# Elementary Calculus

Written by: Michael Corral (Schoolcraft College)

How appropriate that our first new free textbook of 2021 was released on January 1. And, how exciting it was to receive an email from the author letting us know!

Elementary Calculus is Michael Corral’s third open textbook. His other’s are Vector Calculus and Trigonometry. He is an Adjunct faculty member of the Schoolcraft College Department of Mathematics. He earned his BA in Mathematics from the University of California and earned his MA in Mathematics and his MS in Industrial & Operations Engineering from the University of Michigan.

This textbook includes materials for both Calculus I & II. To prepare for these courses, students should have completed high school courses in algebra, geometry and trigonometry. The text is suitable for both high school and college calculus students. Students will get plenty of practice as they work the included 943 exercises – yes, answers and hints are provided for selected exercises.

1. The Derivative
Introduction
The Derivative: Limit Approach
The Derivative: Infinitesimal Approach

Derivatives of Sums, Products and Quotients

The Chain Rule

Higher Order Derivatives

2. Derivatives of Common Functions
Inverse Functions

Trigonometric Functions and Their Inverses

The Exponential and Natural Logarithm Functions

General Exponential and Logarithmic Functions

3. Topics in Differential Calculus
Tangent Lines

Limits: Formal Definition

Continuity

Implicit Differentiation

Related Rates

Differentials

4. Applications of Derivatives
Optimization

Curve Sketching

Numerical Approximation of Roots of Functions

The Mean Value Theorem

5. The Integral
The Indefinite Integral

The Definite Integral

The Fundamental Theorem of Calculus

Integration by Substitution

Improper Integrals

6. Methods of Integration
Integration by Parts

Trigonometric Integrals

Trigonometric Substitutions

Partial Fractions

Miscellaneous Integration Methods

Numerical Integration Methods

7. Analytic Geometry and Plane Curves
Ellipses

Parabolas

Hyperbolas

Translations and Rotations

Hyperbolic Functions

Parametric Equations

Polar Coordinates

8. Applications of Integrals
Area Between Curves

Average Value of a Function

Arc Length and Curvature

Surfaces and Solids of Revolution

Applications in Physics and Statistics

9. Infinite Sequences and Series
Sequences and Series

Tests for Convergence

Alternating Series

Power Series

Appendix A: Answers and Hints to Selected Exercises
History
Index

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Elementary Calculus