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10th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781285057095

Chapter 13, Problem 33RE

Textbook Problem

Using a Differential as an ApproximationIn Exercises 41 ad 42, (a) find

Calculus

Ch. 13.1 - Determining Whether a Graph Is a Function In...Ch. 13.1 - Determine whether graph is a function. Use the...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 7–18, find and...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - 57095-13.1-9E-Question-Digital.docx Evaluating a...Ch. 13.1 - 57095-13.1-10E-Question-Digital.docx Evaluating a...

Ch. 13.1 - Evaluating a Function In Exercises 7–18, find and...Ch. 13.1 - 57095-13.1-12E-Question-Digital.docx Evaluating a...Ch. 13.1 - Evaluating a Function In Exercises 7–18, find and...Ch. 13.1 - 57095-13.1-14E-Question-Digital.docx Evaluating a...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 - 57095-13.1-19E-Question-Digital.docx Finding the...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 - 57095-13.1-24E-Question-Digital.docx Finding the...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 - 57095-13.1-29E-Question-Digital.docx Finding the...Ch. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Think About It The graphs labeled (a), (b). 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Temperature Distribution The temperature T (in...Ch. 13.1 - Electric Potential The electric potential V at any...Ch. 13.1 - Cobb-Douglas Production FunctionUse the...Ch. 13.1 - Cobb-Douglas Production FunctionShow that the...Ch. 13.1 - Ideal Gas Law According to the Ideal Gas Law....Ch. 13.1 - 57095-13.1-82E-Question-Digital.docx Modeling Data...Ch. 13.1 - Meteorology. Meteorologists measure the...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 9194, determine...Ch. 13.1 - True or False? In Exercises 9194, determine...Ch. 13.1 - True or False? In Exercises 87–90, determine...Ch. 13.1 - True or False? In Exercises 91-94, determine...Ch. 13.1 - PUTNAM EXAM CHALLENGE Let f:=2 be a function such...Ch. 13.2 - Verifying a Limit by the Definition In Exercises...Ch. 13.2 - Verifying a Limit by the Definition In Exercises...Ch. 13.2 - Verifying a Limit by the Definition In Exercises...Ch. 13.2 - Verifying a Limit by the Definition In Exercises...Ch. 13.2 - Using Properties of Limits In Exercises 7-10, find...Ch. 13.2 - Using Properties of Limits In Exercises 5–8, find...Ch. 13.2 - Using Properties of Limits In Exercises 7-10, find...Ch. 13.2 - Using Properties of Limits In Exercises 7-10, find...Ch. 13.2 - 57095-13.2-9E-Question-Digital.docx Limit and...Ch. 13.2 - Limit and Continuity In Exercises 9–22, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Limit and Continuity In Exercises 11-24, find the...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - EXPLORING CONCEPTS Limits If...Ch. 13.2 - HOW DO YOU SEE IT? 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)x2yx2+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 63–68, 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 69–72, determine...Ch. 13.2 - True or False? In Exercises 69–72, determine...Ch. 13.2 - True or False? In Exercises 69–72, determine...Ch. 13.2 - 57095-13.2-72E-Question-Digital.docx True or...Ch. 13.2 - Finding a Limit Find the following limit....Ch. 13.2 - Continuity For the function f(x,y)=xy(x2y2x2+y2)...Ch. 13.2 - Proof Prove that...Ch. 13.2 - Proof Prove that if f is continuous and f(a,b)0,...Ch. 13.2 - 57095-13.2-81E-Question-Digital.docx WRITING ABOUT...Ch. 13.2 - 57095-13.2-82E-Question-Digital.docx WRITING ABOUT...Ch. 13.3 - First Partial Derivatives Sketch a surface...Ch. 13.3 - Mixed Partial Derivatives If f is a function of a...Ch. 13.3 - Examining a Partial Derivative In Exercises 5-10,...Ch. 13.3 - Examining a Partial Derivative In Exercises 5-10,...Ch. 13.3 - Examining a Partial Derivative In Exercises 510,...Ch. 13.3 - Examining a Partial Derivative In Exercises 5-10,...Ch. 13.3 - Examining a Partial Derivative In Exercises 5-10,...Ch. 13.3 - Examining a Partial Derivative In Exercises 5-10,...Ch. 13.3 - Finding Partial Derivatives In Exercises 11-40,...Ch. 13.3 - Finding Partial Derivatives In Exercises 11-40,...Ch. 13.3 - 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Laplaces Equation In Exercises 9598, show that the...Ch. 13.3 - Wave EquationIn Exercises 99102, show that the...Ch. 13.3 - Wave EquationIn Exercises 99102, show that the...Ch. 13.3 - Wave Equation In Exercises 99102, show that the...Ch. 13.3 - Wave EquationIn Exercises 99102, 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 - Using First Partial Derivatives In Exercises 107...Ch. 13.3 - Using First Partial Derivatives In Exercises 107...Ch. 13.3 - 57095-13.3-103E-Question-Digital.docx First...Ch. 13.3 - Sketching a Graph Sketch die graph of a function...Ch. 13.3 - Sketching a Graph Sketch the graph of a function...Ch. 13.3 - HOW DO YOU SEE IT? Use the graph of the surface to...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 the Cobb-Douglas...Ch. 13.3 - Think About It Let V 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 - 57095-13.3-119E-Question-Digital.docx Modeling...Ch. 13.3 - 57095-13.3-120E-Question-Digital.docx Modeling...Ch. 13.3 - 57095-13.3-121E-Question-Digital.docx True or...Ch. 13.3 - True or False? In Exercises 121–124, determine...Ch. 13.3 - True or False? In Exercises 121–124, determine...Ch. 13.3 - 57095-13.3-124E-Question-Digital.docx True or...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 line function...Ch. 13.4 - CONCEPT CHECK ApproximationDescribe the change in...Ch. 13.4 - CONCEPT CHECK Linear Approximation What is meant...Ch. 13.4 - 57095-13.4-1E-Question-Digital.docx Finding a...Ch. 13.4 - 57095-13.4-2E-Question-Digital.docx Finding a...Ch. 13.4 - Finding a Total Differential In Exercises 1–10,...Ch. 13.4 - 57095-13.4-4E-Question-Digital.docx Finding a...Ch. 13.4 - 57095-13.4-5E-Question-Digital.docx Finding a...Ch. 13.4 - Finding a Total DifferentialIn Exercises 38, find...Ch. 13.4 - Finding a Total Differential In Exercises 1–10,...Ch. 13.4 - Finding a Total DifferentialIn Exercises 38, find...Ch. 13.4 - Finding a Total Differential In Exercises 1–10,...Ch. 13.4 - Finding a Total Differential In Exercises 1–10,...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 - 57095-13.4-19E-Question-Digital.docx Approximating...Ch. 13.4 - 57095-13.4-23E-Question-Digital.docx Using...Ch. 13.4 - HOW DO YOU SEE IT? Which point has 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 - Numerical Analysis A right circular core of heigh...Ch. 13.4 - Volume The possible error involved in measuring...Ch. 13.4 - Numerical Analysis A right circular core of heigh...Ch. 13.4 - Wind Chill The formula for wind dull C (in degrees...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 - 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 5–10, 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 - 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 - 57095-13.5-13E-Question-Digital.docx Finding...Ch. 13.5 - Finding Partial Derivatives In Exercises 15-18....Ch. 13.5 - Finding Partial Derivatives In Exercises 1518....Ch. 13.5 - Finding Partial DerivativesIn Exercises 1518, 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 - Using Different Methods In Exercises 19-22, find ...Ch. 13.5 - Finding a Derivative ImplicitlyIn Exercises 2326,...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 - 57095-13.5-33E-Question-Digital.docx Finding...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 - 57095-13.5-40E-Question-Digital.docx 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 - 57095-13.5-43E-Question-Digital.docx WRITING ABOUT...Ch. 13.5 - 57095-13.5-44E-Question-Digital.docx WRITING ABOUT...Ch. 13.5 - 57095-13.5-45E-Question-Digital.docx WRITING ABOUT...Ch. 13.5 - 57095-13.5-46E-Question-Digital.docx HOW DO YOU...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 - Finding a Directional DerivativeIn Exercises 36,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 36,...Ch. 13.6 - Finding a Directional Derivative In Exercises 1–4,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 36,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 710,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 710,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 710,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 710,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 1114,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 1114,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 1114,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 1114,...Ch. 13.6 - Finding the Gradient of a FunctionIn Exercises...Ch. 13.6 - Finding the Gradient of a FunctionIn Exercises...Ch. 13.6 - Finding the Gradient of a Function In Exercises...Ch. 13.6 - Finding the Gradient of a FunctionIn Exercises...Ch. 13.6 - 57095-13.6-17E-Question-Digital.docx Finding the...Ch. 13.6 - Finding the Gradient of a FunctionIn Exercises...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2124,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2124,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2124,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2124,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2528,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2528,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2528,...Ch. 13.6 - Finding a Directional DerivativeIn Exercises 2528,...Ch. 13.6 - 57095-13.6-27E-Question-Digital.docx Using...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - 57095-13.6-31E-Question-Digital.docx Using...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Using a Function In Exercises 37-42, consider the...Ch. 13.6 - 57095-13.6-43E-Question-Digital.docx Investigation...Ch. 13.6 - 57095-13.6-44E-Question-Digital.docx Investigation...Ch. 13.6 - InvestigationConsider the function f(x,y)=x2y2 at...Ch. 13.6 - Investigation Consider the function...Ch. 13.6 - Finding a Normal Vector to a Level CurveIn...Ch. 13.6 - Finding a Normal Vector to a Level CurveIn...Ch. 13.6 - Finding a Normal Vector to a Level CurveIn...Ch. 13.6 - Finding a Normal Vector to a Level CurveIn...Ch. 13.6 - Using a FunctionIn Exercises 4346, (a) find the...Ch. 13.6 - Using a FunctionIn Exercises 4346, (a) find the...Ch. 13.6 - Using a FunctionIn Exercises 4346, (a) find the...Ch. 13.6 - Using a FunctionIn Exercises 4346, (a) find the...Ch. 13.6 - 57095-13.6-55E-Question-Digital.docx Directional...Ch. 13.6 - 56. Directional Derivative Write a paragraph...Ch. 13.6 - Using a Function Consider the function...Ch. 13.6 - 57095-13.6-57E-Question-Digital.docx Gradient...Ch. 13.6 - EXPLORING CONCEPTS Think About ItConsider v=3u. Is...Ch. 13.6 - EXPLORING CONCEPTS Sketching a Graph and a...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 IncreaseIn...Ch. 13.6 - Finding the Direction of Maximum IncreaseIn...Ch. 13.6 - Finding the Path of a Heat-Seeking ParticleIn...Ch. 13.6 - Finding the Path of a Heat-Seeking ParticleIn...Ch. 13.6 - True or False? In Exercises 6164, determine...Ch. 13.6 - True or False? In Exercises 6164, determine...Ch. 13.6 - True or False? In Exercises 6164, determine...Ch. 13.6 - True or False? In Exercises 6164, determine...Ch. 13.6 - Finding a FunctionFind 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 DerivativeConsider the function...Ch. 13.7 - Describing a SurfaceIn Exercises 36, describe the...Ch. 13.7 - 57095-13.7-2E-Question-Digital.docx Describing a...Ch. 13.7 - Describing a SurfaceIn Exercises 36, describe the...Ch. 13.7 - Describing a SurfaceIn Exercises 36, describe the...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13.7 - Finding an Equation of a Tangent PlaneIn Exercises...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 - 57095-13.7-16E-Question-Digital.docx Finding an...Ch. 13.7 - Finding an Equation of a Tangent Plane In...Ch. 13.7 - 57095-13.7-18E-Question-Digital.docx Finding an...Ch. 13.7 - 57095-13.7-5E-Question-Digital.docx Finding a Unit...Ch. 13.7 - Finding a Unit Normal Vector In Exercises 5–8,...Ch. 13.7 - Finding a Unit Normal Vector In Exercises 5–8,...Ch. 13.7 - Finding a Unit Normal Vector In Exercises 5–8,...Ch. 13.7 - Finding an Equation of a Tangent Plane and a...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 - 57095-13.7-23E-Question-Digital.docx Finding an...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...Ch. 13.7 - Finding the Equation of a Tangent Line to a...Ch. 13.7 - Finding the Equation of a Tangent Line to a...Ch. 13.7 - Finding the Equation of a Tangent Line to a...Ch. 13.7 - Finding the Equation of a Tangent Line to a...Ch. 13.7 - Finding the Equation of a Tangent Line to a...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 PlaneIn Exercises 3742, find...Ch. 13.7 - Horizontal Tangent PlaneIn Exercises 3742, find...Ch. 13.7 - Horizontal Tangent PlaneIn Exercises 3742, find...Ch. 13.7 - Horizontal Tangent PlaneIn Exercises 3742, find...Ch. 13.7 - Horizontal Tangent PlaneIn Exercises 3742, find...Ch. 13.7 - Horizontal Tangent PlaneIn Exercises 3742, find...Ch. 13.7 - Tangent SurfacesIn Exercises 43 and 44, show that...Ch. 13.7 - Tangent SurfacesIn Exercises 43 and 44, show that...Ch. 13.7 - Perpendicular Tangent PlanesIn Exercises 45 and...Ch. 13.7 - Perpendicular Tangent PlanesIn Exercises 45 and...Ch. 13.7 - Using a HyperboloidFind a point on the hyperboloid...Ch. 13.7 - Using an EllipsoidFind a point on the ellipsoid...Ch. 13.7 - 57095-13.7-53E-Question-Digital.docx Tangent Plane...Ch. 13.7 - 54. Normal Lines For some surfaces, the normal...Ch. 13.7 - 55. Tangent Plane Discuss the relationship between...Ch. 13.7 - HOW DO YOU SEE IT? The graph shows the ellipsoid...Ch. 13.7 - InvestigationConsider the function...Ch. 13.7 - InvestigationConsider the function f(x,y)=sinyx on...Ch. 13.7 - Using FunctionsConsider the functions...Ch. 13.7 - Using FunctionsConsider the functions...Ch. 13.7 - Writing a Tangent PlaneIn Exercises 57 and 58,...Ch. 13.7 - Writing a Tangent PlaneIn Exercises 57 and 58,...Ch. 13.7 - Tangent Planes of a ConeShow that any tangent...Ch. 13.7 - Tangent PlanesLet f be a differentiable function...Ch. 13.7 - Approximation Consider the following...Ch. 13.7 - ApproximationRepeat Exercise 61 for the function...Ch. 13.7 - ProofProve that the angle of inclination ? of the...Ch. 13.7 - Proof Prove theorem 13.14.Ch. 13.8 - Finding Relative ExtremaIn Exercises 38, identify...Ch. 13.8 - Finding Relative ExtremaIn Exercises 38, identify...Ch. 13.8 - Finding Relative ExtremaIn Exercises 38, identify...Ch. 13.8 - 57095-13.8-4E-Question-Digital.docx Finding...Ch. 13.8 - Finding Relative ExtremaIn Exercises 38, identify...Ch. 13.8 - Finding Relative ExtremaIn Exercises 38, identify...Ch. 13.8 - 57095-13.8-7E-Question-Digital.docx Using the...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - 57095-13.8-9E-Question-Digital.docx Using the...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials Test In Exercises 7–20,...Ch. 13.8 - 57095-13.8-12E-Question-Digital.docx Using the...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Using the Second Partials TestIn Exercises 924,...Ch. 13.8 - Finding Relative Extrema and Saddle Points Using...Ch. 13.8 - 57095-13.8-22E-Question-Digital.docx Finding...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 ExtremaIn Exercises 29 and 30,...Ch. 13.8 - Finding Relative ExtremaIn Exercises 29 and 30,...Ch. 13.8 - Think About ItIn Exercises 3134, determine whether...Ch. 13.8 - Think About ItIn Exercises 3134, determine whether...Ch. 13.8 - Think About ItIn Exercises 3134, determine whether...Ch. 13.8 - Think About ItIn Exercises 3134, determine whether...Ch. 13.8 - 31. Using the Second Partials Test A function f...Ch. 13.8 - 57095-13.8-32E-Question-Digital.docx Using the...Ch. 13.8 - Finding Relative Extrema and Saddle PointsIn...Ch. 13.8 - Finding Relative Extrema and Saddle PointsIn...Ch. 13.8 - Finding Relative Extrema and Saddle Points In...Ch. 13.8 - 57095-13.8-36E-Question-Digital.docx Finding...Ch. 13.8 - 57095-13.8-37E-Question-Digital.docx Finding...Ch. 13.8 - Finding Relative Extrema and Saddle PointsIn...Ch. 13.8 - Examining a FunctionIn Exercises 47 and 48, find...Ch. 13.8 - Examining a Function In Exercises 47 and 48, find...Ch. 13.8 - Finding Absolute ExtremaIn Exercises 3946, 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 ExtremaIn Exercises 3946, find...Ch. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Finding Absolute ExtremaIn Exercises 3946, find...Ch. 13.8 - 57095-13.8-49E-Question-Digital.docx Defining...Ch. 13.8 - EXPLORING CONCEPTS Sketching a GraphIn Exercises...Ch. 13.8 - EXPLORING CONCEPTS Sketching a Graph In Exercises...Ch. 13.8 - 57095-13.8-52E-Question-Digital.docx Sketching a...Ch. 13.8 - Comparing Functions Consider the functions...Ch. 13.8 - 57095-13.8-54E-Question-Digital.docx HOW DO YOU...Ch. 13.8 - True or False? In Exercises 5558, determine...Ch. 13.8 - True or False? In Exercises 5558, determine...Ch. 13.8 - True or False? In Exercises 5558, determine...Ch. 13.8 - True or False? In Exercises 5558, determine...Ch. 13.9 - CONCEPT CHECK Applied Optimization ProblemsIn your...Ch. 13.9 - CONCEPT CHECK Method of Least SquaresIn your own...Ch. 13.9 - Finding Minimum Distance In Exercises 1 and 2,...Ch. 13.9 - 57095-13.9-2E-Question-Digital.docx Finding...Ch. 13.9 - Finding Minimum DistanceIn Exercises 5 and 6, find...Ch. 13.9 - Finding Minimum DistanceIn Exercises 5 and 6, 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 - Finding Positive Numbers In Exercises 7-10, find...Ch. 13.9 - CostA home improvement contractor is painting the...Ch. 13.9 - Maximum Volume The material for constructing the...Ch. 13.9 - Volume and Surface AreaShow that a rectangular box...Ch. 13.9 - Maximum VolumeShow that the rectangular box of...Ch. 13.9 - Maximum RevenueA company manufactures running...Ch. 13.9 - Maximum ProfitA corporation manufactures candles...Ch. 13.9 - Hardy-Weinberg LawCommon blood types are...Ch. 13.9 - Shannon Diversity IndexOne way to measure species...Ch. 13.9 - Minimum CostA water line is to be built from point...Ch. 13.9 - AreaA trough with trapezoidal cross sections is...Ch. 13.9 - Finding the Least Squares Regression LineIn...Ch. 13.9 - Finding the Least Squares Regression LineIn...Ch. 13.9 - Finding the Least Squares Regression LineIn...Ch. 13.9 - Finding the Least Squares Regression LineIn...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 LineIn...Ch. 13.9 - Finding the Least Squares Regression LineIn...Ch. 13.9 - Modeling Data The ages x (in years) and systolic...Ch. 13.9 - 30. Modeling Data The table shows the gross income...Ch. 13.9 - EXPLORING CONCEPTS Method of Least SquaresFind a...Ch. 13.9 - HOW DO YOU SEE IT? Match the regression equation...Ch. 13.9 - Finding the Least Squares Regression QuadraticIn...Ch. 13.9 - Finding the Least Squares Regression QuadraticIn...Ch. 13.9 - Finding the Least Squares Regression QuadraticIn...Ch. 13.9 - Finding the Least Squares Regression QuadraticIn...Ch. 13.9 - Modeling DataAfter a new turbocharger for an...Ch. 13.9 - 57095-13.9-38E-Question-Digital.docx Modeling Data...Ch. 13.9 - Modeling DataA meteorologist measures the...Ch. 13.9 - Modeling Data The endpoints of the interval over...Ch. 13.9 - Using the Second Partial TestUse 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 310, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 310, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 310, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 310, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 310, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 3-10, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 310, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 310, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 1114, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 1114, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 1114, use...Ch. 13.10 - Using Lagrange Multipliers In Exercises 1114, 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 - 57095-13.10-31E-Question-Digital.docx Using...Ch. 13.10 - 57095-13.10-32E-Question-Digital.docx Using...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 - 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 - 57095-13-1RE-Question-Digital.docx Evaluating a...Ch. 13 - Evaluating a FunctionIn Exercises 1 and 2,...Ch. 13 - Finding the Domain and Range of a FunctionIn...Ch. 13 - Finding the Domain and Range of a FunctionIn...Ch. 13 - Sketching a Contour MapIn Exercises 7 and 8,...Ch. 13 - Sketching a Contour MapIn Exercises 7 and 8,...Ch. 13 - ConjectureConsider the function f(x,y)=x2+y2. (a)...Ch. 13 - 8. Investment A principal of $2000 is deposited in...Ch. 13 - Sketching a Level Surface In Exercises 11 and 12,...Ch. 13 - Sketching a Level SurfaceIn Exercises 11 and 12,...Ch. 13 - Limit and ContinuityIn Exercises 1318, find the...Ch. 13 - Limit and ContinuityIn Exercises 1318, find the...Ch. 13 - Limit and Continuity In Exercises 13-18, find the...Ch. 13 - Limit and ContinuityIn Exercises 1318, find the...Ch. 13 - Finding Partial DerivativesIn Exercises 1926, find...Ch. 13 - Finding Partial DerivativesIn Exercises 1926, find...Ch. 13 - Finding Partial DerivativesIn Exercises 1926, find...Ch. 13 - Finding Partial DerivativesIn Exercises 1926, find...Ch. 13 - Finding Partial DerivativesIn Exercises 1926, find...Ch. 13 - Finding Partial DerivativesIn Exercises 1926, find...Ch. 13 - Finding Partial Derivatives In Exercises 15–22,...Ch. 13 - Finding Partial DerivativesIn Exercises 1926, find...Ch. 13 - Finding Second Partial DerivativesIn Exercises...Ch. 13 - Finding Second Partial DerivativesIn Exercises...Ch. 13 - Finding Second Partial DerivativesIn Exercises...Ch. 13 - Finding Second Partial DerivativesIn Exercises...Ch. 13 - Finding the Slopes of a SurfaceFind the slopes of...Ch. 13 - Marginal RevenueA company has two plants that...Ch. 13 - Finding a Total DifferentialIn Exercises 3740,...Ch. 13 - Finding a Total DifferentialIn Exercises 3740,...Ch. 13 - Finding a Total DifferentialIn Exercises 3740,...Ch. 13 - Finding a Total DifferentialIn Exercises 3740,...Ch. 13 - Using a Differential as an ApproximationIn...Ch. 13 - Using a Differential as an ApproximationIn...Ch. 13 - Volume The possible error involved in measuring...Ch. 13 - Lateral Surface AreaApproximate the propagated...Ch. 13 - Using Different MethodsIn Exercises 4750, find...Ch. 13 - Using Different MethodsIn Exercises 4750, find...Ch. 13 - Using Different MethodsIn Exercises 51 and 52,...Ch. 13 - Using Different MethodsIn Exercises 51 and 52,...Ch. 13 - Finding Partial Derivative ImplicitlyIn Exercises...Ch. 13 - Finding Partial Derivative ImplicitlyIn Exercises...Ch. 13 - Finding a Directional DerivativeIn Exercises 57...Ch. 13 - Finding a Directional DerivativeIn Exercises 57...Ch. 13 - Finding a Directional DerivativeIn Exercises 59...Ch. 13 - Finding a Directional DerivativeIn Exercises 59...Ch. 13 - Using Properties of the GradientIn Exercises 6166,...Ch. 13 - Using Properties of the GradientIn Exercises 6166,...Ch. 13 - Using Properties of the GradientIn Exercises 6166,...Ch. 13 - Using Properties of the GradientIn Exercises 6166,...Ch. 13 - Using a Function In Exercises 67 and 68, (a) find...Ch. 13 - Using a FunctionIn Exercises 67 and 68, (a) find...Ch. 13 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13 - Finding an Equation of a Tangent PlaneIn Exercises...Ch. 13 - Finding an Equation of a Tangent PlaneIn Exercises...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 - 60. Approximation Consider the following...Ch. 13 - Using the Second Partial TestIn Exercises 7984,...Ch. 13 - Using the Second Partial TestIn Exercises 7984,...Ch. 13 - Using the Second Partial TestIn Exercises 7984,...Ch. 13 - 57095-13-64RE-Question-Digital.docx Using the...Ch. 13 - Using the Second Partial TestIn Exercises 7984,...Ch. 13 - Using the Second Partial TestIn Exercises 7984,...Ch. 13 - Finding Minimum DistanceFind the minimum distance...Ch. 13 - Finding Positive Numbers Find three positive...Ch. 13 - Maximum RevenueA company manufactures two type of...Ch. 13 - Maximum ProfitA corporation manufactures digital...Ch. 13 - Finding the Least Squares Regression LineIn...Ch. 13 - Finding the Least Squares Regression LineIn...Ch. 13 - Modeling DataAn agronomist used four test plots to...Ch. 13 - Modeling Data The table shows the yield y (in...Ch. 13 - Using Lagrange MultipliersIn Exercises 9398, use...Ch. 13 - Using Lagrange MultipliersIn Exercises 9398, use...Ch. 13 - Using Lagrange MultipliersIn Exercises 9398, use...Ch. 13 - Using Lagrange MultipliersIn Exercises 9398, use...Ch. 13 - Using Lagrange MultipliersIn Exercises 9398, use...Ch. 13 - Using Lagrange MultipliersIn Exercises 9398, use...Ch. 13 - Minimum CostA water line is to be built from point...Ch. 13 - Area Herons Formula states that the area of a...Ch. 13 - Minimizing MaterialAn industrial container is in...Ch. 13 - Tangent PlaneLet P(x0,y0,z0) be a point in the...Ch. 13 - Using FunctionsUse a graphing utility to graph the...Ch. 13 - Finding Maximum and Minimum Values (a) Let...Ch. 13 - Minimizing CostsA heated storage room has the...Ch. 13 - Minimizing Costs Repeat Exercise 6 assuming that...Ch. 13 - TemperatureConsider a circular plate of radius 1...Ch. 13 - Cobb-Douglas Production FunctionConsider the...Ch. 13 - Minimizing AreaConsider the ellipse x2a2+y2b2=1...Ch. 13 - Projectile MotionA 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 PlaneLet f be a differentiable function of...Ch. 13 - Wave EquationShow that u(x,t)=12[sin(xt)+sin(x+t)]...Ch. 13 - Wave EquationShow that u(x,t)=12[f(xct)+f(x+ct)]...Ch. 13 - Verifying EquationsConsider the function w=f(x,y)...Ch. 13 - Using a FunctionDemonstrate the result of Exercise...Ch. 13 - Laplaces EquationRewrite Laplaces equation...

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26.

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0
1
π
It does not exist.

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P(−1....

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