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Index
Calculus I
R. Mayer
Math 111, Reed College
0. Introduction
An Overview of the Course
Prerequisites
Exercises and Entertainments
1. Some Notation for Sets
2. Some Area Calculations
2.1 The Area Under a Power Function
2.2 Some Summation Formulas
2.3 The Area Under a Parabola
2.4 Finite Geometric Series
2.5 Area Under the Curve
2.6
Area of a Snowflake.
3. Propositions and Functions
3.1 Propositions
3.2 Sets Defined by Propositions
3.3 Functions
3.4 Summation Notation
3.5 Mathematical Induction
4. Analytic Geometry
4.1 Addition of Points
4.2 Reflections, Rotations and Translations
4.3 The Pythagorean Theorem and Distance.
5. Area
5.1 Basic Assumptions about Area
5.2 Further Assumptions About Area
5.3 Monotonic Functions
5.4 Logarithms.
5.5
Brouncker's Formula For
5.6 Computer Calculation of Area
6. Limits of Sequences
6.1 Absolute Value
6.2 Approximation
6.3 Convergence of Sequences
6.4 Properties of Limits.
6.5 Illustrations of the Basic Limit Properties.
6.6 Geometric Series
6.7 Calculation of
7. Still More Area Calculations
7.1 Area Under a Monotonic Function
7.2 Calculation of Area under Power Functions
8. Integrable Functions
8.1 Definition of the Integral
8.2 Properties of the Integral
8.3 A Non-integrable Function
8.4
The Ruler Function
8.5 Change of Scale
8.6 Integrals and Area
9. Trigonometric Functions
9.1 Properties of Sine and Cosine
9.2 Calculation of
9.3 Integrals of the Trigonometric Functions
9.4 Indefinite Integrals
10. Definition of the Derivative
10.1 Velocity and Tangents
10.2 Limits of Functions
10.3 Definition of the Derivative.
11. Calculation of Derivatives
11.1 Derivatives of Some Special Functions
11.2 Some General Differentiation Theorems.
11.3 Composition of Functions
12. Extreme Values of Functions
12.1 Continuity
12.2
A Nowhere Differentiable Continuous Function.
12.3 Maxima and Minima
12.4 The Mean Value Theorem
13. Applications
13.1 Curve Sketching
13.2 Optimization Problems.
13.3 Rates of Change
14. The Inverse Function Theorem
14.1 The Intermediate Value Property
14.2 Applications
14.3 Inverse Functions
14.4 The Exponential Function
14.5 Inverse Function Theorems
14.6 Some Derivative Calculations
15. The Second Derivative
15.1 Higher Order Derivatives
15.2 Acceleration
15.3 Convexity
16. Fundamental Theorem of Calculus
17. Antidifferentiation Techniques
17.1 The Antidifferentiation Problem
17.2 Basic Formulas
17.3 Integration by Parts
17.4 Integration by Substitution
17.5 Trigonometric Substitution
17.6 Substitution in Integrals
17.7 Rational Functions
Bibliography
A. Hints and Answers
B. Proofs of Some Area Theorems
C. Prerequisites
Index
About this document ...
Ray Mayer 2007-09-07
Area of a Snowflake.
