Nelson Advanced Functions and Introductory Calculus is designed to help you develop skill at creating and analyzing mathematical models to solve real-world problems involving rates of change. In some cases, you will have the chance to practise familiar mathematical skills like solving equations, graphing data, and modelling. You will also have the opportunity to develop your own strategy for solving new types of problems. Throughout this book, you will be encouraged to communicate what you have learned to others.

Introduction

Part 1

Advanced Functions

Chapter 1

Polynomial Function Models

The Chapter Problem

Chapter Challenges

Getting Ready

1.1 Polynomial Functions

1.2 Investigating the Characteristics of Polynomial Functions

1.3 Creating New Polynomial Functions: An Introduction to Composition

1.4 Dividing Polynomials

1.5 Factoring Polynomials

1.6 Solving Polynomial Equations

1.7 Solving Polynomial Inequalities

Chapter 1 Review

Chapter 1 Performance Task

Chapter 1 Review Test

Chapter 2

Exponential and Logarithmic Function Models

The Chapter Problem

Chapter Challenges

Getting Ready

2.1 Investigating Exponential Models

2.2 Graphs of Exponential Functions

2.3 Various Forms of Exponential Functions

2.4 The Logarithmic Function

2.5 Laws of Logarithms

2.6 Solving Exponential Equations

2.7 Creating Logarithmic Models

2.8 Solving Logarithmic Equations

2.9 Creating and Applying Exponential and Logarithmic Models

Chapter 2 Review

Chapter 2 Performance Task

Chapter 2 Review Test

Cumulative Review Test 1

Part 2

Differential Calculus: Polynomial and

Rational Functions

Chapter 3

Rates of Change in Polynomial Function Models

The Chapter Problem

Chapter Challenges

Getting Ready

3.1 Examining Rates of Change in Polynomial Models

3.2 A Closer Look at Rates of Change: The Tangent Problem

3.3 Limits of Polynomial Functions

3.4 Using Limits to Find Instantaneous Rates of Change -The Derivative

3.5 Finding Some Shortcuts - The Constant and Power Rules

3.6 Finding Some Shortcuts - The Sum and Difference Rules

3.7 Polynomial Function Models and the First Derivative

3.8 Polynomial Function Models and the Second Derivative

Chapter 3 Review

Chapter 3 Performance Task

Chapter 3 Review Test

Chapter 4

Using the Derivative to Analyze Polynomial

Function Models

The Chapter Problem

Chapter Challenges

Getting Ready

4.1 Analyzing a Polynomial Function: Intervals of Increase and Decrease

4.2 Maximum and Minimum Values of a Polynomial Function

4.3 The First Derivative Test

4.4 Finding Some Shortcuts - The Product Rule

4.5 Finding Optimal Values for Polynomial Function Models

4.6 Rates of Change in Business and Economics

4.7 Sketching Graphs of Polynomial Functions: Concavity

Chapter 4 Review

Chapter 4 Performance Task

Chapter 4 Review Test

Chapter 5

Rates of Change in Rational Function Models

The Chapter Problem

Chapter Challenges

Getting Ready

5.1 Graphs of Rational Functions

5.2 Limits and End Behaviour of Rational Functions

5.3 Continuity of Rational Functions

5.4 Rate of Change of a Rational Function - The Quotient Rule

5.5 Differentiability of Rational and Other Functions

5.6 Finding Optimal Values for Rational Function Models

5.7 Sketching Graphs of Rational Functions

Chapter 5 Review

Chapter 5 Performance Task

Chapter 5 Review Test

Cumulative Review Test 2

Part 3

Differential Calculus: Composite,

Exponential, and Logarithmic Functions

Chapter 6

Rates of Change in Composite Function Models

The Chapter Problem

Chapter Challenges

Getting Ready

6.1 Composition of Functions

6.2 Rates of Change for Composite Functions - The Chain Rule

6.3 Differentiation Techniques: Combining the Differentiation Rules

6.4 Finding Optimal Values for Composite Function Models

6.5 Sketching Graphs of Composite Functions

6.6 Implicit Differentiation

6.7 Related Rate Models

Chapter 6 Review

Chapter 6 Performance Task

Chapter 6 Review Test

Chapter 7

Rates of Change in Exponential and

Logarithmic Function Models

The Chapter Problem

Chapter Challenges

Getting Ready

7.1 Introducing a Special Number, e

7.2 The Derivative of y = ex

7.3 The Natural Logarithm and its Derivative

7.4 Differentiating Other Logarithmic and Exponential Functions

7.5 Sketching Graphs of Exponential and Logarithmic Functions

7.6 Models of Exponential Growth

Chapter 7 Review

Chapter 7 Performance Task

Chapter 7 Review Test

Cumulative Review Test 3

Explorations Appendix

Technology Appendix

Trigonometry Appendix

Glossary

Answers

Index

Photo Credits

Summary of Mathematical Terms and Formulas