Calculus: Early Transcendentals (2nd Edition)
2nd Edition
ISBN: 9780321947345
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Textbook Question
Chapter 12.6, Problem 34E
Interpreting directional derivatives A function f and a point P are given. Let θ correspond to the direction of the directional derivative.
- a. Find the gradient and evaluate it at P.
- b. Find the angles θ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change.
- c. Write the directional derivative at P as a function of θ; call this function g.
- d. Find the value of θ that maximizes g(θ) and find the maximum value.
- e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient.
33.
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Chapter 12 Solutions
Calculus: Early Transcendentals (2nd Edition)
Ch. 12.1 - Give two pieces of information which, taken...Ch. 12.1 - Find a vector normal to the plane 2x 3y + 4z =...Ch. 12.1 - Where does the plane 2x 3y + 4z = 12 intersect...Ch. 12.1 - Give an equation of the plane with a normal vector...Ch. 12.1 - To which coordinate axes are the following...Ch. 12.1 - Describe the graph of x = z2 in 3.Ch. 12.1 - What is a trace of a surface?Ch. 12.1 - What is the name of the surface defined by the...Ch. 12.1 - What is the name of the surface defined by the...Ch. 12.1 - What is the name of the surface defined by the...
Ch. 12.1 - Equations of planes Find an equation of the plane...Ch. 12.1 - Prob. 12ECh. 12.1 - Equations of planes Find an equation of the plane...Ch. 12.1 - Equations of planes Find an equation of the plane...Ch. 12.1 - Equation of a plane Find an equation of the plane...Ch. 12.1 - Equation of a plane Find an equation of the plane...Ch. 12.1 - Equations of planes Find an equation of the...Ch. 12.1 - Equations of planes Find an equation of the...Ch. 12.1 - Equations of planes Find an equation of the...Ch. 12.1 - Equations of planes Find an equation of the...Ch. 12.1 - Properties of planes Find the points at which the...Ch. 12.1 - Prob. 22ECh. 12.1 - Properties of planes Find the points at which the...Ch. 12.1 - Prob. 24ECh. 12.1 - Pairs of planes Determine whether the following...Ch. 12.1 - Pairs of planes Determine whether the following...Ch. 12.1 - Pairs of planes Determine whether the following...Ch. 12.1 - Pairs of planes Determine whether the following...Ch. 12.1 - Equations of planes For the following sets of...Ch. 12.1 - Equations of planes For the following sets of...Ch. 12.1 - Parallel planes Find an equation of the plane...Ch. 12.1 - Prob. 32ECh. 12.1 - Prob. 33ECh. 12.1 - Prob. 34ECh. 12.1 - Intersecting planes Find an equation of the line...Ch. 12.1 - Intersecting planes Find an equation of the line...Ch. 12.1 - Intersecting planes Find an equation of the line...Ch. 12.1 - Intersecting planes Find an equation of the line...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Cylinders in 3 Consider the following cylinders in...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Prob. 54ECh. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Prob. 58ECh. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Prob. 64ECh. 12.1 - Prob. 65ECh. 12.1 - Prob. 66ECh. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Prob. 68ECh. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Quadric surfaces Consider the following equations...Ch. 12.1 - Explain why or why not Determine whether the...Ch. 12.1 - Prob. 72ECh. 12.1 - Lines normal to planes Find an equation of the...Ch. 12.1 - Lines normal to planes Find an equation of the...Ch. 12.1 - Prob. 75ECh. 12.1 - Orthogonal plane Find an equation of the plane...Ch. 12.1 - Three intersecting planes Describe the set of all...Ch. 12.1 - Three intersecting planes Describe the set of all...Ch. 12.1 - Matching graphs with equations Match equations af...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Identifying surfaces Identify and briefly describe...Ch. 12.1 - Prob. 90ECh. 12.1 - Prob. 91ECh. 12.1 - Prob. 92ECh. 12.1 - Prob. 93ECh. 12.1 - Prob. 94ECh. 12.1 - Angle between planes The angle between two planes...Ch. 12.1 - Prob. 96ECh. 12.1 - Light cones The idea of a light cone appears in...Ch. 12.1 - Prob. 100ECh. 12.1 - Prob. 102ECh. 12.1 - Prob. 103ECh. 12.1 - Prob. 104ECh. 12.2 - What is domain of f(x, y) = x2y xy2?Ch. 12.2 - What is the domain of g(x, y) = 1/(xy)?Ch. 12.2 - What is the domain of h(x,y)=xy?Ch. 12.2 - How many axes (or how many dimensions) are needed...Ch. 12.2 - Explain how to graph the level curves of a surface...Ch. 12.2 - Describe in words the level curves of the...Ch. 12.2 - How many axes (or how many dimensions) are needed...Ch. 12.2 - The domain of Q = f(u, v, w, x, y, z) lies in n...Ch. 12.2 - Give two methods for graphically representing a...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Prob. 12ECh. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Domains Find the domain of the following...Ch. 12.2 - Graphs of familiar functions Use what you learned...Ch. 12.2 - Graphs of familiar functions Use what you learned...Ch. 12.2 - Graphs of familiar functions Use what you learned...Ch. 12.2 - Graphs of familiar functions Use what you learned...Ch. 12.2 - Graphs of familiar functions Use what you learned...Ch. 12.2 - Graphs of familiar functions Use what you learned...Ch. 12.2 - Graphs of familiar functions Use what you learned...Ch. 12.2 - Prob. 28ECh. 12.2 - Matching surfaces Match functions ad with surfaces...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Level curves Graph several level curves of the...Ch. 12.2 - Matching level curves with surfaces Match surfaces...Ch. 12.2 - Prob. 39ECh. 12.2 - Earned run average A baseball pitchers earned run...Ch. 12.2 - Electric potential function The electric potential...Ch. 12.2 - Cobb-Douglas production function The output Q of...Ch. 12.2 - Resistors in parallel Two resistors wired in...Ch. 12.2 - Water waves A snapshot of a water wave moving...Ch. 12.2 - Approximate mountains Suppose the elevation of...Ch. 12.2 - Domains of functions of three or more variables...Ch. 12.2 - Domains of functions of three or more variables...Ch. 12.2 - Domains of functions of three or more variables...Ch. 12.2 - Domains of functions of three or more variables...Ch. 12.2 - Domains of functions of three or more variables...Ch. 12.2 - Domains of functions of three or more variables...Ch. 12.2 - Prob. 52ECh. 12.2 - Explain why or why not Determine whether the...Ch. 12.2 - Graphing functions a.Determine the domain and...Ch. 12.2 - Prob. 55ECh. 12.2 - Prob. 56ECh. 12.2 - Graphing functions a.Determine the domain and...Ch. 12.2 - Graphing functions a.Determine the domain and...Ch. 12.2 - Prob. 59ECh. 12.2 - Prob. 60ECh. 12.2 - Prob. 61ECh. 12.2 - Prob. 62ECh. 12.2 - Peaks and valleys The following functions have...Ch. 12.2 - Prob. 64ECh. 12.2 - Prob. 65ECh. 12.2 - Level surfaces Find an equation for the family of...Ch. 12.2 - Level surfaces Find an equation for the family of...Ch. 12.2 - Level surfaces Find an equation for the family of...Ch. 12.2 - Level surfaces Find an equation for the family of...Ch. 12.2 - Level curves of a savings account Suppose you make...Ch. 12.2 - Level curves of a savings plan Suppose you make...Ch. 12.2 - Prob. 72ECh. 12.2 - Ideal Gas Law Many gases can be modeled by the...Ch. 12.2 - Prob. 74ECh. 12.2 - Challenge domains Find the domains of the...Ch. 12.2 - Prob. 76ECh. 12.2 - Prob. 77ECh. 12.2 - Prob. 78ECh. 12.3 - Prob. 1ECh. 12.3 - Explain why f(x, y) must approach a unique number...Ch. 12.3 - What does it mean to say that limits of...Ch. 12.3 - Suppose (a, b) is on the boundary of the domain of...Ch. 12.3 - Explain how examining limits along multiple paths...Ch. 12.3 - Explain why evaluating a limit along a finite...Ch. 12.3 - What three conditions must be met for a function f...Ch. 12.3 - Let R be the unit disk {(x, y): x2 + y2 1} with...Ch. 12.3 - At what points of 2 is a rational function of two...Ch. 12.3 - Prob. 10ECh. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Limits of functions Evaluate the following limits....Ch. 12.3 - Prob. 19ECh. 12.3 - Limits at boundary points Evaluate the following...Ch. 12.3 - Limits at boundary points Evaluate the following...Ch. 12.3 - Limits at boundary points Evaluate the following...Ch. 12.3 - Limits at boundary points Evaluate the following...Ch. 12.3 - Prob. 24ECh. 12.3 - Limits at boundary points Evaluate the following...Ch. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Nonexistence of limits Use the Two-Path Test to...Ch. 12.3 - Prob. 30ECh. 12.3 - Nonexistence of limits Use the Two-Path Test to...Ch. 12.3 - Nonexistence of limits Use the Two-Path Test to...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity At what points of 2 are the following...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Continuity of composite functions At what points...Ch. 12.3 - Limits of functions of three variables Evaluate...Ch. 12.3 - Limits of functions of three variables Evaluate...Ch. 12.3 - Limits of functions of three variables Evaluate...Ch. 12.3 - Limits of functions of three variables Evaluate...Ch. 12.3 - Limits of functions of three variables Evaluate...Ch. 12.3 - Limits of functions of three variables Evaluate...Ch. 12.3 - Prob. 59ECh. 12.3 - Miscellaneous limits Use the method of your choice...Ch. 12.3 - Miscellaneous limits Use the method of your choice...Ch. 12.3 - Miscellaneous limits Use the method of your choice...Ch. 12.3 - Prob. 63ECh. 12.3 - Miscellaneous limits Use the method of your choice...Ch. 12.3 - Miscellaneous limits Use the method of your choice...Ch. 12.3 - Prob. 66ECh. 12.3 - Miscellaneous limits Use the method of your choice...Ch. 12.3 - Prob. 68ECh. 12.3 - Prob. 69ECh. 12.3 - Prob. 70ECh. 12.3 - Prob. 71ECh. 12.3 - Prob. 72ECh. 12.3 - Piecewise function Let...Ch. 12.3 - Prob. 74ECh. 12.3 - Nonexistence of limits Show that...Ch. 12.3 - Prob. 76ECh. 12.3 - Limits of composite functions Evaluate the...Ch. 12.3 - Prob. 78ECh. 12.3 - Limits of composite functions Evaluate the...Ch. 12.3 - Limits of composite functions Evaluate the...Ch. 12.3 - Prob. 81ECh. 12.3 - Limit proof Use the formal definition of a limit...Ch. 12.3 - Limit proof Use the formal definition of a limit...Ch. 12.3 - Proof of Limit Law 1 Use the formal definition of...Ch. 12.3 - Proof of Limit Law 3 Use the formal definition of...Ch. 12.4 - Suppose you are standing on the surface z = f(x,...Ch. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12.4 - The volume of a right circular cylinder with...Ch. 12.4 - Prob. 7ECh. 12.4 - Prob. 8ECh. 12.4 - Prob. 9ECh. 12.4 - Prob. 10ECh. 12.4 - Prob. 11ECh. 12.4 - Prob. 12ECh. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - Prob. 18ECh. 12.4 - Prob. 19ECh. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Prob. 25ECh. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Prob. 28ECh. 12.4 - Prob. 29ECh. 12.4 - Prob. 30ECh. 12.4 - Prob. 31ECh. 12.4 - Prob. 32ECh. 12.4 - Prob. 33ECh. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Prob. 36ECh. 12.4 - Prob. 37ECh. 12.4 - Prob. 38ECh. 12.4 - Prob. 39ECh. 12.4 - Prob. 40ECh. 12.4 - Prob. 41ECh. 12.4 - Prob. 42ECh. 12.4 - Prob. 43ECh. 12.4 - Equality of mixed partial derivatives Verify that...Ch. 12.4 - Prob. 45ECh. 12.4 - Prob. 46ECh. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - Prob. 49ECh. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Prob. 53ECh. 12.4 - Prob. 54ECh. 12.4 - Gas law calculations Consider the Ideal Gas Law PV...Ch. 12.4 - Prob. 56ECh. 12.4 - Nondifferentiability? Consider the following...Ch. 12.4 - Nondifferentiability? Consider the following...Ch. 12.4 - Prob. 59ECh. 12.4 - Prob. 60ECh. 12.4 - Prob. 61ECh. 12.4 - Prob. 62ECh. 12.4 - Prob. 63ECh. 12.4 - Prob. 64ECh. 12.4 - Prob. 65ECh. 12.4 - Miscellaneous partial derivatives Compute the...Ch. 12.4 - Prob. 67ECh. 12.4 - Prob. 68ECh. 12.4 - Prob. 69ECh. 12.4 - Spherical caps The volume of the cap of a sphere...Ch. 12.4 - Prob. 71ECh. 12.4 - Body mass index The body mass index (BMI) for an...Ch. 12.4 - Electric potential function The electric potential...Ch. 12.4 - Prob. 74ECh. 12.4 - Resistors in parallel Two resistors in an...Ch. 12.4 - Wave on a string Imagine a string that is fixed at...Ch. 12.4 - Wave equation Traveling waves (for example, water...Ch. 12.4 - Wave equation Traveling waves (for example, water...Ch. 12.4 - Wave equation Traveling waves (for example, water...Ch. 12.4 - Laplaces equation A classical equation of...Ch. 12.4 - Laplaces equation A classical equation of...Ch. 12.4 - Laplaces equation A classical equation of...Ch. 12.4 - Laplaces equation A classical equation of...Ch. 12.4 - Heat equation The flow of hear along a thin...Ch. 12.4 - Heat equation The flow of hear along a thin...Ch. 12.4 - Prob. 86ECh. 12.4 - Heat equation The flow of hear along a thin...Ch. 12.4 - Prob. 88ECh. 12.4 - Differentiability Use the definition of...Ch. 12.4 - Nondifferentiability? Consider the following...Ch. 12.4 - Nondifferentiability? Consider the following...Ch. 12.4 - Prob. 92ECh. 12.4 - Derivatives of an integral Let h be continuous for...Ch. 12.4 - Prob. 94ECh. 12.4 - Prob. 95ECh. 12.5 - Suppose z = f(x, y), where x and y are functions...Ch. 12.5 - Let z be a function of x and y, while x and y are...Ch. 12.5 - Suppose w is a function of x, y and z, which are...Ch. 12.5 - Let z = f(x, y), x = g(s, t), and y = h(s, t)....Ch. 12.5 - Given that w = F(x, y, z), and x, y, and z are...Ch. 12.5 - Suppose F(x, y) = 0 and y is a differentiable...Ch. 12.5 - Chain Rule with one independent variable Use...Ch. 12.5 - Prob. 8ECh. 12.5 - Chain Rule with one independent variable Use...Ch. 12.5 - Chain Rule with one independent variable Use...Ch. 12.5 - Chain Rule with one independent variable Use...Ch. 12.5 - Prob. 12ECh. 12.5 - Chain Rule with one independent variable Use...Ch. 12.5 - Prob. 14ECh. 12.5 - Chain Rule with one independent variable Use...Ch. 12.5 - Chain Rule with one independent variable Use...Ch. 12.5 - Changing cylinder The volume of a right circular...Ch. 12.5 - Changing pyramid The volume of a pyramid with a...Ch. 12.5 - Chain Rule with several independent variables Find...Ch. 12.5 - Chain Rule with several independent variables Find...Ch. 12.5 - Chain Rule with several independent variables Find...Ch. 12.5 - Chain Rule with several independent variables Find...Ch. 12.5 - Chain Rule with several independent variables Find...Ch. 12.5 - Prob. 24ECh. 12.5 - Chain Rule with several independent variables Find...Ch. 12.5 - Prob. 26ECh. 12.5 - Making trees Use a tree diagram to write the...Ch. 12.5 - Prob. 28ECh. 12.5 - Prob. 29ECh. 12.5 - Prob. 30ECh. 12.5 - Implicit differentiation Given the following...Ch. 12.5 - Implicit differentiation Given the following...Ch. 12.5 - Implicit differentiation Given the following...Ch. 12.5 - Implicit differentiation Given the following...Ch. 12.5 - Implicit differentiation Given the following...Ch. 12.5 - Implicit differentiation Given the following...Ch. 12.5 - Prob. 37ECh. 12.5 - Prob. 38ECh. 12.5 - Prob. 39ECh. 12.5 - Derivative practice two ways Find the indicated...Ch. 12.5 - Derivative practice two ways Find the indicated...Ch. 12.5 - Derivative practice Find the indicated derivative...Ch. 12.5 - Derivative practice Find the indicated derivative...Ch. 12.5 - Derivative practice Find the indicated derivative...Ch. 12.5 - Derivative practice Find the indicated derivative...Ch. 12.5 - Prob. 46ECh. 12.5 - Change on a line Suppose w=(x,y,z) and is the line...Ch. 12.5 - Prob. 48ECh. 12.5 - Prob. 49ECh. 12.5 - Prob. 50ECh. 12.5 - Prob. 51ECh. 12.5 - Prob. 52ECh. 12.5 - Walking on a surface Consider the following...Ch. 12.5 - Walking on a surface Consider the following...Ch. 12.5 - Walking on a surface Consider the following...Ch. 12.5 - Walking on a surface Consider the following...Ch. 12.5 - Conservation of energy A projectile with mass m is...Ch. 12.5 - Utility functions in economics Economists use...Ch. 12.5 - Constant volume tori The volume of a solid torus...Ch. 12.5 - Body surface area One of several empirical...Ch. 12.5 - Prob. 61ECh. 12.5 - Prob. 62ECh. 12.5 - Prob. 63ECh. 12.5 - Change of coordinates Recall that Cartesian and...Ch. 12.5 - Change of coordinates continued An important...Ch. 12.5 - Prob. 67ECh. 12.5 - Prob. 68ECh. 12.5 - Prob. 69ECh. 12.6 - Prob. 1ECh. 12.6 - How do you compute the gradient of the functions...Ch. 12.6 - Prob. 3ECh. 12.6 - Prob. 4ECh. 12.6 - Given a function f, explain the relationship...Ch. 12.6 - The level curves of the surface z=x2+y2 are...Ch. 12.6 - Directional derivatives Consider the function...Ch. 12.6 - Directional derivatives Consider the function...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing gradients Compute the gradient of the...Ch. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Prob. 19ECh. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Prob. 22ECh. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Computing directional derivatives with the...Ch. 12.6 - Direction of steepest ascent and descent Consider...Ch. 12.6 - Direction of steepest ascent and descent Consider...Ch. 12.6 - Direction of steepest ascent and descent Consider...Ch. 12.6 - Direction of steepest ascent and descent Consider...Ch. 12.6 - Direction of steepest ascent and descent Consider...Ch. 12.6 - Direction of steepest ascent and descent Consider...Ch. 12.6 - Interpreting directional derivatives A function f...Ch. 12.6 - Interpreting directional derivatives A function f...Ch. 12.6 - Interpreting directional derivatives A function f...Ch. 12.6 - Interpreting directional derivatives A function f...Ch. 12.6 - Interpreting directional derivatives A function f...Ch. 12.6 - Interpreting directional derivatives A function f...Ch. 12.6 - Prob. 39ECh. 12.6 - Prob. 40ECh. 12.6 - Prob. 41ECh. 12.6 - Prob. 42ECh. 12.6 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 12.6 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 12.6 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 12.6 - Level curves Consider the paraboloid f(x, y) = 16 ...Ch. 12.6 - Level curves Consider the upper half of the...Ch. 12.6 - Level curves Consider the upper half of the...Ch. 12.6 - Level curves Consider the upper half of the...Ch. 12.6 - Prob. 50ECh. 12.6 - Path of steepest descent Consider each of the...Ch. 12.6 - Path of steepest descent Consider each of the...Ch. 12.6 - Path of steepest descent Consider each of the...Ch. 12.6 - Path of steepest descent Consider each of the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Gradients in three dimensions Consider the...Ch. 12.6 - Explain why or why not Determine whether the...Ch. 12.6 - Gradient of a composite function Consider the...Ch. 12.6 - Directions of zero change Find the directions in...Ch. 12.6 - Prob. 66ECh. 12.6 - Directions of zero change Find the directions in...Ch. 12.6 - Directions of zero change Find the directions in...Ch. 12.6 - Steepest ascent on a plane Suppose a long sloping...Ch. 12.6 - Gradient of a distance function Let (a, b) be a...Ch. 12.6 - Looking aheadtangent planes Consider the following...Ch. 12.6 - Prob. 72ECh. 12.6 - Looking aheadtangent planes Consider the following...Ch. 12.6 - Prob. 74ECh. 12.6 - Prob. 75ECh. 12.6 - Prob. 76ECh. 12.6 - Prob. 77ECh. 12.6 - Prob. 78ECh. 12.6 - Prob. 79ECh. 12.6 - Prob. 80ECh. 12.6 - Rules for gradients Use the definition of the...Ch. 12.6 - Prob. 82ECh. 12.6 - Prob. 83ECh. 12.6 - Prob. 84ECh. 12.6 - Prob. 85ECh. 12.6 - Prob. 86ECh. 12.6 - Prob. 87ECh. 12.7 - Suppose n is a vector normal to the tangent plane...Ch. 12.7 - Write the explicit function z = xy2 + x2y 10 in...Ch. 12.7 - Write an equation for the plane tangent to the...Ch. 12.7 - Prob. 4ECh. 12.7 - Explain how to approximate a function f at a point...Ch. 12.7 - Explain how to approximate the change in a...Ch. 12.7 - Write the approximate change formula for a...Ch. 12.7 - Write the differential dw for the function w =...Ch. 12.7 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 12.7 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 12.7 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 12.7 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 12.7 - Tangent planes for F(x,y,z) = 0 Find an equation...Ch. 12.7 - Tangent planes for F(x, y, z) = 0 Find an equation...Ch. 12.7 - Tangent planes for F(x, y, z) = 0 Find an equation...Ch. 12.7 - Tangent planes for F(x, y, z) = 0 Find an equation...Ch. 12.7 - Tangent planes for z = f (x, y) Find an equation...Ch. 12.7 - Tangent planes for z = f (x, y) Find an equation...Ch. 12.7 - Tangent planes for z = f (x, y) Find an equation...Ch. 12.7 - Tangent planes for z = f (x, y) Find an equation...Ch. 12.7 - Tangent planes for z = f (x, y) Find an equation...Ch. 12.7 - Prob. 22ECh. 12.7 - Tangent planes for z = f (x, y) Find an equation...Ch. 12.7 - Tangent planes for z = f (x, y) Find an equation...Ch. 12.7 - Linear approximation a.Find the linear...Ch. 12.7 - Linear approximation a.Find the linear...Ch. 12.7 - Linear approximation a.Find the linear...Ch. 12.7 - Linear approximation a.Find the linear...Ch. 12.7 - Linear approximation a.Find the linear...Ch. 12.7 - Prob. 30ECh. 12.7 - Approximate function change Use differentials to...Ch. 12.7 - Approximate function change Use differentials to...Ch. 12.7 - Approximate function change Use differentials to...Ch. 12.7 - Approximate function change Use differentials to...Ch. 12.7 - Changes in torus surface area The surface area of...Ch. 12.7 - Changes in cone volume The volume of a right...Ch. 12.7 - Area of an ellipse The area of an ellipse with...Ch. 12.7 - Volume of a paraboloid The volume of a segment of...Ch. 12.7 - Differentials with more than two variables Write...Ch. 12.7 - Differentials with more than two variables Write...Ch. 12.7 - Differentials with more than two variables Write...Ch. 12.7 - Differentials with more than two variables Write...Ch. 12.7 - Law of Cosines The side lengths of any triangle...Ch. 12.7 - Explain why or why not Determine whether the...Ch. 12.7 - Tangent planes Find an equation of the plane...Ch. 12.7 - Tangent planes Find an equation of the plane...Ch. 12.7 - Tangent planes Find an equation of the plane...Ch. 12.7 - Tangent planes Find an equation of the plane...Ch. 12.7 - Horizontal tangent planes Find the points at which...Ch. 12.7 - Horizontal tangent planes Find the points at which...Ch. 12.7 - Horizontal tangent planes Find the points at which...Ch. 12.7 - Horizontal tangent planes Find the points at which...Ch. 12.7 - Prob. 54ECh. 12.7 - Surface area of a cone A cone with height h and...Ch. 12.7 - Line tangent to an intersection curve Consider the...Ch. 12.7 - Water-level changes A conical tank with radius...Ch. 12.7 - Prob. 59ECh. 12.7 - Floating-point operations In general, real numbers...Ch. 12.7 - Probability of at least one encounter Suppose that...Ch. 12.7 - Prob. 62ECh. 12.7 - Prob. 63ECh. 12.7 - Prob. 64ECh. 12.7 - Logarithmic differentials Let f be a...Ch. 12.8 - Describe the appearance of a smooth surface with a...Ch. 12.8 - Describe the usual appearance of a smooth surface...Ch. 12.8 - What are the conditions for a critical point of a...Ch. 12.8 - If fx (a, b) = fy (a, b) = 0, does it follow the f...Ch. 12.8 - Consider the function z = f(x, y). What is the...Ch. 12.8 - Prob. 6ECh. 12.8 - What is an absolute minimum value of a function f...Ch. 12.8 - What is the procedure for locating absolute...Ch. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Prob. 16ECh. 12.8 - Critical points Find all critical points of the...Ch. 12.8 - Prob. 18ECh. 12.8 - Prob. 19ECh. 12.8 - Prob. 20ECh. 12.8 - Prob. 21ECh. 12.8 - Prob. 22ECh. 12.8 - Prob. 23ECh. 12.8 - Prob. 24ECh. 12.8 - Analyzing critical points Find the critical points...Ch. 12.8 - Analyzing critical points Find the critical points...Ch. 12.8 - Prob. 27ECh. 12.8 - Prob. 28ECh. 12.8 - Analyzing critical points Find the critical points...Ch. 12.8 - Analyzing critical points Find the critical points...Ch. 12.8 - Prob. 31ECh. 12.8 - Prob. 32ECh. 12.8 - Analyzing critical points Find the critical points...Ch. 12.8 - Prob. 34ECh. 12.8 - Shipping regulations A shipping company handles...Ch. 12.8 - Cardboard boxes A lidless box is to be made using...Ch. 12.8 - Cardboard boxes A lidless cardboard box is to be...Ch. 12.8 - Optimal box Find the dimensions of the largest...Ch. 12.8 - Prob. 39ECh. 12.8 - Inconclusive tests Show that the Second Derivative...Ch. 12.8 - Prob. 41ECh. 12.8 - Inconclusive tests Show that the Second Derivative...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Prob. 51ECh. 12.8 - Absolute maxima and minima Find the absolute...Ch. 12.8 - Absolute extrema on open and / or unbounded...Ch. 12.8 - Absolute extrema on open and / or unbounded...Ch. 12.8 - Absolute extrema on open and / or unbounded...Ch. 12.8 - Absolute extrema on open and / or unbounded...Ch. 12.8 - Prob. 57ECh. 12.8 - Prob. 58ECh. 12.8 - Prob. 59ECh. 12.8 - Absolute extrema on open and / or unbounded...Ch. 12.8 - Explain why or why not Determine whether the...Ch. 12.8 - Prob. 62ECh. 12.8 - Extreme points from contour plots Based on the...Ch. 12.8 - Optimal box Find the dimensions of the rectangular...Ch. 12.8 - Lease distance What point on the plane x y + z =...Ch. 12.8 - Maximum/minimum of linear functions Let R be a...Ch. 12.8 - Magic triples Let x, y, and z be nonnegative...Ch. 12.8 - Powers and roots Assume that x + y + z = 1 with x ...Ch. 12.8 - Prob. 69ECh. 12.8 - Least squares approximation In its many guises,...Ch. 12.8 - Prob. 71ECh. 12.8 - Prob. 72ECh. 12.8 - Prob. 73ECh. 12.8 - Second Derivative Test Suppose the conditions of...Ch. 12.8 - Maximum area triangle Among all triangles with a...Ch. 12.8 - Ellipsoid inside a tetrahedron (1946 Putnam Exam)...Ch. 12.8 - Slicing plane Find an equation of the plane...Ch. 12.8 - Two mountains without a saddle Show that the...Ch. 12.8 - Solitary critical points A function of one...Ch. 12.9 - Explain why, at a point that maximizes or...Ch. 12.9 - Prob. 2ECh. 12.9 - Prob. 3ECh. 12.9 - Prob. 4ECh. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Prob. 11ECh. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Prob. 13ECh. 12.9 - Lagrange multipliers in two variables Use Lagrange...Ch. 12.9 - Lagrange multipliers in three variables Use...Ch. 12.9 - Lagrange multipliers in three variables Use...Ch. 12.9 - Lagrange multipliers in three variables Use...Ch. 12.9 - Lagrange multipliers in three variables Use...Ch. 12.9 - Prob. 19ECh. 12.9 - Lagrange multipliers in three variables Use...Ch. 12.9 - Lagrange multipliers in three variables Use...Ch. 12.9 - Prob. 22ECh. 12.9 - Prob. 23ECh. 12.9 - Lagrange multipliers in three variables Use...Ch. 12.9 - Applications of Lagrange multipliers Use Lagrange...Ch. 12.9 - Prob. 26ECh. 12.9 - Prob. 27ECh. 12.9 - Prob. 28ECh. 12.9 - Prob. 29ECh. 12.9 - Prob. 30ECh. 12.9 - Prob. 31ECh. 12.9 - Prob. 32ECh. 12.9 - Prob. 33ECh. 12.9 - Applications of Lagrange multipliers Use Lagrange...Ch. 12.9 - Maximizing utility functions Find the values of l...Ch. 12.9 - Maximizing utility functions Find the values of l...Ch. 12.9 - Maximizing utility functions Find the values of l...Ch. 12.9 - Maximizing utility functions Find the values of l...Ch. 12.9 - Explain why or why not Determine whether the...Ch. 12.9 - Prob. 40ECh. 12.9 - Prob. 41ECh. 12.9 - Prob. 42ECh. 12.9 - Prob. 43ECh. 12.9 - Prob. 44ECh. 12.9 - Prob. 45ECh. 12.9 - Prob. 46ECh. 12.9 - Prob. 47ECh. 12.9 - Prob. 48ECh. 12.9 - Prob. 49ECh. 12.9 - Graphical Lagrange multipliers The following...Ch. 12.9 - Graphical Lagrange multipliers The following...Ch. 12.9 - Extreme points on flattened spheres The equation...Ch. 12.9 - Production functions Economists model the output...Ch. 12.9 - Production functions Economists model the output...Ch. 12.9 - Production functions Economists model the output...Ch. 12.9 - Temperature of an elliptical plate The temperature...Ch. 12.9 - Maximizing a sum 57.Find the maximum value of x1 +...Ch. 12.9 - Prob. 58ECh. 12.9 - Prob. 59ECh. 12.9 - Geometric and arithmetic means Given positive...Ch. 12.9 - Problems with two constraints Given a...Ch. 12.9 - Prob. 62ECh. 12.9 - Two-constraint problems Use the result of Exercise...Ch. 12.9 - Prob. 64ECh. 12.9 - Two-constraint problems Use the result of Exercise...Ch. 12 - Prob. 1RECh. 12 - Prob. 2RECh. 12 - Equations of planes Consider the plane passing...Ch. 12 - Intersecting planes Find an equation of the line...Ch. 12 - Intersecting planes Find an equation of the line...Ch. 12 - Prob. 6RECh. 12 - Equations of planes Find an equation of the...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Prob. 21RECh. 12 - Identifying surfaces Consider the surfaces defined...Ch. 12 - Prob. 23RECh. 12 - Prob. 24RECh. 12 - Prob. 25RECh. 12 - Prob. 26RECh. 12 - Prob. 27RECh. 12 - Level curves Make a sketch of several level curves...Ch. 12 - Prob. 29RECh. 12 - Matching level curves with surfaces Match level...Ch. 12 - Prob. 31RECh. 12 - Prob. 32RECh. 12 - Prob. 33RECh. 12 - Prob. 34RECh. 12 - Prob. 35RECh. 12 - Prob. 36RECh. 12 - Prob. 37RECh. 12 - Prob. 38RECh. 12 - Prob. 39RECh. 12 - Prob. 40RECh. 12 - Prob. 41RECh. 12 - Prob. 42RECh. 12 - Prob. 43RECh. 12 - Prob. 44RECh. 12 - Prob. 45RECh. 12 - Prob. 46RECh. 12 - Prob. 47RECh. 12 - Laplaces equation Verify that the following...Ch. 12 - Prob. 49RECh. 12 - Prob. 50RECh. 12 - Prob. 51RECh. 12 - Chain Rule Use the Chain Rule to evaluate the...Ch. 12 - Prob. 53RECh. 12 - Implicit differentiation Find dy/dx for the...Ch. 12 - Implicit differentiation Find dy/dx for the...Ch. 12 - Walking on a surface Consider the following...Ch. 12 - Walking on a surface Consider the following...Ch. 12 - Constant volume cones Suppose the radius of a...Ch. 12 - Directional derivatives Consider the function f(x,...Ch. 12 - Computing gradients Compute the gradient of the...Ch. 12 - Computing gradients Compute the gradient of the...Ch. 12 - Computing gradients Compute the gradient of the...Ch. 12 - Computing gradients Compute the gradient of the...Ch. 12 - Prob. 64RECh. 12 - Prob. 65RECh. 12 - Direction of steepest ascent and descent a.Find...Ch. 12 - Prob. 67RECh. 12 - Level curves Let f(x, y) = 8 2x2 y2. For the...Ch. 12 - Level curves Let f(x, y) = 8 2x2 y2. For the...Ch. 12 - Prob. 70RECh. 12 - Prob. 71RECh. 12 - Tangent planes Find an equation of the plane...Ch. 12 - Tangent planes Find an equation of the plane...Ch. 12 - Prob. 74RECh. 12 - Prob. 75RECh. 12 - Prob. 76RECh. 12 - Prob. 77RECh. 12 - Linear approximation a.Find the linear...Ch. 12 - Linear approximation a.Find the linear...Ch. 12 - Changes in a function Estimate the change in the...Ch. 12 - Volume of a cylinder The volume of a cylinder with...Ch. 12 - Volume of an ellipsoid The volume of an ellipsoid...Ch. 12 - Water-level changes A hemispherical tank with a...Ch. 12 - Prob. 84RECh. 12 - Analyzing critical points Identify the critical...Ch. 12 - Analyzing critical points Identify the critical...Ch. 12 - Analyzing critical points Identify the critical...Ch. 12 - Absolute maxima and minima Find the absolute...Ch. 12 - Absolute maxima and minima Find the absolute...Ch. 12 - Prob. 90RECh. 12 - Absolute maxima and minima Find the absolute...Ch. 12 - Prob. 92RECh. 12 - Lagrange multipliers Use Lagrange multipliers to...Ch. 12 - Prob. 94RECh. 12 - Lagrange multipliers Use Lagrange multipliers to...Ch. 12 - Lagrange multipliers Use Lagrange multipliers to...Ch. 12 - Maximum perimeter rectangle Use Lagrange...Ch. 12 - Minimum surface area cylinder Use Lagrange...Ch. 12 - Minimum distance to a cone Find the point(s) on...Ch. 12 - Prob. 100RE
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