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Chapter Four

USING THE DERIVATIVE

Contents

4.1 Local Maxima and Minima

What Derivatives Tell Us About a Function and Its Graph

Local Maxima and Minima

Detecting a Local Maximum or Minimum

Testing For Local Maxima and Minima

4.2 Inflection Points

Concavity and Inflection Points

How Do You Locate an Inflection Point?

4.3 Global Maxima and Minima

Global Maxima and Minima

A Graphical Example: Minimizing Gas Consumption

4.4 Profit, Cost, and Revenue

Maximizing Profit

Where MR = MC

Maximizing Revenue

4.5 Average Cost

What Is Average Cost?

Minimizing Average Cost

4.6 Elasticity of Demand

Elasticity of Demand

Revenue and Elasticity of Demand

Elasticity of Demand for Different Products

4.7 Logistic Growth

Modeling the US Population

An Exponential Model

A Logistic Model

The Logistic Function

The Carrying Capacity and the Point of Diminishing Returns

US Population: 1790–2010

Sales Predictions

Dose-Response Curves

4.8 The Surge Function and Drug Concentration

Nicotine in the Blood

The Family of Functions y = ate-bt

Drug Concentration Curves

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: Average and Marginal Costs, Firebreaks, Production and the Price of Raw Materials, Medical Case Study: Impact of Asthma on Breathing

4.1 LOCAL MAXIMA AND MINIMA

What Derivatives Tell Us About a Function and Its Graph

As we saw in Chapter 2, values of a function and its derivatives are related as follows: ...

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