## MAT135 Unit 5: Using Derivatives

**3.9 LINEAR APPROXIMATION AND THE DERIVATIVE**

The Tangent Line Approximation

Estimating the Error in the Approximation

Why Differentiability Makes a Graph Look Straight

**4.1 USING FIRST AND SECOND DERIVATIVES**

What Derivatives Tell Us About a Function and Its Graph

Local Maxima and Minima

Finding Local Maxima and Minima

Critical Points: Testing for Local Maxima and Minima

The Second-Derivative Test

Concavity and Inflection Points

How Do We Detect an Inflection Point?

Showing Local Extrema Are at Critical Points

**4.2 OPTIMIZATION**

Global Maxima and Minima

How Do We Find Global Extrema?

Finding Upper and Lower Bounds

**4.3 OPTIMIZATION AND MODELLING**

Optimization and Modelling

Graphing Examples

**4.4 FAMILIES OF FUNCTIONS AND MODELLING**

The Bell-Shaped Curve

Motion Under Gravity

Exponential Model with a Limit

The Logistic Model

**4.6 RATES AND RELATED RATES**

Rates and Related Rates

**4.7 L’HOPITAL’S RULE, GROWTH, AND DOMINANCE**

L'Hopital's Rule

Dominance

Recognizing the Form of a Limit