# Elementary Calculus: An Approach Using Infinitesimals by Jerome H. Keisler

• Calculus
• bcrowell
In summary, "Elementary Calculus: An Approach Using Infinitesimals" by Jerome H. Keisler is a unique calculus textbook that focuses on infinitesimals instead of limits. It covers topics such as real and hyperreal numbers, differentiation, continuous functions, integration, limits, analytic geometry, and vector calculus. The book also includes sections on trigonometric, exponential, and logarithmic functions, as well as infinite series and differential equations. It is available for download under a CC-BY-NC-SA license.

## For those who have used this book

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Code:
[LIST]
[*] Introduction
[*] Real and Hyperreal Numbers
[LIST]
[*] The Real Line
[*] Functions of Real Numbers
[*] Straight Lines
[*] Slopes and Velocity; The Hyperreal Line
[*] Infinitesimal, Finite and Infinite Numbers
[*] Standard Parts
[*] Extra Problems
[/LIST]
[*] Differentiation
[LIST]
[*] Derivatives
[*] Differentials and Tangent Lines
[*] Derivatives of Rational Functions
[*] Inverse Functions
[*] Transcendental Functions
[*] Chain Rule
[*] Higher Derivatives
[*] Implicit Functions
[*] Extra Problems
[/LIST]
[*] Continuous Functions
[LIST]
[*] How to Set Up a Problem
[*] Related Rates
[*] Limits
[*] Continuity
[*] Maxima and Minima
[*] Maxima and Minima - Applications
[*] Derivatives and Curve Sketching
[*] Properties of Continuous Functions
[*] Extra Problems
[/LIST]
[*] Integration
[LIST]
[*] The Definite Integral
[*] Fundamental Theorem of Calculus
[*] Indefinite Integrals
[*] Integration by Change of Variables
[*] Area between Two Curves
[*] Numerical Integration
[*] Extra Problems
[/LIST]
[*] Limits, Analytic Geometry, and Approximations
[LIST]
[*] Infinite Limits
[*] L'Hospital's Rule
[*] Limits and Curve Sketching
[*] Parabolas
[*] Ellipses and Hyperbolas
[*] Second Degree Curves
[*] Rotation of Axes
[*] The $\varepsilon$, $\delta$ Condition for Limits
[*] Newton's Method
[*] Derivatives and Increments
[*] Extra Problems
[/LIST]
[*] Applications of the Integral
[LIST]
[*] Infinite Sum Theorem
[*] Volumes of Solids of Revolution
[*] Length of a Curve
[*] Area of a Surface of Revolution
[*] Averages
[*] Some Applications to Physics
[*] Improper Integrals
[*] Extra Problems
[/LIST]
[*] Trigonometric Functions
[LIST]
[*] Trigonometry
[*] Derivatives of Trigonometric Functions
[*] Inverse Trigonometric Functions
[*] Integration by Parts
[*] Integrals of Powers of Trigonometric Functions
[*] Trigonometric Substitutions
[*] Polar Coordinates
[*] Slopes and Curve Sketching in Polar Coordinates
[*] Area in Polar Coordinates
[*] Length of a Curve in Polar Coordinates
[*] Extra Problems
[/LIST]
[*] Exponential and Logarithmic Functions
[LIST]
[*] Exponential Functions
[*] Logarithmic Functions
[*] Derivatives of Exponential Functions and the Number $e$
[*] Some Uses of Exponential Functions
[*] Natural Logarithms
[*] Some Differential Equations
[*] Derivatives and Integrals Involving $\ln x$
[*] Integration of Rational Functions
[*] Methods of Integration
[*] Extra Problems
[/LIST]
[*] Infinite Series
[LIST]
[*] Sequences
[*] Series
[*] Properties of Infinite Series
[*] Series with Positive Terms
[*] Alternating Series
[*] Absolute and Conditional Convergence
[*] Power Series
[*] Derivatives and Integrals of Power Series
[*] Approximations by Power Series
[*] Taylor's Formula
[*] Taylor Series
[*] Extra Problems
[/LIST]
[*] Vectors
[LIST]
[*] Vector Algebra
[*] Vectors and Plane Geometry
[*] Vectors and Lines in Space
[*] Products of Vectors
[*] Planes in Space
[*] Vector Values Functions
[*] Vector Derivatives
[*] Hyperreal Vectors
[*] Extra Problems
[/LIST]
[*] Partial Differentiation
[LIST]
[*] Surfaces
[*] Continuous Functions of Two or More Variables
[*] Partial Derivatives
[*] Total Differentials and Tangent Planes
[*] Chain Rule
[*] Implicit Functions
[*] Maxima and Minima
[*] Higher Partial Derivatives
[*] Extra Problems
[/LIST]
[*] Multiple Integrals
[LIST]
[*] Double Integrals
[*] Iterated Integrals
[*] Infinite Sum Theorem and Volume
[*] Applications to Physics
[*] Double Integrals in Polar Coordinates
[*] Triple Integrals
[*] Cylindrical and Spherical Coordinates
[*] Extra Problems
[/LIST]
[*] Vector Calculus
[LIST]
[*] Line Integrals
[*] Independence of Path
[*] Green's Theorem
[*] Surface Area and Surface Integrals
[*] Theorems of Stokes and Gauss
[*] Extra Problems
[/LIST]
[*] Differential Equations
[LIST]
[*] Equations with Separable Variables
[*] First Order Homogeneous Linear Equations
[*] First Order Linear Equations
[*] Existence and Approximation of Solutions
[*] Complex Numbers
[*] Second Order Homogeneous Linear Equations
[*] Second Order Linear Equations
[*] Extra Problems
[/LIST]
[*] Epilogue
[*] Appendix: Tables
[LIST]
[*] Trigonometric Functions
[*] Greek Alphabet
[*] Exponential Functions
[*] Natural Logarithms
[*] Powers and Roots
[/LIST]
[*] Index
[/LIST]

Last edited by a moderator:
This is an unusual freshman calculus textbook from 1976. It emphasizes infinitesimals rather than limits. When the book went out of print, the rights reverted to Keisler, who made it available online under the CC-BY-NC-SA license.

## 1. What is the main concept behind infinitesimals in elementary calculus?

The main concept behind infinitesimals in elementary calculus is that they represent infinitely small quantities that are smaller than any real number, yet still have a magnitude. This allows for a more intuitive and rigorous approach to calculus, as opposed to the traditional limit-based approach.

## 2. How is this approach different from traditional calculus?

This approach differs from traditional calculus in that it uses infinitesimals instead of limits to define derivatives and integrals. This leads to a more intuitive and geometric understanding of these concepts, rather than a computational one.

## 3. How does this book explain the fundamental theorem of calculus?

The book explains the fundamental theorem of calculus using the concept of infinitesimals. It shows how the derivative and integral operations are inverse to each other, and how infinitesimals are used to bridge the gap between these two operations.

## 4. Is this book accessible to someone with no prior knowledge of calculus?

Yes, this book is designed to be accessible to someone with no prior knowledge of calculus. It introduces the necessary concepts and techniques in a clear and concise manner, making it suitable for beginners.

## 5. Are there any real-world applications of this approach to calculus?

Yes, there are many real-world applications of this approach to calculus. It has been used in physics, engineering, and economics, among other fields, to solve problems that would be difficult or impossible to solve using traditional calculus.

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