xy2 + x2y 10 in...Ch. 15.6 - Write an equation for the plane tangent to the...Ch. 15.6 - Prob. 4E Ch. 15.6 - Explain how to approximate a function f at a point...Ch. 15.6 - Explain how to approximate the change in a...Ch. 15.6 - Write the approximate change formula for a...Ch. 15.6 - Write the differential dw for the function w =...Ch. 15.6 - Suppose f(1, 2) = 4, fx(1, 2) = 5, and fy(1, 2) =...Ch. 15.6 - Suppose f(l, 2) = 4, fx(1, 2) = 5, and fy(1, 2) =...Ch. 15.6 - Suppose F(0, 2, 1) = 0, Fx(0, 2, 1) = 3, Fy(0, 2,...Ch. 15.6 - Suppose F(0, 2, 1) = 0, Fx(0, 2, 1) = 3, Fy(0, 2,...Ch. 15.6 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 15.6 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 15.6 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 15.6 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 15.6 - Tangent planes for z = f (x, y) Find an equation...Ch. 15.6 - Tangent planes for z = f (x, y) Find an equation...Ch. 15.6 - Tangent planes for z = f (x, y) Find an equation...Ch. 15.6 - Tangent planes for z = f (x, y) Find an equation...Ch. 15.6 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 15.6 - Tangent planes for F(x, y, z) = 0 Find an equation...Ch. 15.6 - Tangent planes for F(x, y, z) = 0 Find an equation...Ch. 15.6 - Tangent planes for F(x, y, z) = 0 Find an equation...Ch. 15.6 - Tangent planes for z = f (x, y) Find an equation...Ch. 15.6 - Prob. 26E Ch. 15.6 - Tangent planes for z = f (x, y) Find an equation...Ch. 15.6 - Tangent planes for z = f (x, y) Find an equation...Ch. 15.6 - Tangent planes Find an equation of the plane...Ch. 15.6 - Tangent planes Find an equation of the plane...Ch. 15.6 - Tangent planes Find an equation of the plane...Ch. 15.6 - Tangent planes Find an equation of the plane...Ch. 15.6 - Linear approximation a.Find the linear...Ch. 15.6 - Linear approximation a.Find the linear...Ch. 15.6 - Linear approximation a.Find the linear...Ch. 15.6 - Linear approximation a.Find the linear...Ch. 15.6 - Linear Approximation a. Find the linear...Ch. 15.6 - Linear Approximation a. Find the linear...Ch. 15.6 - Approximate function change Use differentials to...Ch. 15.6 - Approximate function change Use differentials to...Ch. 15.6 - Approximate function change Use differentials to...Ch. 15.6 - Approximate function change Use differentials to...Ch. 15.6 - Changes in torus surface area The surface area of...Ch. 15.6 - Changes in cone volume The volume of a right...Ch. 15.6 - Area of an ellipse The area of an ellipse with...Ch. 15.6 - Volume of a paraboloid The volume of a segment of...Ch. 15.6 - Differentials with more than two variables Write...Ch. 15.6 - Differentials with more than two variables Write...Ch. 15.6 - Differentials with more than two variables Write...Ch. 15.6 - Differentials with more than two variables Write...Ch. 15.6 - Law of Cosines The side lengths of any triangle...Ch. 15.6 - Explain why or why not Determine whether the...Ch. 15.6 - Horizontal tangent planes Find the points at which...Ch. 15.6 - Horizontal tangent planes Find the points at which...Ch. 15.6 - Horizontal tangent planes Find the points at which...Ch. 15.6 - Horizontal tangent planes Find the points at which...Ch. 15.6 - Prob. 58E Ch. 15.6 - Surface area of a cone A cone with height h and...Ch. 15.6 - Line tangent to an intersection curve Consider the...Ch. 15.6 - Water-level changes A conical tank with radius...Ch. 15.6 - Prob. 63E Ch. 15.6 - Floating-point operations In general, real numbers...Ch. 15.6 - Probability of at least one encounter Suppose that...Ch. 15.6 - Two electrical resistors When two electrical...Ch. 15.6 - Three electrical resistors Extending Exercise 66,...Ch. 15.6 - Prob. 68E Ch. 15.6 - Logarithmic differentials Let f be a...Ch. 15.7 - The parabola z = x2 + y2 4x + 2y + 5 has a local...Ch. 15.7 - Consider the plane tangent to a surface at a...Ch. 15.7 - Compute the discriminant D(x, y) of f(x, y) =...Ch. 15.7 - Does the linear function f(x, y) = 2x + 3y have an...Ch. 15.7 - Describe the appearance of a smooth surface with a...Ch. 15.7 - Describe the usual appearance of a smooth surface...Ch. 15.7 - What are the conditions for a critical point of a...Ch. 15.7 - If fx (a, b) = fy (a, b) = 0, does it follow the f...Ch. 15.7 - Consider the function z = f(x, y). What is the...Ch. 15.7 - Prob. 6E Ch. 15.7 - What is an absolute minimum value of a function f...Ch. 15.7 - What is the procedure for locating absolute...Ch. 15.7 - Assume the second derivatives of fare continuous...Ch. 15.7 - Assume the second derivatives of fare continuous...Ch. 15.7 - Assume the second derivatives of fare continuous...Ch. 15.7 - Assume the second derivatives of fare continuous...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Critical points Find all critical points of the...Ch. 15.7 - Prob. 23E Ch. 15.7 - Prob. 24E Ch. 15.7 - Prob. 25E Ch. 15.7 - Prob. 26E Ch. 15.7 - Prob. 27E Ch. 15.7 - Prob. 28E Ch. 15.7 - Analyzing critical points Find the critical points...Ch. 15.7 - Prob. 30E Ch. 15.7 - Analyzing critical points Find the critical points...Ch. 15.7 - Analyzing critical points Find the critical points...Ch. 15.7 - Prob. 33E Ch. 15.7 - Prob. 34E Ch. 15.7 - Analyzing critical points Find the critical points...Ch. 15.7 - Analyzing critical points Find the critical points...Ch. 15.7 - Prob. 37E Ch. 15.7 - Prob. 38E Ch. 15.7 - Analyzing critical points Find the critical points...Ch. 15.7 - Prob. 40E Ch. 15.7 - Inconclusive tests Show that the Second Derivative...Ch. 15.7 - Inconclusive tests Show that the Second Derivative...Ch. 15.7 - Shipping regulations A shipping company handles...Ch. 15.7 - Cardboard boxes A lidless box is to be made using...Ch. 15.7 - Cardboard boxes A lidless cardboard box is to be...Ch. 15.7 - Optimal box Find the dimensions of the largest...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Prob. 55E Ch. 15.7 - Absolute maxima and minima Find the absolute...Ch. 15.7 - Pectin Extraction An increase in world production...Ch. 15.7 - Absolute extrema on open and / or unbounded...Ch. 15.7 - Absolute extrema on open and / or unbounded...Ch. 15.7 - Absolute extrema on open and / or unbounded...Ch. 15.7 - Absolute extrema on open and / or unbounded...Ch. 15.7 - Absolute extrema on open and/or unbounded regions...Ch. 15.7 - Lease distance What point on the plane x y + z =...Ch. 15.7 - Absolute extrema on open and/or unbounded regions...Ch. 15.7 - Absolute extrema on open and/or unbounded regions...Ch. 15.7 - Absolute extrema on open and / or unbounded...Ch. 15.7 - Explain why or why not Determine whether the...Ch. 15.7 - Prob. 68E Ch. 15.7 - Extreme points from contour plots Based on the...Ch. 15.7 - Optimal box Find the dimensions of the rectangular...Ch. 15.7 - Magic triples Let x, y, and z be nonnegative...Ch. 15.7 - Maximum/minimum of linear functions Let R be a...Ch. 15.7 - Prob. 73E Ch. 15.7 - Least squares approximation In its many guises,...Ch. 15.7 - Least squares approximation In its many guises,...Ch. 15.7 - Prob. 76E Ch. 15.7 - Prob. 77E Ch. 15.7 - Second Derivative Test Suppose the conditions of...Ch. 15.7 - Maximum area triangle Among all triangles with a...Ch. 15.7 - Slicing plane Find an equation of the plane...Ch. 15.7 - Solitary critical points A function of one...Ch. 15.7 - Two mountains without a saddle Show that the...Ch. 15.7 - Powers and roots Assume that x + y + z = 1 with x ...Ch. 15.7 - Ellipsoid inside a tetrahedron (1946 Putnam Exam)...Ch. 15.8 - It can be shown that the function f(x, y) = x2 +...Ch. 15.8 - Prob. 2QC Ch. 15.8 - Prob. 3QC Ch. 15.8 - In Figure 15.85, explain why, if you move away...Ch. 15.8 - Explain why, at a point that maximizes or...Ch. 15.8 - Describe the steps used to find the absolute...Ch. 15.8 - Prob. 3E Ch. 15.8 - Prob. 4E Ch. 15.8 - Graphical Lagrange multipliers The following...Ch. 15.8 - Graphical Lagrange multipliers The following...Ch. 15.8 - Lagrange multipliers in two variables Use Lagrange...Ch. 15.8 - Lagrange multipliers in two variables Use Lagrange...Ch. 15.8 - Lagrange multipliers in two variables Use Lagrange...Ch. 15.8 - Lagrange multipliers in two variables Use Lagrange...Ch. 15.8 - Lagrange multipliers in two variables Use Lagrange...Ch. 15.8 - Lagrange multipliers in two variables Use Lagrange...Ch. 15.8 - Lagrange multipliers Each function f has an...Ch. 15.8 - Lagrange multipliers Each function f has an...Ch. 15.8 - Lagrange multipliers Each function f has an...Ch. 15.8 - Lagrange multipliers Each function f has an...Ch. 15.8 - Lagrange multipliers in three variables Use...Ch. 15.8 - Lagrange multipliers in three variables Use...Ch. 15.8 - Lagrange multipliers in three variables Use...Ch. 15.8 - Lagrange multipliers in three variables Use...Ch. 15.8 - Lagrange multipliers Each function f has an...Ch. 15.8 - Lagrange multipliers in three variables Use...Ch. 15.8 - Lagrange multipliers in three variables Use...Ch. 15.8 - Lagrange multipliers Each function f has an...Ch. 15.8 - Lagrange multipliers Each function f has an...Ch. 15.8 - Lagrange multipliers in three variables Use...Ch. 15.8 - Applications of Lagrange multipliers Use Lagrange...Ch. 15.8 - Prob. 28E Ch. 15.8 - Prob. 29E Ch. 15.8 - Prob. 30E Ch. 15.8 - Prob. 31E Ch. 15.8 - Prob. 32E Ch. 15.8 - Prob. 33E Ch. 15.8 - Prob. 34E Ch. 15.8 - Prob. 35E Ch. 15.8 - Applications of Lagrange multipliers Use Lagrange...Ch. 15.8 - Maximizing utility functions Find the values of l...Ch. 15.8 - Maximizing utility functions Find the values of l...Ch. 15.8 - Maximizing utility functions Find the values of l...Ch. 15.8 - Maximizing utility functions Find the values of l...Ch. 15.8 - Explain why or why not Determine whether the...Ch. 15.8 - Prob. 42E Ch. 15.8 - Alternative method Solve the following problem...Ch. 15.8 - Prob. 44E Ch. 15.8 - Prob. 45E Ch. 15.8 - Prob. 46E Ch. 15.8 - Alternative method Solve the following problems...Ch. 15.8 - Prob. 48E Ch. 15.8 - Absolute maximum and minimum values Find the...Ch. 15.8 - Prob. 50E Ch. 15.8 - Absolute maximum and minimum values Find the...Ch. 15.8 - Extreme points on flattened spheres The equation...Ch. 15.8 - Production functions Economists model the output...Ch. 15.8 - Production functions Economists model the output...Ch. 15.8 - Production functions Economists model the output...Ch. 15.8 - Temperature of an elliptical plate The temperature...Ch. 15.8 - Maximizing a sum 57.Find the maximum value of x1 +...Ch. 15.8 - Prob. 58E Ch. 15.8 - Prob. 59E Ch. 15.8 - Geometric and arithmetic means Given positive...Ch. 15.8 - Problems with two constraints Given a...Ch. 15.8 - Prob. 62E Ch. 15.8 - Two-constraint problems Use the result of Exercise...Ch. 15.8 - Two-constraint problems Use the result of Exercise...Ch. 15.8 - Check assumptions Consider the function f(x, y) =...Ch. 15 - Prob. 1RE Ch. 15 - Prob. 2RE Ch. 15 - Prob. 3RE Ch. 15 - Prob. 4RE Ch. 15 - Prob. 5RE Ch. 15 - Graphs Describe the graph of the following...Ch. 15 - Graphs Describe the graph of the following...Ch. 15 - Level curves Make a sketch of several level curves...Ch. 15 - Level curves Make a sketch of several level curves...Ch. 15 - Matching level curves with surfaces Match level...Ch. 15 - Prob. 11RE Ch. 15 - Prob. 12RE Ch. 15 - Prob. 13RE Ch. 15 - Prob. 14RE Ch. 15 - Prob. 15RE Ch. 15 - Prob. 16RE Ch. 15 - Prob. 17RE Ch. 15 - Prob. 18RE Ch. 15 - Prob. 19RE Ch. 15 - Prob. 20RE Ch. 15 - Prob. 21RE Ch. 15 - Prob. 22RE Ch. 15 - Prob. 23RE Ch. 15 - Prob. 24RE Ch. 15 - Prob. 25RE Ch. 15 - Prob. 26RE Ch. 15 - Prob. 27RE Ch. 15 - Prob. 28RE Ch. 15 - Prob. 29RE Ch. 15 - Laplaces equation Verify that the following...Ch. 15 - Prob. 31RE Ch. 15 - Chain Rule Use the Chain Rule to evaluate the...Ch. 15 - Chain Rule Use the Chain Rule to evaluate the...Ch. 15 - Chain Rule Use the Chain Rule to evaluate the...Ch. 15 - Prob. 35RE Ch. 15 - Implicit differentiation Find dy/dx for the...Ch. 15 - Implicit differentiation Find dy/dx for the...Ch. 15 - Walking on a surface Consider the following...Ch. 15 - Walking on a surface Consider the following...Ch. 15 - Constant volume cones Suppose the radius of a...Ch. 15 - Directional derivatives Consider the function f(x,...Ch. 15 - Computing gradients Compute the gradient of the...Ch. 15 - Computing gradients Compute the gradient of the...Ch. 15 - Computing gradients Compute the gradient of the...Ch. 15 - Computing gradients Compute the gradient of the...Ch. 15 - Computing directional derivatives Compute the...Ch. 15 - Computing directional derivatives Compute the...Ch. 15 - Direction of steepest ascent and descent a.Find...Ch. 15 - Prob. 49RE Ch. 15 - Level curves Let f(x, y) = 8 2x2 y2. For the...Ch. 15 - Level curves Let f(x, y) = 8 2x2 y2. For the...Ch. 15 - Prob. 52RE Ch. 15 - Prob. 53RE Ch. 15 - Tangent planes Find an equation of the plane...Ch. 15 - Tangent planes Find an equation of the plane...Ch. 15 - Prob. 56RE Ch. 15 - Tangent planes Find an equation of the plane...Ch. 15 - Tangent planes Find an equation of the plane...Ch. 15 - Tangent planes Find an equation of the plane...Ch. 15 - Linear approximation a.Find the linear...Ch. 15 - Linear approximation a.Find the linear...Ch. 15 - Changes in a function Estimate the change in the...Ch. 15 - Volume of a cylinder The volume of a cylinder with...Ch. 15 - Volume of an ellipsoid The volume of an ellipsoid...Ch. 15 - Water-level changes A hemispherical tank with a...Ch. 15 - Prob. 66RE Ch. 15 - Analyzing critical points Identify the critical...Ch. 15 - Analyzing critical points Identify the critical...Ch. 15 - Analyzing critical points Identify the critical...Ch. 15 - Absolute maxima and minima Find the absolute...Ch. 15 - Absolute maxima and minima Find the absolute...Ch. 15 - Prob. 72RE Ch. 15 - Absolute maxima and minima Find the absolute...Ch. 15 - Prob. 74RE Ch. 15 - Lagrange multipliers Use Lagrange multipliers to...Ch. 15 - Prob. 76RE Ch. 15 - Lagrange multipliers Use Lagrange multipliers to...Ch. 15 - Lagrange multipliers Use Lagrange multipliers to...Ch. 15 - Maximum perimeter rectangle Use Lagrange...Ch. 15 - Minimum surface area cylinder Use Lagrange...Ch. 15 - Minimum distance to a cone Find the point(s) on...Ch. 15 - Prob. 82RE Ch. 15 - Prob. 83RE

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