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(revised edition)

**William Anthony Granville,**
*with extra material added by David Joyner*

**Date:** 2008-11-22

- Contents
- Preface
- Collection of formulas
- Formulas for reference
- Greek alphabet
- Rules for signs of the trigonometric functions
- Natural values of the trigonometric functions
- Logarithms of numbers and trigonometric functions

- Variables and functions
- Variables and constants
- Interval of a variable
- Continuous variation
- Functions
- Independent and dependent variables
- Notation of functions
- Values of the independent variable for which a function is defined
- Exercises

- Theory of limits
- Limit of a variable
- Division by zero excluded
- Infinitesimals
- The concept of infinity ()
- Limiting value of a function
- Continuous and discontinuous functions
- Continuity and discontinuity of functions illustrated by their graphs
- Fundamental theorems on limits
- Special limiting values
- Show that
- The number
- Expressions assuming the form
- Exercises

- Differentiation
- Introduction
- Increments
- Comparison of increments
- Derivative of a function of one variable
- Symbols for derivatives
- Differentiable functions
- General rule for differentiation
- Exercises
- Applications of the derivative to Geometry
- Exercises

- Rules for differentiating standard elementary forms
- Importance of General Rule
- Differentiation of a constant
- Differentiation of a variable with respect to itself
- Differentiation of a sum
- Differentiation of the product of a constant and a function
- Differentiation of the product of two functions
- Differentiation of the product of any finite number of functions
- Differentiation of a function with a constant exponent
- Differentiation of a quotient
- Examples
- Differentiation of a function of a function
- Differentiation of inverse functions
- Differentiation of a logarithm
- Differentiation of the simple exponential function
- Differentiation of the general exponential function
- Logarithmic differentiation
- Examples
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Exercises
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Example
- Implicit functions
- Differentiation of implicit functions
- Exercises
- Miscellaneous Exercises

- Simple applications of the derivative
- Direction of a curve
- Exercises
- Equations of tangent and normal lines
- Exercises
- Parametric equations of a curve
- Exercises
- Angle between the radius vector and tangent
- Lengths of polar subtangent and polar subnormal
- Examples
- Solution of equations having multiple roots
- Examples
- Applications of the derivative in mechanics
- Component velocities. Curvilinear motion
- Acceleration. Rectilinear motion
- Component accelerations. Curvilinear motion
- Examples
- Application: Newton's method

- Successive differentiation
- Definition of successive derivatives
- Notation
- The -th derivative
- Leibnitz's Formula for the -th derivative of a product
- Successive differentiation of implicit functions
- Exercises

- Maxima, minima and inflection points
- Introduction
- Increasing and decreasing functions
- Tests for determining when a function is increasing or decreasing
- Maximum and minimum values of a function
- Examining a function for extremal values: first method
- Examining a function for extremal values: second method
- Problems
- Points of inflection
- Examples
- Curve tracing
- Exercises

- Differentials
- Introduction
- Definitions
- Infinitesimals
- Derivative of the arc in rectangular coordinates
- Derivative of the arc in polar coordinates
- Exercises
- Formulas for finding the differentials of functions
- Successive differentials
- Examples

- Rates

- Change of variable
- Interchange of dependent and independent variables
- Change of the dependent variable
- Change of the independent variable
- Simultaneous change of both independent and dependent variables
- Exercises

- Curvature; radius of curvature
- Curvature
- Curvature of a circle
- Curvature at a point
- Formulas for curvature
- Radius of curvature
- Circle of curvature
- Exercises

- Theorem of mean value; indeterminant forms
- Rolle's Theorem
- The Mean-value Theorem
- The Extended Mean Value Theorem
- Exercises
- Maxima and minima treated analytically
- Exercises
- Indeterminate forms
- Evaluation of a function taking on an indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate forms , ,
- Exercises
- Application: Using Taylor's Theorem to Approximate Functions.
- Example/Application: Finite Difference Schemes

- Circle of curvature. Center of Curvature
- Circle of curvature
- Second method for finding center of curvature
- Center of curvature
- Evolutes
- Properties of the evolute
- Exercises

- References
- Bibliography
- Index
- About this document ...

david joyner 2008-11-22