Chapter 4 - Differential Calculus and Its Uses
- About this Chapter
- Section 4.1 Derivatives and Graphs
- 4.1.1 Local Maximum and Minimum Values
4.1.2 Global Maximum and Minimum Values- 4.1.3 Minimizing Production Cost
- 4.1.4 The Derivative of the Reciprocal Function
- 4.1.5 Completion of the Minimal Cost Calculation
- Section Summary
- Exercises
- Problems
- Section 4.2 Second Derivatives and Graphs
- 4.2.1 The Second Derivative
- 4.2.2 Graphs and the Second Derivative
- 4.2.3 Terminology: Zeros, Critical Values,
and Inflection Points
- Section Summary
- Exercises
- Problems
- Section 4.3 Solving Nonlinear Equations by Linearization:
Newton's Method
- 4.3.1 Local Linearity
- 4.3.2 Newton's Method
- 4.3.3 Examples of Newton's Method
- Section Summary
- Exercises
- Problems
- Section 4.4 The Product Rule
- 4.4.1 The Growth Rate of Energy Consumption
- 4.4.2 A General Rule for Differentiating Products
- 4.4.3 More on the Growth Rate of Energy Consumption
- Section Summary
- Exercises
- Problems
- Section 4.5 The Chain Rule
- 4.5.1 Optimization in Nature
- 4.5.2 The Derivative of the Square Root Function
- 4.5.3 Differentiating a Function-of-a-Function: The Chain Rule
- 4.5.4 Analysis of Reflection
- Section Summary
- Exercises
- Problems
- Section 4.6 Derivatives of Functions Defined Implicitly
- 4.6.1 An Example
- 4.6.2 The Derivative of the Natural Logarithm
- 4.6.3 Derivatives of Inverse Functions
- Section Summary
- Exercises
- Problems
- Section 4.7 The General Power Rule
- 4.7.1 Special Cases
- 4.7.2 The Power Rule for Rational Exponents
- 4.7.3 Power Functions with Irrational Exponents
- 4.7.4 Symbols
- Section Summary
- Exercises
- Problems
- Section 4.8 Differentials and Leibniz Notation
- 4.8.1 Background and a Calculation
- 4.8.2 Zooming In — Again
- 4.8.3 An Application of the Differential
- Section Summary
- Exercises
- Problems
- Chapter Summary
- Chapter Review
- Concepts and Applications
- Formulas
- Project 1. Air Traffic Control
- Project 2. Refraction