- #1
- 6,724
- 431
- Author: Jerome H. Keisler
- Title: Elementary Calculus: An Approach Using Infinitesimals
- Download Link: http://www.math.wisc.edu/~keisler/calc.html
- Prerequisities: High School Mathematics
Table of Contents:
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 [itex]\varepsilon[/itex], [itex]\delta[/itex] 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 [itex]e[/itex]
[*] Some Uses of Exponential Functions
[*] Natural Logarithms
[*] Some Differential Equations
[*] Derivatives and Integrals Involving [itex]\ln x[/itex]
[*] 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]
[*] Directional Derivatives and Gradients
[*] 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]
[*] Answers to Selected Problems
[*] Index
[/LIST]
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