# Calculus

**Calculus**

This wikibook aims to be a high quality **calculus** textbook through which users can master the discipline. Standard topics such as *limits*, *differentiation* and *integration* are covered, as well as several others. Please contribute wherever you feel the need. You can simply help by rating individual sections of the book that you feel were inappropriately rated!

## PrecalculusEdit

1.1 Algebra 1.2 Trigonometric functions 1.3 Functions 1.4 Graphing linear functions 1.5 Exercises

## LimitsEdit

2.1 An Introduction to Limits 2.2 Finite Limits 2.3 Infinite Limits 2.4 Continuity 2.5 Formal Definition of the Limit 2.6 Proofs of Some Basic Limit Rules 2.7 Exercises

## DifferentiationEdit

### Basics of Differentiation Edit

3.1 Differentiation Defined 3.2 Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 Chain Rule 3.5 Higher Order Derivatives: an introduction to second order derivatives 3.6 Implicit Differentiation 3.7 Derivatives of Exponential and Logarithm Functions 3.8 Some Important Theorems 3.9 Exercises

### Applications of Derivatives Edit

3.10 L'Hôpital's Rule 3.11 Extrema and Points of Inflection 3.12 Newton's Method 3.13 Related Rates 3.14 Optimization 3.15 Euler's Method 3.16 Exercises

## IntegrationEdit

### Basics of IntegrationEdit

4.1 Definite integral 4.2 Fundamental Theorem of Calculus 4.3 Indefinite integral 4.4 Improper Integrals

### Integration TechniquesEdit

4.5 Infinite Sums 4.6 Derivative Rules and the Substitution Rule 4.7 Integration by Parts 4.8 Trigonometric Substitutions 4.9 Trigonometric Integrals 4.10 Rational Functions by Partial Fraction Decomposition 4.11 Tangent Half Angle Substitution 4.12 Reduction Formula 4.13 Irrational Functions 4.14 Numerical Approximations 4.15 Exercises

### Applications of IntegrationEdit

4.16 Area 4.17 Volume 4.18 Volume of solids of revolution 4.19 Arc length

## Parametric and Polar EquationsEdit

### Parametric EquationsEdit

- Introduction to Parametric Equations
- Differentiation and Parametric Equations
- Integration and Parametric Equations
- Exercises

### Polar EquationsEdit

## Sequences and SeriesEdit

### SequencesEdit

### SeriesEdit

- Definition of a Series
- Series
- Limit Test for Convergence
- Comparison Test for Convergence
- Integral Test for Convergence

### Series and calculusEdit

### ExercisesEdit

## Multivariable and Differential CalculusEdit

- Vectors
- Lines and Planes in Space
- Multivariable Calculus
- Derivatives of multivariate functions
- The chain rule and Clairaut's theorem
- Inverse function theorem, implicit function theorem
- Vector calculus
- Vector calculus identities
- Inverting vector calculus operators
- Points, paths, surfaces, and volumes
- Helmholtz Decomposition Theorem
- Discrete analog to Vector calculus
- Exercises

## Differential EquationsEdit

## ExtensionsEdit

### Advanced Integration TechniquesEdit

### Further AnalysisEdit

### Formal Theory of CalculusEdit

## ReferencesEdit

- Lester R. Ford, Sr. & Jr. (1963) Calculus, McGraw-Hill via HathiTrust
- w:Mellen W. Haskell (1895) On the introduction of the notion of hyperbolic functions
*Bulletin of the American Mathematical Society*1(6):155–9.