Multivariable Calculus
Multivariable Calculus
11th Edition
ISBN: 9781337275378
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Concept explainers

Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 13, Problem 87RE

Maximum RevenueA company manufactures two type of bicycles, a racing bicycle and a mountain bicycle. The total revenue (in thousands of dollars) from x 1 units of racing bicycles and x 2 units of mountain bicycles is

R = − 6 x 1 2 − 10 x 2 2 − 2 x 1 x 2 + 32 x 1 + 84 x 2

where x 1 and x 2 are in thousands of units. Find x 1 and x 2 so as to maximize the revenue.

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The solutions are 1 where x1 x2- ● Question 11 Solve: x 54 Give your answer as an interval. Question 12

Chapter 13 Solutions

Multivariable Calculus

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 - Prob. 3ECh. 13.1 - Prob. 4ECh. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Prob. 6ECh. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Prob. 9ECh. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...
Ch. 13.1 - Prob. 11ECh. 13.1 - Prob. 12ECh. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Prob. 14ECh. 13.1 - Prob. 15ECh. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Prob. 18ECh. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Prob. 20ECh. 13.1 - Prob. 21ECh. 13.1 - Prob. 22ECh. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Prob. 24ECh. 13.1 - Prob. 25ECh. 13.1 - Prob. 26ECh. 13.1 - Prob. 27ECh. 13.1 - Prob. 28ECh. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Prob. 30ECh. 13.1 - Finding the Domain and Range of a Function In...Ch. 13.1 - Prob. 32ECh. 13.1 - Prob. 33ECh. 13.1 - Prob. 34ECh. 13.1 - Prob. 35ECh. 13.1 - Prob. 36ECh. 13.1 - Prob. 37ECh. 13.1 - Prob. 38ECh. 13.1 - Prob. 39ECh. 13.1 - Prob. 40ECh. 13.1 - Prob. 41ECh. 13.1 - Prob. 42ECh. 13.1 - Prob. 43ECh. 13.1 - Graphing a Function Using Technology In Exercises...Ch. 13.1 - Prob. 45ECh. 13.1 - Prob. 46ECh. 13.1 - Prob. 47ECh. 13.1 - Prob. 48ECh. 13.1 - Prob. 49ECh. 13.1 - Prob. 50ECh. 13.1 - Prob. 51ECh. 13.1 - Prob. 52ECh. 13.1 - Prob. 53ECh. 13.1 - Prob. 54ECh. 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 - Prob. 59ECh. 13.1 - Sraphing Level Curves Using Technology In...Ch. 13.1 - Sraphing Level Curves Using Technology In...Ch. 13.1 - Sraphing Level Curves Using Technology In...Ch. 13.1 - Prob. 63ECh. 13.1 - Prob. 64ECh. 13.1 - Creating a FunctionConstruct a function whose...Ch. 13.1 - Prob. 66ECh. 13.1 - Prob. 67ECh. 13.1 - Prob. 68ECh. 13.1 - Prob. 69ECh. 13.1 - Prob. 70ECh. 13.1 - Sketching a Level Surface In Exercises 71-76,...Ch. 13.1 - Prob. 72ECh. 13.1 - Prob. 73ECh. 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 - Prob. 77ECh. 13.1 - Prob. 78ECh. 13.1 - Prob. 79ECh. 13.1 - Electric Potential The electric potential V at any...Ch. 13.1 - Prob. 81ECh. 13.1 - Prob. 82ECh. 13.1 - Prob. 83ECh. 13.1 - Cobb-Douglas Production FunctionShow that the...Ch. 13.1 - Ideal Gas Law According to the Ideal Gas Law....Ch. 13.1 - Modeling Data The table shows the net sales x (in...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 - Prob. 91ECh. 13.1 - Prob. 92ECh. 13.1 - Prob. 93ECh. 13.1 - True or False? In Exercises 91-94, determine...Ch. 13.1 - Prob. 95ECh. 13.2 - CONCEPT CHECK Describing Notation Write a brief...Ch. 13.2 - CONCEPT CHECK Limits Explain how examining limits...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 - Prob. 7ECh. 13.2 - Prob. 8ECh. 13.2 - Prob. 9ECh. 13.2 - Prob. 10ECh. 13.2 - Prob. 11ECh. 13.2 - Prob. 12ECh. 13.2 - Prob. 13ECh. 13.2 - Prob. 14ECh. 13.2 - Prob. 15ECh. 13.2 - Prob. 16ECh. 13.2 - Prob. 17ECh. 13.2 - Prob. 18ECh. 13.2 - Prob. 19ECh. 13.2 - Prob. 20ECh. 13.2 - Prob. 21ECh. 13.2 - Prob. 22ECh. 13.2 - Prob. 23ECh. 13.2 - Prob. 24ECh. 13.2 - Prob. 25ECh. 13.2 - Prob. 26ECh. 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 - Prob. 30ECh. 13.2 - Finding a Limit In Exercises 25-36, find the limit...Ch. 13.2 - Prob. 32ECh. 13.2 - Prob. 33ECh. 13.2 - Prob. 34ECh. 13.2 - Prob. 35ECh. 13.2 - Prob. 36ECh. 13.2 - Prob. 37ECh. 13.2 - Prob. 38ECh. 13.2 - Prob. 39ECh. 13.2 - Prob. 40ECh. 13.2 - Prob. 41ECh. 13.2 - Prob. 42ECh. 13.2 - Prob. 43ECh. 13.2 - Prob. 44ECh. 13.2 - Prob. 45ECh. 13.2 - Prob. 46ECh. 13.2 - Prob. 47ECh. 13.2 - Prob. 48ECh. 13.2 - Comparing Continuity In Exercises 49 and 50,...Ch. 13.2 - Prob. 50ECh. 13.2 - Prob. 51ECh. 13.2 - Prob. 52ECh. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Prob. 54ECh. 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 - Prob. 58ECh. 13.2 - Finding a Limit Using Polar Coordinates In...Ch. 13.2 - Prob. 60ECh. 13.2 - Prob. 61ECh. 13.2 - Prob. 62ECh. 13.2 - Continuity In Exercises 61-66, discuss the...Ch. 13.2 - Prob. 64ECh. 13.2 - Prob. 65ECh. 13.2 - Prob. 66ECh. 13.2 - Prob. 67ECh. 13.2 - Prob. 68ECh. 13.2 - Prob. 69ECh. 13.2 - Prob. 70ECh. 13.2 - Prob. 71ECh. 13.2 - Prob. 72ECh. 13.2 - Finding a Limit In Exercises 71-76, find each...Ch. 13.2 - Prob. 74ECh. 13.2 - Prob. 75ECh. 13.2 - Prob. 76ECh. 13.2 - Finding a Limit Using Spherical Coordinates In...Ch. 13.2 - Prob. 78ECh. 13.2 - Prob. 79ECh. 13.2 - Prob. 80ECh. 13.2 - Prob. 81ECh. 13.2 - Prob. 82ECh. 13.2 - Prob. 83ECh. 13.2 - Prob. 84ECh. 13.2 - Prob. 85ECh. 13.2 - Prob. 86ECh. 13.3 - Prob. 1ECh. 13.3 - Prob. 2ECh. 13.3 - Higher-Order Partial Derivatives Describe the...Ch. 13.3 - Prob. 4ECh. 13.3 - Prob. 5ECh. 13.3 - Prob. 6ECh. 13.3 - Prob. 7ECh. 13.3 - Prob. 8ECh. 13.3 - Prob. 9ECh. 13.3 - Prob. 10ECh. 13.3 - Prob. 11ECh. 13.3 - Prob. 12ECh. 13.3 - Prob. 13ECh. 13.3 - Prob. 14ECh. 13.3 - Prob. 15ECh. 13.3 - Prob. 16ECh. 13.3 - Prob. 17ECh. 13.3 - Prob. 18ECh. 13.3 - Prob. 19ECh. 13.3 - Prob. 20ECh. 13.3 - Prob. 21ECh. 13.3 - Prob. 22ECh. 13.3 - Prob. 23ECh. 13.3 - Prob. 24ECh. 13.3 - Prob. 25ECh. 13.3 - Prob. 26ECh. 13.3 - Prob. 27ECh. 13.3 - Prob. 28ECh. 13.3 - Prob. 29ECh. 13.3 - Prob. 30ECh. 13.3 - Prob. 31ECh. 13.3 - Prob. 32ECh. 13.3 - Prob. 33ECh. 13.3 - Prob. 34ECh. 13.3 - Prob. 35ECh. 13.3 - Prob. 36ECh. 13.3 - Prob. 37ECh. 13.3 - Prob. 38ECh. 13.3 - Prob. 39ECh. 13.3 - Prob. 40ECh. 13.3 - Prob. 41ECh. 13.3 - Prob. 42ECh. 13.3 - Prob. 43ECh. 13.3 - Prob. 44ECh. 13.3 - Prob. 45ECh. 13.3 - Prob. 46ECh. 13.3 - Prob. 47ECh. 13.3 - Prob. 48ECh. 13.3 - Prob. 49ECh. 13.3 - Prob. 50ECh. 13.3 - Prob. 51ECh. 13.3 - Prob. 52ECh. 13.3 - Prob. 53ECh. 13.3 - Prob. 54ECh. 13.3 - Prob. 55ECh. 13.3 - Prob. 56ECh. 13.3 - Prob. 57ECh. 13.3 - Prob. 58ECh. 13.3 - Prob. 59ECh. 13.3 - Prob. 60ECh. 13.3 - Prob. 61ECh. 13.3 - Prob. 62ECh. 13.3 - Prob. 63ECh. 13.3 - Prob. 64ECh. 13.3 - Prob. 65ECh. 13.3 - Prob. 66ECh. 13.3 - Prob. 67ECh. 13.3 - Prob. 68ECh. 13.3 - Prob. 69ECh. 13.3 - Prob. 70ECh. 13.3 - Prob. 71ECh. 13.3 - Prob. 72ECh. 13.3 - Prob. 73ECh. 13.3 - Prob. 74ECh. 13.3 - Prob. 75ECh. 13.3 - Prob. 76ECh. 13.3 - Prob. 77ECh. 13.3 - Prob. 78ECh. 13.3 - Prob. 79ECh. 13.3 - Prob. 80ECh. 13.3 - Prob. 81ECh. 13.3 - Prob. 82ECh. 13.3 - Prob. 83ECh. 13.3 - Prob. 84ECh. 13.3 - Prob. 85ECh. 13.3 - Prob. 86ECh. 13.3 - Prob. 87ECh. 13.3 - Prob. 88ECh. 13.3 - Prob. 89ECh. 13.3 - Prob. 90ECh. 13.3 - Prob. 91ECh. 13.3 - Prob. 92ECh. 13.3 - Prob. 93ECh. 13.3 - Prob. 94ECh. 13.3 - Prob. 95ECh. 13.3 - Prob. 96ECh. 13.3 - Prob. 97ECh. 13.3 - Prob. 98ECh. 13.3 - Prob. 99ECh. 13.3 - Prob. 100ECh. 13.3 - Prob. 101ECh. 13.3 - Prob. 102ECh. 13.3 - Prob. 103ECh. 13.3 - Prob. 104ECh. 13.3 - Prob. 105ECh. 13.3 - Prob. 106ECh. 13.3 - Prob. 107ECh. 13.3 - Prob. 108ECh. 13.3 - Prob. 109ECh. 13.3 - Prob. 110ECh. 13.3 - Prob. 111ECh. 13.3 - Prob. 112ECh. 13.3 - Prob. 113ECh. 13.3 - Prob. 114ECh. 13.3 - Prob. 115ECh. 13.3 - Prob. 116ECh. 13.3 - Prob. 117ECh. 13.3 - Prob. 118ECh. 13.3 - Prob. 119ECh. 13.3 - Prob. 120ECh. 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 - Prob. 123ECh. 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 - Prob. 126ECh. 13.3 - Prob. 127ECh. 13.3 - Prob. 128ECh. 13.3 - Prob. 129ECh. 13.3 - Prob. 130ECh. 13.3 - Prob. 131ECh. 13.4 - CONCEPT CHECK ApproximationDescribe the change in...Ch. 13.4 - Prob. 2ECh. 13.4 - Prob. 3ECh. 13.4 - Prob. 4ECh. 13.4 - Finding a Total DifferentialIn Exercises 38, find...Ch. 13.4 - Finding a Total DifferentialIn Exercises 38, find...Ch. 13.4 - Prob. 7ECh. 13.4 - Prob. 8ECh. 13.4 - Prob. 9ECh. 13.4 - Prob. 10ECh. 13.4 - Prob. 11ECh. 13.4 - Prob. 12ECh. 13.4 - Using a Differential as an Approximation In...Ch. 13.4 - Prob. 14ECh. 13.4 - Prob. 15ECh. 13.4 - Prob. 16ECh. 13.4 - Prob. 17ECh. 13.4 - Prob. 18ECh. 13.4 - Prob. 19ECh. 13.4 - Prob. 20ECh. 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 - Prob. 23ECh. 13.4 - Prob. 24ECh. 13.4 - Prob. 25ECh. 13.4 - Prob. 26ECh. 13.4 - Wind Chill The formula for wind dull C (in degrees...Ch. 13.4 - Prob. 28ECh. 13.4 - Prob. 29ECh. 13.4 - Prob. 30ECh. 13.4 - Volume A trough is 16 feet long (see figure). Its...Ch. 13.4 - Prob. 32ECh. 13.4 - Prob. 33ECh. 13.4 - Prob. 34ECh. 13.4 - Prob. 35ECh. 13.4 - Prob. 36ECh. 13.4 - Prob. 37ECh. 13.4 - Prob. 38ECh. 13.4 - Prob. 39ECh. 13.4 - Prob. 40ECh. 13.5 - Prob. 1ECh. 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...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 - Prob. 15ECh. 13.5 - Prob. 16ECh. 13.5 - Prob. 17ECh. 13.5 - Prob. 18ECh. 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 - Prob. 22ECh. 13.5 - Finding a Derivative ImplicitlyIn Exercises 2326,...Ch. 13.5 - Prob. 24ECh. 13.5 - Finding a Derivative Implicitly In Exercises...Ch. 13.5 - Prob. 26ECh. 13.5 - Prob. 27ECh. 13.5 - Prob. 28ECh. 13.5 - Prob. 29ECh. 13.5 - Prob. 30ECh. 13.5 - Prob. 31ECh. 13.5 - Prob. 32ECh. 13.5 - Prob. 33ECh. 13.5 - Prob. 34ECh. 13.5 - Prob. 35ECh. 13.5 - Prob. 36ECh. 13.5 - Prob. 37ECh. 13.5 - Prob. 38ECh. 13.5 - Homogeneous Functions A function f is homogeneous...Ch. 13.5 - Prob. 40ECh. 13.5 - Prob. 41ECh. 13.5 - Prob. 42ECh. 13.5 - Prob. 43ECh. 13.5 - Prob. 44ECh. 13.5 - Prob. 45ECh. 13.5 - Prob. 46ECh. 13.5 - Prob. 47ECh. 13.5 - Prob. 48ECh. 13.5 - Prob. 49ECh. 13.5 - Prob. 50ECh. 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 - Prob. 1ECh. 13.6 - Prob. 2ECh. 13.6 - Prob. 3ECh. 13.6 - Prob. 4ECh. 13.6 - Finding a Directional DerivativeIn Exercises 36,...Ch. 13.6 - Prob. 6ECh. 13.6 - Prob. 7ECh. 13.6 - Prob. 8ECh. 13.6 - Prob. 9ECh. 13.6 - Prob. 10ECh. 13.6 - Prob. 11ECh. 13.6 - Prob. 12ECh. 13.6 - Prob. 13ECh. 13.6 - Prob. 14ECh. 13.6 - Prob. 15ECh. 13.6 - Prob. 16ECh. 13.6 - Finding the Gradient of a FunctionIn Exercises...Ch. 13.6 - Prob. 18ECh. 13.6 - Prob. 19ECh. 13.6 - Prob. 20ECh. 13.6 - Prob. 21ECh. 13.6 - Prob. 22ECh. 13.6 - Prob. 23ECh. 13.6 - Prob. 24ECh. 13.6 - Prob. 25ECh. 13.6 - Prob. 26ECh. 13.6 - Prob. 27ECh. 13.6 - Prob. 28ECh. 13.6 - Prob. 29ECh. 13.6 - Prob. 30ECh. 13.6 - Using Properties of the GradientIn Exercises 2938,...Ch. 13.6 - Prob. 32ECh. 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 - Prob. 36ECh. 13.6 - Prob. 37ECh. 13.6 - Prob. 38ECh. 13.6 - Prob. 39ECh. 13.6 - Prob. 40ECh. 13.6 - Prob. 41ECh. 13.6 - Prob. 42ECh. 13.6 - Using a FunctionIn Exercises 4346, (a) find the...Ch. 13.6 - Prob. 44ECh. 13.6 - Prob. 45ECh. 13.6 - Prob. 46ECh. 13.6 - Prob. 47ECh. 13.6 - Prob. 48ECh. 13.6 - Prob. 49ECh. 13.6 - Prob. 50ECh. 13.6 - Prob. 51ECh. 13.6 - Prob. 52ECh. 13.6 - Topography The surface of a mountain is modeled by...Ch. 13.6 - Prob. 54ECh. 13.6 - Temperature The temperature at the point (x,y) on...Ch. 13.6 - Prob. 56ECh. 13.6 - Prob. 57ECh. 13.6 - Prob. 58ECh. 13.6 - Prob. 59ECh. 13.6 - Prob. 60ECh. 13.6 - Prob. 61ECh. 13.6 - Prob. 62ECh. 13.6 - Prob. 63ECh. 13.6 - Prob. 64ECh. 13.6 - Prob. 65ECh. 13.6 - Ocean Floor A team of oceanographers is mapping...Ch. 13.6 - Prob. 67ECh. 13.6 - Prob. 68ECh. 13.7 - CONCEPT CHECK Tangent VectorConsider a point...Ch. 13.7 - Prob. 2ECh. 13.7 - Prob. 3ECh. 13.7 - Prob. 4ECh. 13.7 - Prob. 5ECh. 13.7 - Prob. 6ECh. 13.7 - Prob. 7ECh. 13.7 - Prob. 8ECh. 13.7 - Prob. 9ECh. 13.7 - Prob. 10ECh. 13.7 - Prob. 11ECh. 13.7 - Prob. 12ECh. 13.7 - Prob. 13ECh. 13.7 - Prob. 14ECh. 13.7 - Prob. 15ECh. 13.7 - Prob. 16ECh. 13.7 - Prob. 17ECh. 13.7 - Prob. 18ECh. 13.7 - Prob. 19ECh. 13.7 - Prob. 20ECh. 13.7 - Prob. 21ECh. 13.7 - Prob. 22ECh. 13.7 - Prob. 23ECh. 13.7 - Prob. 24ECh. 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 - Prob. 27ECh. 13.7 - Prob. 28ECh. 13.7 - Prob. 29ECh. 13.7 - Prob. 30ECh. 13.7 - Prob. 31ECh. 13.7 - Prob. 32ECh. 13.7 - Prob. 33ECh. 13.7 - Prob. 34ECh. 13.7 - Finding the Angle of Inclination of a Tangent...Ch. 13.7 - Prob. 36ECh. 13.7 - Prob. 37ECh. 13.7 - Prob. 38ECh. 13.7 - Horizontal Tangent PlaneIn Exercises 3742, find...Ch. 13.7 - Prob. 40ECh. 13.7 - Prob. 41ECh. 13.7 - Prob. 42ECh. 13.7 - Prob. 43ECh. 13.7 - Prob. 44ECh. 13.7 - Prob. 45ECh. 13.7 - Prob. 46ECh. 13.7 - Prob. 47ECh. 13.7 - Prob. 48ECh. 13.7 - Using an EllipsoidFind a point on the ellipsoid...Ch. 13.7 - Using a HyperboloidFind a point on the hyperboloid...Ch. 13.7 - Prob. 51ECh. 13.7 - HOW DO YOU SEE IT? The graph shows the ellipsoid...Ch. 13.7 - Prob. 53ECh. 13.7 - Prob. 54ECh. 13.7 - Prob. 55ECh. 13.7 - Prob. 56ECh. 13.7 - Prob. 57ECh. 13.7 - Prob. 58ECh. 13.7 - Prob. 59ECh. 13.7 - Prob. 60ECh. 13.7 - Prob. 61ECh. 13.7 - ApproximationRepeat Exercise 61 for the function...Ch. 13.7 - Prob. 63ECh. 13.7 - Prob. 64ECh. 13.8 - CONCEPT CHECK Function of Two VariablesFor a...Ch. 13.8 - Prob. 2ECh. 13.8 - Prob. 3ECh. 13.8 - Prob. 4ECh. 13.8 - Prob. 5ECh. 13.8 - Prob. 6ECh. 13.8 - Prob. 7ECh. 13.8 - Prob. 8ECh. 13.8 - Prob. 9ECh. 13.8 - Prob. 10ECh. 13.8 - Prob. 11ECh. 13.8 - Prob. 12ECh. 13.8 - Prob. 13ECh. 13.8 - Prob. 14ECh. 13.8 - Prob. 15ECh. 13.8 - Prob. 16ECh. 13.8 - Prob. 17ECh. 13.8 - Prob. 18ECh. 13.8 - Prob. 19ECh. 13.8 - Prob. 20ECh. 13.8 - Prob. 21ECh. 13.8 - Prob. 22ECh. 13.8 - Prob. 23ECh. 13.8 - Prob. 24ECh. 13.8 - Prob. 25ECh. 13.8 - Prob. 26ECh. 13.8 - Prob. 27ECh. 13.8 - Prob. 28ECh. 13.8 - Prob. 29ECh. 13.8 - Prob. 30ECh. 13.8 - Prob. 31ECh. 13.8 - Prob. 32ECh. 13.8 - Prob. 33ECh. 13.8 - Prob. 34ECh. 13.8 - Prob. 35ECh. 13.8 - Prob. 36ECh. 13.8 - Prob. 37ECh. 13.8 - Prob. 38ECh. 13.8 - Finding Absolute ExtremaIn Exercises 3946, find...Ch. 13.8 - Prob. 40ECh. 13.8 - Finding Absolute Extrema In Exercises 39-46, find...Ch. 13.8 - Prob. 42ECh. 13.8 - Prob. 43ECh. 13.8 - Prob. 44ECh. 13.8 - Prob. 45ECh. 13.8 - Prob. 46ECh. 13.8 - Prob. 47ECh. 13.8 - Prob. 48ECh. 13.8 - Prob. 49ECh. 13.8 - Prob. 50ECh. 13.8 - Prob. 51ECh. 13.8 - Prob. 52ECh. 13.8 - Prob. 53ECh. 13.8 - Prob. 54ECh. 13.8 - Prob. 55ECh. 13.8 - Prob. 56ECh. 13.8 - Prob. 57ECh. 13.8 - Prob. 58ECh. 13.9 - Prob. 1ECh. 13.9 - CONCEPT CHECK Method of Least SquaresIn your own...Ch. 13.9 - Prob. 3ECh. 13.9 - Prob. 4ECh. 13.9 - Prob. 5ECh. 13.9 - Prob. 6ECh. 13.9 - Prob. 7ECh. 13.9 - Finding Positive Numbers In Exercises 7-10, find...Ch. 13.9 - Prob. 9ECh. 13.9 - Prob. 10ECh. 13.9 - CostA home improvement contractor is painting the...Ch. 13.9 - Prob. 12ECh. 13.9 - Prob. 13ECh. 13.9 - Prob. 14ECh. 13.9 - Prob. 15ECh. 13.9 - Prob. 16ECh. 13.9 - Prob. 17ECh. 13.9 - Shannon Diversity IndexOne way to measure species...Ch. 13.9 - Prob. 19ECh. 13.9 - Prob. 20ECh. 13.9 - Prob. 21ECh. 13.9 - Prob. 22ECh. 13.9 - Prob. 23ECh. 13.9 - Prob. 24ECh. 13.9 - Prob. 25ECh. 13.9 - Prob. 26ECh. 13.9 - Prob. 27ECh. 13.9 - Prob. 28ECh. 13.9 - Prob. 29ECh. 13.9 - Prob. 30ECh. 13.9 - EXPLORING CONCEPTS Method of Least SquaresFind a...Ch. 13.9 - Prob. 32ECh. 13.9 - Prob. 33ECh. 13.9 - Prob. 34ECh. 13.9 - Prob. 35ECh. 13.9 - Prob. 36ECh. 13.9 - Prob. 37ECh. 13.9 - Prob. 38ECh. 13.9 - Modeling DataA meteorologist measures the...Ch. 13.9 - Prob. 40ECh. 13.9 - Prob. 41ECh. 13.10 - CONCEPT CHECK Constrained Optimization Problems...Ch. 13.10 - Prob. 2ECh. 13.10 - Using Lagrange Multipliers In Exercises 310, use...Ch. 13.10 - Prob. 4ECh. 13.10 - Prob. 5ECh. 13.10 - Prob. 6ECh. 13.10 - Prob. 7ECh. 13.10 - Prob. 8ECh. 13.10 - Prob. 9ECh. 13.10 - Prob. 10ECh. 13.10 - Prob. 11ECh. 13.10 - Prob. 12ECh. 13.10 - Prob. 13ECh. 13.10 - Prob. 14ECh. 13.10 - Prob. 15ECh. 13.10 - Using Lagrange Multipliers In Exercises 15 and 16,...Ch. 13.10 - Using Lagrange Multipliers In Exercises 17 and 18,...Ch. 13.10 - Prob. 18ECh. 13.10 - Finding Minimum Distance In Exercises 19-28, use...Ch. 13.10 - Prob. 20ECh. 13.10 - Prob. 21ECh. 13.10 - Prob. 22ECh. 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 - Prob. 26ECh. 13.10 - Prob. 27ECh. 13.10 - Prob. 28ECh. 13.10 - Prob. 29ECh. 13.10 - Prob. 30ECh. 13.10 - Prob. 31ECh. 13.10 - Using Lagrange Multipliers In Exercises 3138, use...Ch. 13.10 - Prob. 33ECh. 13.10 - Prob. 34ECh. 13.10 - Prob. 35ECh. 13.10 - Prob. 36ECh. 13.10 - Prob. 37ECh. 13.10 - Prob. 38ECh. 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 - Prob. 41ECh. 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 - Prob. 47ECh. 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 FunctionIn Exercises 1 and 2,...Ch. 13 - Prob. 2RECh. 13 - Finding the Domain and Range of a FunctionIn...Ch. 13 - Finding the Domain and Range of a FunctionIn...Ch. 13 - Sketching a SurfaceIn Exercises 5 and 6, describe...Ch. 13 - Prob. 6RECh. 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 - Cobb-Douglas Production Function A manufacturer...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 - Prob. 14RECh. 13 - Prob. 15RECh. 13 - Limit and ContinuityIn Exercises 1318, find the...Ch. 13 - Limit and ContinuityIn Exercises 1318, find the...Ch. 13 - Prob. 18RECh. 13 - Prob. 19RECh. 13 - Prob. 20RECh. 13 - Prob. 21RECh. 13 - Prob. 22RECh. 13 - Prob. 23RECh. 13 - Prob. 24RECh. 13 - Prob. 25RECh. 13 - Prob. 26RECh. 13 - Prob. 27RECh. 13 - Prob. 28RECh. 13 - Prob. 29RECh. 13 - Prob. 30RECh. 13 - Prob. 31RECh. 13 - Prob. 32RECh. 13 - Prob. 33RECh. 13 - Prob. 34RECh. 13 - Finding the Slopes of a SurfaceFind the slopes of...Ch. 13 - Prob. 36RECh. 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 - DifferentiabilityIn Exercises 45 and 46, show that...Ch. 13 - DifferentiabilityIn Exercises 45 and 46, show that...Ch. 13 - Using Different MethodsIn Exercises 4750, find...Ch. 13 - Using Different MethodsIn Exercises 4750, find...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 a Derivative ImplicitlyIn Exercises 53 and...Ch. 13 - Finding a Derivative ImplicitlyIn Exercises 53 and...Ch. 13 - Prob. 55RECh. 13 - Prob. 56RECh. 13 - Prob. 57RECh. 13 - Prob. 58RECh. 13 - Prob. 59RECh. 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 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 - Finding the Angle of Inclination of a Tangent...Ch. 13 - Prob. 77RECh. 13 - Horizontal Tangent PlaneIn Exercises 77 and 78,...Ch. 13 - Prob. 79RECh. 13 - Prob. 80RECh. 13 - Prob. 81RECh. 13 - Prob. 82RECh. 13 - Prob. 83RECh. 13 - Prob. 84RECh. 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 - Prob. 88RECh. 13 - Prob. 89RECh. 13 - Prob. 90RECh. 13 - Prob. 91RECh. 13 - Prob. 92RECh. 13 - Prob. 93RECh. 13 - Prob. 94RECh. 13 - Using Lagrange MultipliersIn Exercises 9398, use...Ch. 13 - Prob. 96RECh. 13 - Prob. 97RECh. 13 - Prob. 98RECh. 13 - Prob. 99RECh. 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 - Prob. 4PSCh. 13 - Finding Maximum and Minimum Values (a) Let...Ch. 13 - Minimizing CostsA heated storage room has the...Ch. 13 - Prob. 7PSCh. 13 - TemperatureConsider a circular plate of radius 1...Ch. 13 - Prob. 9PSCh. 13 - Prob. 10PSCh. 13 - Prob. 11PSCh. 13 - Prob. 12PSCh. 13 - Prob. 13PSCh. 13 - Prob. 14PSCh. 13 - Prob. 15PSCh. 13 - Prob. 16PSCh. 13 - Prob. 17PSCh. 13 - Prob. 18PSCh. 13 - Prob. 19PSCh. 13 - Prob. 20PSCh. 13 - Prob. 21PS
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Finding Local Maxima and Minima by Differentiation; Author: Professor Dave Explains;https://www.youtube.com/watch?v=pvLj1s7SOtk;License: Standard YouTube License, CC-BY