# Chapter 8: Introduction to Calculus

## Introduction

This final chapter introduces the concept of limits: values that a function may approach, even get extremely close to, but never actually hit. You will learn the meaning of one-sided limits, and how to calculate the limit of a polynomial function.

The real point to this chapter, though, is an introduction to derivatives and integrals.

You have had practice in the past with identifying average speed over a certain distance or time, but what if you needed to know the speed of an object *right now*? In other words, how could it be possible to calculate the instantaneous speed of an object? When you calculate the average speed of something, you divide the distance traveled by the time it takes, but when calculating the instantaneous speed of something, there is no distance, and no time either! This is where the derivative of a function comes in, learning to calculate the derivative allows you to calculate instantaneous speeds, among other things.

## Chapter Outline

- 8.1. Definition of a Limit
- 8.2. One-Sided Limits
- 8.3. Infinite Limits
- 8.4. Polynomial Function Limits
- 8.5. Rational Function Limits
- 8.6. Applications of One-Sided Limits
- 8.7. Tangents to a Curve
- 8.8. Instantaneous Rates of Change
- 8.9. Constant Derivatives and the Power Rule
- 8.10. Derivatives of Sums and Differences
- 8.11. Quotient Rule and Higher Derivatives
- 8.12. Area Under the Curve
- 8.13. Fundamental Theorem of Calculus

### Chapter Summary

## Summary

This final chapter covers the concepts of limits, including one-sided limits. Limits are studied in order to provide a platform from which to introduce derivatives and integrals.

Students are taught about tangent lines and rates of change, and the comparison between the tangent to a curve and the instantaneous speed of an object.

The text wraps up with a discussion of calculating the area under a curve with integrals and the definition of the Fundamental Theorem of Calculus.