Math

CalculusCalculus: Early Transcendental FunctionsShannon Diversity Index One way to measure species diversity is to use the Shannon diversity index H . If a habitat consists of three species. A, B, and C, then its Shannon diversity index is H = − x ln x − y ln y − z ln z where x is the percent of species A in the habitat, y is the percent of species B in the habitat, and z is the percent of species C in the habitat. Use the fact that x + y + z = 1 to show that the maximum value of H occurs when x = y = z = 1 3 . What is the maximum value of H ?BuyFind*launch*

7th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337552516

Chapter 13.9, Problem 18E

Textbook Problem

Shannon Diversity Index One way to measure species diversity is to use the Shannon diversity index *H*. If a habitat consists of three species. A, B, and C, then its Shannon diversity index is

where *x* is the percent of species A in the habitat, *y* is the percent of species B in the habitat, and *z* is the percent of species *C* in the habitat. Use the fact that

to show that the maximum value of *H* occurs when
*H*?

Calculus: Early Transcendental Functions

Ch. 13.1 - Think About It Explain why z2=x+3y is not a...Ch. 13.1 - Function of Two Variables What is a graph of a...Ch. 13.1 - Determine whether graph is a function. Use the...Ch. 13.1 - Contour Map Explain how to sketch a contour map of...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...

Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Think About It The graphs labeled (a), (b). (c)....Ch. 13.1 - Think About It Use the function given in Exercise...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Sketching a Surface In Exercises 35-42, describe...Ch. 13.1 - Graphing a Function Using Technology In Exercises...Ch. 13.1 - Graphing a Function Using Technology In Exercises...Ch. 13.1 - Graphing a Function Using Technology In Exercises...Ch. 13.1 - Graphing a Function Using Technology In Exercises...Ch. 13.1 - Matching In Exercises 47-50, match the graph of...Ch. 13.1 - Matching In Exercises 47-50, match the graph of...Ch. 13.1 - Matching In Exercises 47-50, match the graph of...Ch. 13.1 - Matching In Exercises 47-50, match the graph of...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Sketching a Contour Map In Exercises 51-58,...Ch. 13.1 - Graphing Level Curves Using Technology In...Ch. 13.1 - Graphing Level Curves Using Technology In...Ch. 13.1 - Graphing Level Curves Using Technology In...Ch. 13.1 - Graphing Level Curves Using Technology In...Ch. 13.1 - Vertical Line Test Does die Vertical Line Test...Ch. 13.1 - Using Level Curves All of the level curves of the...Ch. 13.1 - Creating a Function Construct a function whose...Ch. 13.1 - Conjecture Consider the function f(x,y)=xy, for...Ch. 13.1 - Writing In Exercises 67 and 68, use the graphs of...Ch. 13.1 - Writing In Exercises 67 and 68, use the graphs of...Ch. 13.1 - Investment In 2016, an investment of S1000 was...Ch. 13.1 - Investment A principal of $5000 is deposited in a...Ch. 13.1 - Sketching a Level Surface In Exercises 71-76....Ch. 13.1 - Sketching a Level Surface In Exercises 71-76....Ch. 13.1 - Sketching a Level Surface In Exercises 71-76....Ch. 13.1 - Sketching a Level Surface In Exercises 71-76....Ch. 13.1 - Sketching a Level Surface In Exercises 71-76....Ch. 13.1 - Sketching a Level Surface In Exercises 71-76....Ch. 13.1 - Forestry The Doyle Lux Rule is one of several...Ch. 13.1 - Queuing Model The average length of time that a...Ch. 13.1 - Temperature Distribution The temperature T (in...Ch. 13.1 - Electric Potential The electric potential V at any...Ch. 13.1 - Cobb-Douglas Production Function In Exercises 81...Ch. 13.1 - Cobb-Douglas Production Function In Exercises 81...Ch. 13.1 - Cobb-Douglas Production Function Use the...Ch. 13.1 - Cobb-Douglas Production Function Show that the...Ch. 13.1 - Ideal Gas Law According to the Ideal Gas Law, PV=...Ch. 13.1 - Modeling Data The table shows the net sales x (in...Ch. 13.1 - Meteorology Meteorologists measure die atmospheric...Ch. 13.1 - Acid Rain The acidity of rainwater is measured in...Ch. 13.1 - Construction Cost A rectangular storage box with...Ch. 13.1 - HOW DO YOU SEE IT? The contour map of the Southern...Ch. 13.1 - True or False? In Exercises 91-94, determine...Ch. 13.1 - True or False? In Exercises 91-94, determine...Ch. 13.1 - True or False? In Exercises 91-94, determine...Ch. 13.1 - True or False? 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The figure shows the graph of...Ch. 13.2 - Continuity In Exercises 41 and 42, discuss the...Ch. 13.2 - Continuity In Exercises 41 and 42, discuss the...Ch. 13.2 - Limit and Continuity In Exercises 43-46, use a...Ch. 13.2 - Limit and Continuity In Exercises 43-46, use a...Ch. 13.2 - Limit and Continuity In Exercises 43-46, use a...Ch. 13.2 - Limit and Continuity In Exercises 43-46, use a...Ch. 13.2 - Limit Consider lim(x,y)(0,0)x2+y2xy (see figure)....Ch. 13.2 - Limit Consider lim(x,y)(0,0)x2yx4+y2 (see figure).Ch. 13.2 - Comparing Continuity In Exercises 49 and 50,...Ch. 13.2 - Comparing Continuity In Exercises 49 and 50,...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Continuity In Exercises 61-66, discuss the...Ch. 13.2 - Continuity In Exercises 61-66, discuss the...Ch. 13.2 - Continuity In Exercises 61-66, discuss the...Ch. 13.2 - Continuity In Exercises 61-66, discuss the...Ch. 13.2 - Continuity In Exercises 61-66, discuss the...Ch. 13.2 - Continuity In Exercises 61-66, discuss the...Ch. 13.2 - Continuity of a Composite Function In Exercises...Ch. 13.2 - Continuity of a Composite Function In Exercises...Ch. 13.2 - Continuity of a Composite Function In Exercises...Ch. 13.2 - Continuity of a Composite Function In Exercises...Ch. 13.2 - Finding a Limit In Exercises 71-76, find each...Ch. 13.2 - Finding a Limit In Exercises 71-76, find each...Ch. 13.2 - Finding a Limit In Exercises 71-76, find each...Ch. 13.2 - Finding a Limit In Exercises 71-76, find each...Ch. 13.2 - Finding a Limit In Exercises 71-76, find each...Ch. 13.2 - Finding a Limit In Exercises 71-76, find each...Ch. 13.2 - Finding a Limit Using Spherical Coordinates In...Ch. 13.2 - Finding a Limit Using Spherical Coordinates In...Ch. 13.2 - True or False? In Exercises 79-82, determine...Ch. 13.2 - True or False? In Exercises 79-82, determine...Ch. 13.2 - True or False? In Exercises 79-82, determine...Ch. 13.2 - True or False? 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Wave Equation In Exercises 99-102, show that the...Ch. 13.3 - Wave Equation In Exercises 99-102, show that the...Ch. 13.3 - Wave Equation In Exercises 99-102, show that the...Ch. 13.3 - Wave Equation In Exercises 99-102, show that the...Ch. 13.3 - Heat Equation In Exercises 103 and 104, show that...Ch. 13.3 - Heat Equation In Exercises 103 and 104, show that...Ch. 13.3 - Cauchy-Riemann Equations In Exercises 105 and 106,...Ch. 13.3 - Cauchy-Riemann Equations In Exercises 105 and 106,...Ch. 13.3 - Using First Partial Derivatives In Exercises 107...Ch. 13.3 - Using First Partial Derivatives In Exercises 107...Ch. 13.3 - Think About It Consider z=f(x,y) such that zx=zy....Ch. 13.3 - First Partial Derivatives Given z=f(x)g(y), find...Ch. 13.3 - Sketching a Graph Sketch the graph of a function...Ch. 13.3 - Sketching a Graph Sketch the graph of a function...Ch. 13.3 - Think About It The price P (in dollars) of a used...Ch. 13.3 - HOW DO YOU SEE IT? Use the graph of the surface to...Ch. 13.3 - Area The area of a triangle is represented by...Ch. 13.3 - Volume The volume of a right circular cone of...Ch. 13.3 - Marginal Revenue A pharmaceutical corporation has...Ch. 13.3 - Marginal Costs A company manufactures two types of...Ch. 13.3 - Psychology Early in the twentieth century, an...Ch. 13.3 - Marginal Productivity Consider die Cobb-Douglas...Ch. 13.3 - Think About It Let N be the number of applicants...Ch. 13.3 - Investment The value of an investment of $1000...Ch. 13.3 - Temperature Distribution The temperature at any...Ch. 13.3 - Apparent Temperature A measure of how hot weather...Ch. 13.3 - Ideal Gas Law The Ideal Gas Law states that PV=nRT...Ch. 13.3 - Marginal Utility The utility function U=f(x,y) is...Ch. 13.3 - Modeling Data Personal consumption expenditures...Ch. 13.3 - Modeling Data The table shows the national health...Ch. 13.3 - Using a Function Consider the function defined by...Ch. 13.3 - Using a Function Consider die function...Ch. 13.3 - Using a Function Consider the function...Ch. 13.4 - CONCEPT CHECK Approximation Describe the change in...Ch. 13.4 - CONCEPT CHECK Linear Approximation What is meant...Ch. 13.4 - Finding a Total Differential find the total...Ch. 13.4 - Finding a Total Differential find the total...Ch. 13.4 - Finding a Total Differential find the total...Ch. 13.4 - Finding a Total Differential find the total...Ch. 13.4 - Finding a Total Differential find the total...Ch. 13.4 - Finding a Total Differential find the total...Ch. 13.4 - Using a Differential as an Approximation In...Ch. 13.4 - Using a Differential as an Approximation In...Ch. 13.4 - Using a Differential as an Approximation In...Ch. 13.4 - Using a Differential as an Approximation In...Ch. 13.4 - Using a Differential as an Approximation In...Ch. 13.4 - Using a Differential as an Approximation In...Ch. 13.4 - Approximating an Expression In Exercises 15-18,...Ch. 13.4 - Approximating an Expression In Exercises 15-18,...Ch. 13.4 - Approximating an Expression In Exercises 15-18,...Ch. 13.4 - Approximating an Expression In Exercises 15-18,...Ch. 13.4 - Continuity If fx. and fy are each continuous in an...Ch. 13.4 - HOW DO YOU SEE IT? Which point has a greater...Ch. 13.4 - Area The area of the shaded rectangle in the...Ch. 13.4 - Volume The volume of the red right circular...Ch. 13.4 - Volume The possible error involved in measuring...Ch. 13.4 - Volume The possible error involved in measuring...Ch. 13.4 - Numerical Analysis A right circular core of height...Ch. 13.4 - Numerical Analysis A right circular core of height...Ch. 13.4 - Wind Chill The formula for wind chill C (in...Ch. 13.4 - Resistance The total resistance R (in ohms) of two...Ch. 13.4 - Power Electrical power P is given by P=E2R where...Ch. 13.4 - Acceleration The centripetal acceleration of a...Ch. 13.4 - Volume A trough is 16 feet long (see figure). Its...Ch. 13.4 - Sports A baseball player in center field is...Ch. 13.4 - Inductance The inductance L (in microhenrys) of a...Ch. 13.4 - Pendulum The period T of a pendulum of length L is...Ch. 13.4 - Differentiability In Exercises 35-38, show that...Ch. 13.4 - Differentiability In Exercises 35-38, show that...Ch. 13.4 - Differentiability In Exercises 35-38, show that...Ch. 13.4 - Differentiability In Exercises 35-38, show that...Ch. 13.4 - Differentiability In Exercises 39 and 40, use the...Ch. 13.4 - Differentiability In Exercises 39 and 40, use the...Ch. 13.5 - Chain Rule Consider w=f(x,y), where x=g(s,t) and...Ch. 13.5 - Implicit Differentiation Why is using the Chain...Ch. 13.5 - Using the Chain Rule In Exercises 3-6, find dw/dt...Ch. 13.5 - Using the Chain Rule In Exercises 3-6, find dw/dt...Ch. 13.5 - Using the Chain Rule In Exercises 3-6, find dw/dt...Ch. 13.5 - Using the Chain Rule In Exercises 3-6, find dw/dt...Ch. 13.5 - Using Different Methods In Exercises 7-12, find d...Ch. 13.5 - Using Different Methods In Exercises 7-12, find...Ch. 13.5 - Using Different Methods In Exercises 7-12, find...Ch. 13.5 - Using Different Methods In Exercises 7-12, find...Ch. 13.5 - Using Different Methods In Exercises 7-12, find...Ch. 13.5 - Using Different Methods In Exercises 7-12, find...Ch. 13.5 - Projectile Motion In Exercises 13 and 14, the...Ch. 13.5 - Projectile Motion In Exercises 13 and 14, the...Ch. 13.5 - Finding Partial Derivatives In Exercises 15-18,...Ch. 13.5 - Finding Partial Derivatives In Exercises 15-18,...Ch. 13.5 - Finding Partial Derivatives In Exercises 15-18....Ch. 13.5 - Finding Partial Derivatives In Exercises 15-18,...Ch. 13.5 - Using Different Methods In Exercises 19-22, find...Ch. 13.5 - Using Different Methods In Exercises 19-22, find...Ch. 13.5 - Using Different Methods In Exercises 19-22, find...Ch. 13.5 - Using Different Methods In Exercises 19-22, find...Ch. 13.5 - Finding a Derivative Implicitly In Exercises...Ch. 13.5 - Finding a Derivative Implicitly In Exercises...Ch. 13.5 - Finding a Derivative Implicitly In Exercises...Ch. 13.5 - Finding a Derivative Implicitly In Exercises...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Finding Partial Derivatives Implicitly In...Ch. 13.5 - Homogeneous Functions A function f is homogeneous...Ch. 13.5 - Homogeneous Functions A function f is homogeneous...Ch. 13.5 - Homogeneous Functions A function f is homogeneous...Ch. 13.5 - Homogeneous Functions A function f is homogeneous...Ch. 13.5 - Using a Table of Values Let w=f(x,y),x=g(t), and...Ch. 13.5 - Using a Table of Values Let w=f(x,y),x=g(s,t), and...Ch. 13.5 - Using the Chain Rule Show that wu+wv=0 for...Ch. 13.5 - Using the Chain Rule Demonstrate the result of...Ch. 13.5 - Using the Chain Rule Let F (u, v) be a function of...Ch. 13.5 - HOW DO YOU SEE IT? The path of an object...Ch. 13.5 - Volume and Surface Area The radius of a right...Ch. 13.5 - Ideal Gas Law The Ideal Gas Law is PV=mRT where P...Ch. 13.5 - Moment of Inertia An annular cylinder has an...Ch. 13.5 - Volume and Surface Area The two radii of the...Ch. 13.5 - Cauchy-Riemann Equations Given the functions u(x,...Ch. 13.5 - Cauchy-Riemann Equations Demonstrate the result of...Ch. 13.5 - Homogeneous Function Show that if f(x, y) is...Ch. 13.6 - CONCEPT CHECK Directional Derivative For a...Ch. 13.6 - CONCEPT CHECK Gradient What is the meaning of the...Ch. 13.6 - Finding a Directional Derivative In Exercises 3-6,...Ch. 13.6 - Finding a Directional Derivative In Exercises 3-6,...Ch. 13.6 - Finding a Directional Derivative In Exercises 3-6,...Ch. 13.6 - Finding a Directional Derivative In Exercises 3-6,...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding the Gradient of a Function In Exercises...Ch. 13.6 - Finding the Gradient of a Function In Exercises...Ch. 13.6 - Finding the Gradient of a Function In Exercises...Ch. 13.6 - Finding the Gradient of a Function In Exercises...Ch. 13.6 - Finding the Gradient of a Function In Exercises...Ch. 13.6 - Finding the Gradient of a Function In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Finding a Directional Derivative In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Using Properties of the Gradient In Exercises...Ch. 13.6 - Finding a Normal Vector to a Level Curve In...Ch. 13.6 - Finding a Normal Vector to a Level Curve In...Ch. 13.6 - Finding a Normal Vector to a Level Curve In...Ch. 13.6 - Finding a Normal Vector to a Level Curve In...Ch. 13.6 - Using a Function In Exercises 43-46, (a) find the...Ch. 13.6 - Using a Function In Exercises 43-46, (a) find the...Ch. 13.6 - Using a Function In Exercises 43-46, (a) find the...Ch. 13.6 - Using a Function In Exercises 43-46, (a) find the...Ch. 13.6 - Using a Function Consider the function...Ch. 13.6 - Using a Function Consider the function...Ch. 13.6 - Investigation Consider the function f(x,y)=x2y2 at...Ch. 13.6 - Investigation Consider the function...Ch. 13.6 - EXPLORING CONCEPTS Think About It Consider v=3u....Ch. 13.6 - EXPLORING CONCEPTS Sketching a Graph and a Vector...Ch. 13.6 - Topography The surface of a mountain is modeled by...Ch. 13.6 - HOW DO YOU SEE IT? The figure shows a topographic...Ch. 13.6 - Temperature The temperature at the point (x, y) on...Ch. 13.6 - Temperature The temperature at the point (x, y) on...Ch. 13.6 - Finding the Direction of Maximum Increase In...Ch. 13.6 - Finding the Direction of Maximum Increase In...Ch. 13.6 - Finding the Path of a Heat-Seeking Particle In...Ch. 13.6 - Finding the Path of a Heat-Seeking Particle In...Ch. 13.6 - True or False? In Exercises 61-64, determine...Ch. 13.6 - True or False? In Exercises 61-64, determine...Ch. 13.6 - True or False? In Exercises 61-64, determine...Ch. 13.6 - True or False? In Exercises 61-64, determine...Ch. 13.6 - Finding a Function Find a function f such that...Ch. 13.6 - Ocean Floor A team of oceanographers is mapping...Ch. 13.6 - Using a Function Consider the function f(x,y)=xy3....Ch. 13.6 - Directional Derivative Consider the function...Ch. 13.7 - CONCEPT CHECK Tangent Vector Consider a point...Ch. 13.7 - CONCEPT CHECK Normal Line Consider a point...Ch. 13.7 - Describing a Surface In Exercises 3-6, describe...Ch. 13.7 - Describing a Surface In Exercises 3-6, describe...Ch. 13.7 - Describing a Surface In Exercises 3-6, describe...Ch. 13.7 - Describing a Surface In Exercises 3-6, describe...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...Ch. 13.7 - Finding the Equation of a Tangent Line to a Curve...Ch. 13.7 - Finding the Equation of a Tangent Line to a Curve...Ch. 13.7 - Finding the Equation of a Tangent Line to a Curve...Ch. 13.7 - Finding the Equation of a Tangent Line to a Curve...Ch. 13.7 - Finding the Equation of a Tangent Line to a Curve...Ch. 13.7 - Finding the Equation of a Tangent Line to a Curve...Ch. 13.7 - Finding the Angle of Inclination of a Tangent...Ch. 13.7 - Finding the Angle of Inclination of a Tangent...Ch. 13.7 - Finding the Angle of Inclination of a Tangent...Ch. 13.7 - Finding the Angle of Inclination of a Tangent...Ch. 13.7 - Horizontal Tangent Plane In Exercises 37-42, find...Ch. 13.7 - Horizontal Tangent Plane In Exercises 37-42, find...Ch. 13.7 - Horizontal Tangent Plane In Exercises 37-42, find...Ch. 13.7 - Horizontal Tangent Plane In Exercises 37-42, find...Ch. 13.7 - Horizontal Tangent Plane In Exercises 37-42, find...Ch. 13.7 - Horizontal Tangent Plane In Exercises 37-42, find...Ch. 13.7 - Tangent Surfaces In Exercises 43 and 44, show that...Ch. 13.7 - Tangent Surfaces In Exercises 43 and 44, show that...Ch. 13.7 - Perpendicular Tangent Planes In Exercises 45 and...Ch. 13.7 - Perpendicular Tangent Planes In Exercises 45 and...Ch. 13.7 - EXPLORING CONCEPTS Tangent Plane The tangent plane...Ch. 13.7 - EXPLORING CONCEPTS Normal Lines For some surfaces,...Ch. 13.7 - Using an Ellipsoid Find a point on the ellipsoid...Ch. 13.7 - Using a Hyperboloid Find a point on the...Ch. 13.7 - Using an Ellipsoid Find a point on the ellipsoid...Ch. 13.7 - HOW DO YOU SEE IT? The graph shows the ellipsoid...Ch. 13.7 - Investigation Consider the function...Ch. 13.7 - Investigation Consider the function f(x,y)=sinyx...Ch. 13.7 - Using Functions Consider the functions...Ch. 13.7 - Using Functions Consider the functions...Ch. 13.7 - Writing a Tangent Plane In Exercises 57 and 58,...Ch. 13.7 - Writing a Tangent Plane In Exercises 57 and 58,...Ch. 13.7 - Tangent Planes of a Cone Show that any tangent...Ch. 13.7 - Tangent Planes Let f be a differentiable function...Ch. 13.7 - Approximation Consider the following...Ch. 13.7 - Approximation Repeat Exercise 61 for the function...Ch. 13.7 - Proof Prove that the angle of inclination of the...Ch. 13.7 - Proof Prove Theorem 13.14.Ch. 13.8 - CONCEPT CHECK Function of Two Variables For a...Ch. 13.8 - CONCEPT CHECK Second Partials Test Under what...Ch. 13.8 - Finding Relative Extrema In Exercises 5-8,...Ch. 13.8 - Finding Relative Extrema In Exercises 5-8,...Ch. 13.8 - Finding Relative Extrema In Exercises 5-8,...Ch. 13.8 - Finding Relative Extrema In Exercises 3-8,...Ch. 13.8 - Finding Relative Extrema In Exercises 3-8,...Ch. 13.8 - Finding Relative Extrema In Exercises 3-8,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Using the Second Partials Test In Exercises 9-24,...Ch. 13.8 - Finding Relative Extrema and Saddle Points Using...Ch. 13.8 - Finding Relative Extrema and Saddle Points Using...Ch. 13.8 - Finding Relative Extrema and Saddle Points Using...Ch. 13.8 - Finding Relative Extrema and Saddle Points Using...Ch. 13.8 - Finding Relative Extrema In Exercises 29 and 30,...Ch. 13.8 - Finding Relative Extrema In Exercises 29 and 30,...Ch. 13.8 - Think About It In Exercises 31-34, determine...Ch. 13.8 - Think About It In Exercises 31-34, determine...Ch. 13.8 - Think About It In Exercises 31-34, determine...Ch. 13.8 - Think About It In Exercises 31-34, determine...Ch. 13.8 - Finding Relative Extrema and Saddle Points In...Ch. 13.8 - Finding Relative Extrema and Saddle Points In...Ch. 13.8 - Finding Relative Extrema and Saddle Points In...Ch. 13.8 - Finding Relative Extrema and Saddle Points In...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Examining a Function In Exercises 47 and 48, find...Ch. 13.8 - Examining a Function In Exercises 47 and 48, find...Ch. 13.8 - EXPLORING CONCEPTS Using the Second Partials Test...Ch. 13.8 - EXPLORING CONCEPTS Using the Second Partials Test...Ch. 13.8 - EXPLORING CONCEPTS Sketching a Graph In Exercises...Ch. 13.8 - EXPLORING CONCEPTS Sketching a Graph In Exercises...Ch. 13.8 - Comparing Functions Consider the functions...Ch. 13.8 - HOW DO YOU SEE IT? Determine whether each labeled...Ch. 13.8 - True or False? In Exercises 55-58, determine...Ch. 13.8 - True or False? In Exercises 55-58, determine...Ch. 13.8 - True or False? In Exercises 55-58, determine...Ch. 13.8 - True or False? In Exercises 55-58, determine...Ch. 13.9 - CONCEPT CHECK Applied Optimization Problems In...Ch. 13.9 - CONCEPT CHECK Method of Least Squares In your own...Ch. 13.9 - Finding Minimum Distance In Exercises 3 and 4,...Ch. 13.9 - Finding Minimum Distance In Exercises 3 and 4,...Ch. 13.9 - Finding Minimum Distance In Exercises 5 and 6,...Ch. 13.9 - Finding Minimum Distance In Exercises 5 and 6,...Ch. 13.9 - Finding Positive Numbers In Exercises 7-10, find...Ch. 13.9 - Finding Positive Numbers In Exercises 7-10, find...Ch. 13.9 - Finding Positive Numbers In Exercises 7-10, find...Ch. 13.9 - Finding Positive Numbers In Exercises 7-10, find...Ch. 13.9 - Cost A home improvement contractor is painting the...Ch. 13.9 - Maximum Volume The material for constructing the...Ch. 13.9 - Volume and Surface Area Show that a rectangular...Ch. 13.9 - Maximum Volume Show that the rectangular box of...Ch. 13.9 - Maximum Revenue A company manufactures running...Ch. 13.9 - Maximum Profit A corporation manufactures candles...Ch. 13.9 - Hardy-Weinberg Law Common blood types are...Ch. 13.9 - Shannon Diversity Index One way to measure species...Ch. 13.9 - Minimum Cost A water line is to be built from...Ch. 13.9 - Area A trough with trapezoidal cross sections is...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Finding the Least Squares Regression Line In...Ch. 13.9 - Modeling Data The table shows the gross income tax...Ch. 13.9 - Modeling Data The ages x (in years) and systolic...Ch. 13.9 - EXPLORING CONCEPTS Method of Least Squares Find a...Ch. 13.9 - HOW DO YOU SEE IT? Match the regression equation...Ch. 13.9 - Finding the Least Squares Regression Quadratic In...Ch. 13.9 - Finding the Least Squares Regression Quadratic In...Ch. 13.9 - Finding the Least Squares Regression Quadratic In...Ch. 13.9 - Finding the Least Squares Regression Quadratic In...Ch. 13.9 - Modeling Data After a new turbocharger for an...Ch. 13.9 - Modeling Data The table shows the total numbers of...Ch. 13.9 - Modeling Data A meteorologist measures the...Ch. 13.9 - Modeling Data The endpoints of the interval over...Ch. 13.9 - Using the Second Partials Test Use the Second...Ch. 13.10 - CONCEPT CHECK Constrained Optimization Problems...Ch. 13.10 - CONCEPT CHECK Method of Lagrange Multipliers In...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 11-14. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 11-14. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 11-14. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 11-14. use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 15 and 16....Ch. 13.10 - Using Lagrange Multipliers In Exercises 15 and 16....Ch. 13.10 - Using Lagrange Multipliers In Exercises 17 and 18,...Ch. 13.10 - Using Lagrange Multipliers In Exercises 17 and 18,...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Intersection of Surfaces In Exercises 29 and 30,...Ch. 13.10 - Intersection of Surfaces In Exercises 29 and 30,...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 31-38, use...Ch. 13.10 - Maximum Volume Use Lagrange multipliers to find...Ch. 13.10 - HOW DO YOU SEE IT? The graphs show the constraint...Ch. 13.10 - EXPLORING CONCEPTS Method of Lagrange Multipliers...Ch. 13.10 - EXPLORING CONCEPTS^ Method of Lagrange Multipliers...Ch. 13.10 - Minimum Cost A cargo container (in the shape of a...Ch. 13.10 - Geometric and Arithmetic Means (a) Use Lagrange...Ch. 13.10 - Minimum Surface Area Use Lagrange multipliers to...Ch. 13.10 - Temperature Let T(x,y,z)=100+x2+y2 represent the...Ch. 13.10 - Refraction of Light When light waves traveling in...Ch. 13.10 - Area and Perimeter A semicircle is on top of a...Ch. 13.10 - Production Level In Exercises 49 and 50, use...Ch. 13.10 - Production Level In Exercises 49 and 50, use...Ch. 13.10 - Cost In Exercises 51 and 52, use Lagrange...Ch. 13.10 - Cost In Exercises 51 and 52, use Lagrange...Ch. 13.10 - A can buoy is to be made of three pieces, namely,...Ch. 13 - Evaluating a Function In Exercises 1 and 2,...Ch. 13 - Evaluating a Function In Exercises 1 and 2,...Ch. 13 - Finding the Domain and Range of a Function In...Ch. 13 - Finding the Domain and Range of a Function In...Ch. 13 - Sketching a Surface In Exercises 5 and 6, describe...Ch. 13 - Sketching a Surface In Exercises 5 and 6, describe...Ch. 13 - Sketching a Contour Map In Exercises 7 and 8,...Ch. 13 - Sketching a Contour Map In Exercises 7 and 8,...Ch. 13 - Conjecture Consider the function f(x,y)=x2+y2. (a)...Ch. 13 - Cobb-Douglas Production Function A manufacturer...Ch. 13 - Sketching a Level Surface In Exercises 11 and 12,...Ch. 13 - Sketching a Level Surface In Exercises 11 and 12,...Ch. 13 - Limit and Continuity In Exercises 13-18, find the...Ch. 13 - Limit and Continuity In Exercises 13-18, find the...Ch. 13 - Limit and Continuity In Exercises 13-18, find the...Ch. 13 - Limit and Continuity In Exercises 13-18, find the...Ch. 13 - Limit and Continuity In Exercises 13-18, find the...Ch. 13 - Limit and Continuity In Exercises 13-18, find the...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding Partial Derivatives In Exercises 19-26,...Ch. 13 - Finding and Evaluating Partial Derivatives In...Ch. 13 - Finding and Evaluating Partial Derivatives In...Ch. 13 - Finding and Evaluating Partial Derivatives In...Ch. 13 - Finding and Evaluating Partial Derivatives In...Ch. 13 - Finding Second Partial Derivatives In Exercises...Ch. 13 - Finding Second Partial Derivatives In Exercises...Ch. 13 - Finding Second Partial Derivatives In Exercises...Ch. 13 - Finding Second Partial Derivatives In Exercises...Ch. 13 - Finding the Slopes of a Surface Find the slopes of...Ch. 13 - Marginal Revenue A company has two plants that...Ch. 13 - Finding a Total Differential In Exercises 37-40,...Ch. 13 - Finding a Total Differential In Exercises 37-40,...Ch. 13 - Finding a Total Differential In Exercises 37-40,...Ch. 13 - Finding a Total Differential In Exercises 37-40,...Ch. 13 - Using a Differential as an Approximation In...Ch. 13 - Using a Differential as an Approximation In...Ch. 13 - Volume The possible error involved in measuring...Ch. 13 - Lateral Surface Area Approximate the propagated...Ch. 13 - Differentiability In Exercises 45 and 46, show...Ch. 13 - Differentiability In Exercises 45 and 46, show...Ch. 13 - Using Different Methods In Exercises 47-50, find...Ch. 13 - Using Different Methods In Exercises 47-50, find...Ch. 13 - Using Different Methods In Exercises 47-50, find...Ch. 13 - Using Different Methods In Exercises 47-50, find...Ch. 13 - Using Different Methods In Exercises 51 and 52,...Ch. 13 - Using Different Methods In Exercises 51 and 52,...Ch. 13 - Finding a Derivative Implicitly In Exercises 53...Ch. 13 - Finding a Derivative Implicitly In Exercises 53...Ch. 13 - Finding Partial Derivatives Implicitly In...Ch. 13 - Finding Partial Derivatives Implicitly In...Ch. 13 - Finding a Directional Derivative In Exercises 57...Ch. 13 - Finding a Directional Derivative In Exercises 57...Ch. 13 - Finding a Directional Derivative In Exercises 59...Ch. 13 - Finding a Directional Derivative In Exercises 59...Ch. 13 - Using Properties of the Gradient In Exercises...Ch. 13 - Using Properties of the Gradient In Exercises...Ch. 13 - Using Properties of the Gradient In Exercises...Ch. 13 - Using Properties of the Gradient In Exercises...Ch. 13 - Using Properties of the Gradient In Exercises...Ch. 13 - Using Properties of the Gradient In Exercises...Ch. 13 - Using a Function In Exercises 67 and 68, (a) find...Ch. 13 - Using a Function In Exercises 67 and 68, (a) find...Ch. 13 - Finding an Equation of a Tangent Plane In...Ch. 13 - Finding an Equation of a Tangent Plane In...Ch. 13 - Finding an Equation of a Tangent Plane In...Ch. 13 - Finding an Equation of a Tangent Plane In...Ch. 13 - Finding an Equation of a Tangent Plane and a...Ch. 13 - Finding an Equation of a Tangent Plane and a...Ch. 13 - Finding the Angle of Inclination of a Tangent...Ch. 13 - Finding the Angle of Inclination of a Tangent...Ch. 13 - Horizontal Tangent Plane In Exercises 77 and 78,...Ch. 13 - Horizontal Tangent Plane In Exercises 77 and 78,...Ch. 13 - Using the Second Partials Test In Exercises 79-84,...Ch. 13 - Using the Second Partials Test In Exercises 79-84,...Ch. 13 - Using the Second Partials Test In Exercises 79-84,...Ch. 13 - Using the Second Partials Test In Exercises 79-84,...Ch. 13 - Using the Second Partials Test In Exercises 79-84,...Ch. 13 - Using the Second Partials Test In Exercises 79-84,...Ch. 13 - Finding Minimum Distance Find the minimum distance...Ch. 13 - Finding Positive Numbers Find three positive...Ch. 13 - Maximum Revenue A company manufactures two types...Ch. 13 - Maximum Profit A corporation manufactures digital...Ch. 13 - Finding the Least Squares Regression Line In...Ch. 13 - Finding the Least Squares Regression Line In...Ch. 13 - Modeling Data An agronomist used four test plots...Ch. 13 - Modeling Data The table shows the yield y (in...Ch. 13 - Using Lagrange Multipliers In Exercises 93-98, use...Ch. 13 - Using Lagrange Multipliers In Exercises 93-98, use...Ch. 13 - Using Lagrange Multipliers In Exercises 93-98, use...Ch. 13 - Using Lagrange Multipliers In Exercises 93-98, use...Ch. 13 - Using Lagrange Multipliers In Exercises 93-98, use...Ch. 13 - Using Lagrange Multipliers In Exercises 93-98, use...Ch. 13 - Minimum Cost A water line is to be built from...Ch. 13 - Area Herons Formula states that the area of a...Ch. 13 - Minimizing Material An industrial container is in...Ch. 13 - Tangent Plane Let P(x0,y0,z0) be a point in the...Ch. 13 - Using Functions Use a graphing utility to graph...Ch. 13 - Finding Maximum and Minimum Values (a) Let...Ch. 13 - Minimizing Costs A heated storage room has the...Ch. 13 - Minimizing Costs Repeat Exercise 6 assuming that...Ch. 13 - Temperature Consider a circular plate of radius 1...Ch. 13 - Cobb-Douglas Production Function Consider the...Ch. 13 - Minimizing Area Consider the ellipse x2a2+y2b2=1...Ch. 13 - Projectile Motion A projectile is launched at an...Ch. 13 - Distance Consider die distance d between the...Ch. 13 - Finding Extrema and Saddle Points Using Technology...Ch. 13 - Proof Prove that if f is a differentiable function...Ch. 13 - Area The figure shows a rectangle that is...Ch. 13 - Tangent Planes Let f be a differentiable function...Ch. 13 - Wave Equation Show that...Ch. 13 - Wave Equation Show that u(x,t)=12[f(xct)+f(x+ct)]...Ch. 13 - Verifying Equations Consider the function...Ch. 13 - Using a Function Demonstrate the result of...Ch. 13 - Laplaces Equation Rewrite Laplaces equation...

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