Calculus Now

by Donovan H. Van Osdol, University of New Hampshire

Copyright © 1998-2003 by Donovan H. Van Osdol

Table of Contents

1. Tangent Lines to Polynomial Functions

1.2 First Computations

• First Computations
• First Example

1.4 Derivatives for Polynomial Functions of Higher Degree

1.5 Some Theory and Applications of the Derivative for Polynomials: the Product Rule, Rolle's Theorem, the Mean Value Theorem, and Graphing Techniques

• First, some theory
• Graphing techniques, and the second derivative
• Behavior for large values of x, and applications

1.7 Rational Numbers, Real Numbers, Complex Numbers,....Always know where you are!

• Complex numbers, addition, and subtraction
• Multiplication, and interpretation of the derivative for complex polynomial functions

Appendices

• Rationale for our approach to finding the slope
• The "Whole Part" Computations
• The fundamental theorem of algebra implies the intermediate value property
• For polynomials, the intermediate value property implies the orderly property
• An example of the proof of Rolle's Theorem
• Point-slope and Slope-Intercept Forms for the Equation of a Line
• How to solve the quadratic ax² + bx + c = 0  whenever we can divide and take square roots
• x² = 2 has no rational solution, or Ö2 is irrational
• Q, R, C, R[x], and C[x] are Rings; N is a Rig

Projects

• Interpolation

2. Tangent Lines to Rational Functions and Inverses, and "Rules" for Differentiation

2.2 Derivatives of Rational Functions

2.3 Applications of the Derivative for Rational Functions

• First, some theory
• Examples and Applications

2.5 Derivatives of Inverse Functions

• Inverse functions, and the inverse function theorem
• Laws of rational exponents, and differentiation rules
• The bisection method for estimating real roots, and how to find powers and roots of complex numbers geometrically

Appendices

• Proof that rational functions have derivatives, and formulas for them
• The intermediate value property for polynomials implies the intermediate value property for rational functions, and rational functions are orderly
• Calculation of Ranges for n-th Power Functions
• Partial Fraction Decompositions
• Q(x), R(x), and C(x) are Rings

3. Derivatives of Power Series and Their Applications

3.2 Series of Numbers and Geometric Series

3.3 More General Series

3.4 Derivatives of Power Series

3.5 The Exponential Function

• Definition and Properties
• Applications

• First, the Natural Logarithm
• Other Logarithms and Exponentials
• Applications

• Definitions of the Trigonometric Functions
• Properties of the Trigonometric Functions, their Derivatives, and Applications
• Further Information on Trigonometric Functions, and Inverse Trigonometric Functions

Appendices

• How to Expand a Number in an Arbitrary Base; Continued Fractions and Greatest Common Divisors
• Proof of the Derivative Formula for Power Series
• The Interval (or Disk) of Convergence for a Power Series
• Some Arithmetic Properties Generalized to Power Series
• Some Theory about Derivatives for Power Series
• The Intermediate Value Property for Power Series Functions, and Power Series Functions are Orderly
• The Generalized Binomial Theorem and the Power Series Expansion for arcsin

Projects

• Lambert's Function W
• Hyperbolic and Inverse Hyperbolic Functions

4. Miscellania

4.2 Lograithmic Differentiation

4.3 Related Rates

4.4 l'Hôpital's Rules

4.5 Error Estimation

5. Integarals and Area

5.2 Axioms for Signed Area

5.3 Antiderivatives for Rational Functions, and Integration by Substitution

5.4 Some Applications of the Integral

5.5 Integration by Parts and Other Techniques of Integration

5.6 Slope Fields and Euler's Method

5.7 A Few More Differential Equations

5.8 Some Numerical Methods of Integration

Appendices

• Proof that sarea(f) Exists for f  in UC, and that the Definite Integral Satisfies the Axioms
• The Proof of Axiom 3 for the Riemann Integral
• Discovering the Antiderivative of Ö(1 - x²)
• How to Integrate General Rational Functions
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Projects

• Average Value of a Continuous Function
• The Henstock-Kurtzweil Integral

6 . General Continuous Functions, Derivatives and Integrals

6.2 Continuity, and Properties of Continuous Functions

6.3 Derivatives of Continuous Functions, and General Rules for Differentiation; Equivalence with Earlier Definitions

6.4 Continuity, Riemann Sums, and Integrals

6.5 The Fundamental Theorem of Calculus