Applied Calculus 5th Edition

Chapter Eight

FUNCTIONS OF SEVERAL VARIABLES

Contents

8.1 Understanding Functions of Two Variables

Functions Given Numerically

Functions Given Algebraically

Strategy to Investigate Functions of Two Variables: Vary One Variable at a Time

Concentration of a Drug in the Blood

8.2 Contour Diagrams

Weather Maps

Topographical Maps

Using Contour Diagrams

Cobb-Douglas Production Functions

Contour Diagrams and Tables

Finding Contours Algebraically

8.3 Partial Derivatives

Rate of Change of Airline Revenue

Estimating Partial Derivatives from a Table

Using Partial Derivatives to Estimate Values of the Function

Estimating Partial Derivatives from Contours

Using Units to Interpret Partial Derivatives

8.4 Computing Partial Derivatives Algebraically

Second-Order Partial Derivatives

The Mixed Partial Derivatives Are Equal

8.5 Critical Points and Optimization

Finding Local Maxima/Minima Analytically

How Do We Find Critical Points?

Is a Critical Point a Local Maximum or a Local Minimum?

8.6 Constrained Optimization

Graphical Approach: Maximizing Production Subject to a Budget Constraint

Analytical Approach: Lagrange Multipliers

The Meaning of λ

The Lagrangian Function

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: A Heater in a Room, Optimizing Relative Prices for Adults and Children, Maximizing Production and Minimizing Cost: “Duality”

Focus on Theory

Deriving the Formula for a Regression ...