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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!

Precalculus edit

1.1 Algebra  

1.2 Functions  

1.3 Trigonometric functions  

1.4 Graphing functions  

1.5 Rational functions

1.6 Conic sections  

1.7 Exercises

1.8 Hyperbolic logarithm and angles  

Limits edit


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

Differentiation edit

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 Derivatives of Hyperbolic Functions

3.9 Some Important Theorems

3.10 Exercises

Applications of Derivatives   edit

3.11 L'Hôpital's Rule  

3.12 Extrema and Points of Inflection

3.13 Newton's Method

3.14 Related Rates

3.15 Optimization

3.16 Euler's Method

3.17 Approximating Values of Functions

3.18 Exercises

Integration edit

The definite integral of a function f(x) from x=0 to x=a is equal to the area under the curve from 0 to a.

Basics of Integration edit

4.1 Definite integral  

4.2 Fundamental Theorem of Calculus  

4.3 Indefinite integral  

4.4 Improper Integrals

Integration Techniques edit

From bottom to top:
  • an acceleration function a(t);
  • the integral of the acceleration is the velocity function v(t);
  • and the integral of the velocity is the distance function s(t).

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 Integration edit

4.16 Area

4.17 Volume

4.18 Volume of Solids of Revolution

4.19 Arc Length

4.20 Surface Area

4.21 Work

4.22 Center of Mass

4.23 Pressure and Force

4.24 Probability

Parametric and Polar Equations edit


Parametric Equations edit

5.1 Introduction to Parametric Equations

5.2 Differentiation and Parametric Equations

5.3 Integration and Parametric Equations

Polar Equations edit

5.5 Introduction to Polar Equations

5.6 Differentiation and Polar Equations

5.7 Integration and Polar Equations

Sequences and Series edit

Sequences edit

6.1 Definition of a Sequence

6.2 Sequences

Series and Tests edit

6.3 Definition of a Series

6.4 Series

6.5 Divergence Test

6.6 Ratio Test

6.7 Limit Comparison Test

6.8 Direct Comparison Test

6.9 Integral Test

Convergence edit

6.10 Absolute and Conditional Convergence

Series and Calculus edit

6.11 Taylor series

6.12 Power series

6.13 Leibniz' formula for pi

Exercises edit

6.14 Exercises

Multivariable Calculus edit

This is an example of using spherical coordinates in 3 dimensions to calculate the volume of a given shape

Introduction to Multivariable Calculus edit

7.1 Vectors  

7.2 Curves and Surfaces in Space  

7.3 Vector Functions  

7.4 Introduction to multivariable calculus  

Differentiation edit

7.5 Limits and Continuity

7.6 Partial Derivatives

7.7 The chain rule and Clairaut's theorem  

7.8 Chain Rule

7.9 Directional derivatives and the gradient vector

7.10 Derivatives of Multivariate Functions  

7.11 Inverse Function Theorem, Implicit Function Theorem (optional)  

Integration edit

7.12 Multiple integration

7.13 Change of variables

Vector calculus edit

7.14 Vector Calculus  

7.15 Vector Calculus Identities  

7.16 Inverting Vector Calculus Operators  

7.17 Points, Paths, Surfaces, and Volumes  

7.18 Helmholtz Decomposition Theorem  

7.19 Discrete Analog to Vector Calculus  

7.20 Exercises

Differential Equations edit

8.1 Ordinary Differential Equations  

8.2 Partial Differential Equations  

Extensions edit

Advanced Integration Techniques edit

9.1 Complexifying

Further Analysis edit

9.2 Systems of Ordinary Differential Equations  

Formal Theory of Calculus edit

9.3 Real Numbers  

9.4 Complex Numbers  

9.5 Hyperbolic Angle  

References edit

Acknowledgements and Further Reading edit