Gottfried Wilhelm Leibniz (1646–1716), left, and Sir Isaac Newton (1643–1727), right, are considered founders of calculus.

After a quick review of the skills we need to study calculus (Chapters 1–3), we will begin with the concept of a limit and continuity in Chapter 4. We will then continue to learn about the derivative of a function and how to find it in Chapter 5. Derivatives have many important applications that will be discussed in chapter 6. In chapter 7, we will learn how to find a function if its derivative is given. Chapter 8 will discuss how to find the area under a curve. This process is called integration. Finally, in the last chapter, we will talk about the applications of integration.

There are more than 370 fully solved examples that can help you gain a thorough understanding and master the skills you need to ace your exams in elementary calculus.

## Table of Contents

### Chapter 4: Limits and Continuity

- The Concept of a Limit
- The Precise Definition of a Limit
- One-Sided Limits
- Theorems for Calculating Limits
- The Indeterminate Form 0/0
- Infinite Limits
- Limits at Infinity
- Continuity
- Properties of Continuous Functions
- The Other Indeterminate Forms: Infinity Divided by Infinity, Zero Times Infinity, and Infinity Minus Infinity
- Asymptotes
- The number
*e*

### Chapter 5: Differentiation

- The Derivative Concept
- Geometric Interpretation of the Derivative as a Slope
- Graphing the Derivative
- Differentiability and Continuity of Functions
- One-Sided Derivatives
- When a Function Is Not Differentiable at a Point
- Differentiation Rules
- Derivatives of The Trigonometric Functions
- Higher Derivatives
- * More About the Leibniz Notation for Higher Derivatives
- Implicit Differentiation
- Derivatives of Inverse Functions
- Derivatives of the Inverse Trigonometric Functions
- Derivatives of Logarithmic Functions
- Derivatives of Exponential Functions
- Hyperbolic Functions and Their Derivatives
- Linear Approximations
- Differentials

### Chapter 6: Applications of Differentiation

- Tangents and Normals
- Rectilinear Motion
- Related Rates
- Extreme Values of Functions
- Rolle’s Theorem
- The Mean-Value Theorem for Derivatives
- Increasing and Decreasing Functions
- Concavity and Points of Inflection
- First Derivative Test for Local Extrema
- Second Derivative Test for Local Extrema
- L’Hôpital’s Rule for Indeterminate Limits
- Curve Sketching
- Optimization