Calculus

Welcome to the Wikibook of

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!

• Introduction
• Contributing
• Resources

Precalculus

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

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

Basics of Differentiation

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

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

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

4.1 Definite integral

4.2 Fundamental Theorem of Calculus

4.3 Indefinite integral

4.4 Improper Integrals

Integration Techniques

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

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

Parametric Equations

5.1 Introduction to Parametric Equations

5.2 Differentiation and Parametric Equations

5.3 Integration and Parametric Equations

Polar Equations

5.5 Introduction to Polar Equations

5.6 Differentiation and Polar Equations

5.7 Integration and Polar Equations

Sequences and Series

Sequences

6.1 Definition of a Sequence

6.2 Sequences

Series and Tests

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

6.10 Absolute and Conditional Convergence

Series and Calculus

6.11 Taylor series

6.12 Power series

6.13 Leibniz' formula for pi

Exercises

6.14 Exercises

Multivariable Calculus

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

Introduction to Multivariable Calculus

7.1 Vectors

7.2 Curves and Surfaces in Space

7.3 Vector Functions

7.4 Introduction to multivariable calculus

Differentiation

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

• Multivariate optimisation: The Lagrangian
• Calculus on matrices

7.10 Derivatives of Multivariate Functions

7.11 Inverse Function Theorem, Implicit Function Theorem (optional)

Integration

7.12 Multiple integration

• Old: Double Integrals

7.13 Change of variables

Vector calculus

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

8.1 Ordinary Differential Equations

8.2 Partial Differential Equations

Extensions

Advanced Integration Techniques

9.1 Complexifying

Further Analysis

9.2 Systems of Ordinary Differential Equations

Formal Theory of Calculus

9.3 Real Numbers

9.4 Complex Numbers

9.5 Hyperbolic Angle

References

• 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.

• Table of Trigonometry
• Summation notation
• Tables of Derivatives
• Tables of Integrals