Single Variable Calculus

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

 

  1. The Concept of a Limit
  2. The Precise Definition of a Limit
  3. One-Sided Limits
  4. Theorems for Calculating Limits
  5. The Indeterminate Form 0/0
  6. Infinite Limits
  7. Limits at Infinity
  8. Continuity
  9. Properties of Continuous Functions
  10. The Other Indeterminate Forms: Infinity Divided by Infinity, Zero Times Infinity, and Infinity Minus Infinity
  11. Asymptotes
  12. The number e

Chapter 5: Differentiation

 

  1. The Derivative Concept
  2. Geometric Interpretation of the Derivative as a Slope
  3. Graphing the Derivative
  4. Differentiability and Continuity of Functions
  5. One-Sided Derivatives
  6. When a Function Is Not Differentiable at a Point
  7. Differentiation Rules
  8. Derivatives of The Trigonometric Functions
  9. Higher Derivatives
  10. * More About the Leibniz Notation for Higher Derivatives
  11. Implicit Differentiation
  12. Derivatives of Inverse Functions
  13. Derivatives of the Inverse Trigonometric Functions
  14. Derivatives of Logarithmic Functions
  15. Derivatives of Exponential Functions
  16. Hyperbolic Functions and Their Derivatives
  17. Linear Approximations
  18. Differentials

Chapter 6: Applications of Differentiation

 

  1. Tangents and Normals
  2. Rectilinear Motion
  3. Related Rates
  4. Extreme Values of Functions
  5. Rolle’s Theorem
  6. The Mean-Value Theorem for Derivatives
  7. Increasing and Decreasing Functions
  8. Concavity and Points of Inflection
  9. First Derivative Test for Local Extrema
  10. Second Derivative Test for Local Extrema
  11. L’Hôpital’s Rule for Indeterminate Limits
  12. Curve Sketching
  13. Optimization

Chapter 7: Indefinite Integrals

  1. Definition
  2. Basic Integration Formulas
  3. Constant of Integration
  4. Integration by Substitution
  5. Integrals of Trigonometric Functions
  6. Integration by Parts

Chapter 8: Definite Integrals

  1. Definition of the Definite Integral
  2. Properties of Definite Integrals
  3. The Fundamental Theorem of Calculus
  4. Derivatives of Integrals
  5. Evaluation of Definite Integrals by Substitution

Chapter 9: Applications of Integration

  1. The Area Between Two Curves
  2. Volumes of Solids of Revolution: The Disk and Washer Methods
  3. Volumes of Solids With Known Cross-Sections: The Slice Method
  4. Volumes of Solids of Revolution: The Shell Method
  5. Arc Length
  6. Areas of Surfaces of Revolution
  7. Work