Introduction to Calculus

This ebook provides a comprehensive introduction to fundamental concepts in calculus and their applications, covering all of Calculus 1 and some of Calculus 2. The book starts with functions and limits, followed by differential calculus, and then moves on to integral calculus and a brief discussion of differential equations. It ends with the complete solution for a sample exam based on the AP Calculus AB course. Videos for the topics covered in the ebook, along with practice problems, exercises and quizzes, are available in the interactive Wolfram U course Introduction to Calculus.

Contents

• What Is Calculus?
• Functions
• The Elementary Functions
• The Limit of a Function
• The Laws of Limits
• Continuity
• Derivatives and Rates of Change
• The Derivative as a Function
• Differentiation Formulas
• Derivatives of Trigonometric Functions
• The Chain Rule
• Implicit Differentiation
• Rates of Change in the Sciences
• Related Rates
• Linear Approximations
• Maxima and Minima
• The Mean Value Theorem
• Derivatives and the Shape of Graphs
• Asymptotes
• Curve Sketching
• Optimization
• Antiderivatives
• Riemann Sums
• The Definite Integral
• The Fundamental Theorem of Calculus
• Indefinite Integrals
• The Substitution Rule
• Areas between Curves
• Volumes of Solids
• Volumes by Cylindrical Shells
• Average Value of a Function
• Approximate Integration
• Exponential Functions
• Logarithmic Functions
• L'Hôpital's Rule
• Slope Fields and Euler's Method
• Separable Differential Equations
• Calculus and Programming
• Sample Calculus Exam, Part 1
• Sample Calculus Exam, Part 2

About the Authors

John Michael Clark is a mathematics PhD student at the University of Texas at Austin. He received his BSc in mathematics from the California Institute of Technology in 2016 and later his MSc from Oklahoma State University in 2020.

Devendra Kapadia is the Manager of Mathematical Computation at Wolfram. He has a BSc and an MSc in mathematics from the University of Mumbai, an MSc by Research in mathematics from the University of Oxford and a PhD in mathematics from the University of Pittsburgh. He is actively involved in algorithm development for Wolfram Language and in developing online mathematics courses for Wolfram U.