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Differential Calculus and Sage
(revised edition)

William Anthony Granville,
with extra material added by David Joyner

Date: 2008-11-22

Contents

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 ( $\infty$ )
- 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 $\lim_{x \to 0} \frac{\sin\, x}{x} = 1$
- The number
- Expressions assuming the form $\frac {\infty }{\infty }$
- 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 $\sin v$
- Differentiation of $\cos v$
- Differentiation of $\tan v$
- Differentiation of $\cot v$
- Differentiation of $\sec v$
- Differentiation of $\csc\, v$
- Differentiation of ${\rm vers}\, v$
- Exercises
- Differentiation of $\arcsin v$
- Differentiation of $\arccos v$
- Differentiation of $\arctan\, v$
- Differentiation of ${\rm arccot}u$
- Differentiation of ${\rm arcsec}u$
- Differentiation of ${\rm arccsc}\, v$
- Differentiation of ${\rm arcvers}\, v$
- 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
- The derivative considered as the ratio of two rates
- Exercises
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 $\frac {0}{0}$
  - Rule for evaluating the indeterminate form $\frac {0}{0}$
  - Exercises
- Evaluation of the indeterminate form $\frac {\infty }{\infty }$
- Evaluation of the indeterminate form $0 \cdot \infty$
- Evaluation of the indeterminate form $\infty -\infty$
  - Exercises
- Evaluation of the indeterminate forms , $1^{\infty }$ , $\infty ^0$
- 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