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Calculus Guide
https://github.com/mikeroyal/calculus-guide

calculus calculus-2 calculus-of-variations integrals math mathematics theorems

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Calculus Guide

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Calculus Guide

#### A guide covering Calculus including the Formulas, Equations and Theorems that will help you a better understand practical applications of Calulus.

**Note: You can easily convert this markdown file to a PDF in [VSCode](https://code.visualstudio.com/) using this handy extension [Markdown PDF](https://marketplace.visualstudio.com/items?itemName=yzane.markdown-pdf).**





# Table of Contents

1. [Calculus Learning Resources](https://github.com/mikeroyal/Calculus-Guide#calculus-learning-resources)

2. [Introduction to Calculus](https://github.com/mikeroyal/Calculus-Guide#introduction-to-calculus)

- Velocity and Distance
- Calculus Without Limits
- The Velocity at an Instant
- Circular Motion
- Trigonometry Review
- Thousand Points of Light
- Computing in Calculus

3. [Derivatives](https://github.com/mikeroyal/Calculus-Guide#derivatives)

- The Derivative of a Function
- Powers and Polynomials
- The Slope and the Tangent Line
- Derivative of the Sine and Cosine
- Product and Quotient and Power Rules
- Limits
- Continuous Functions

4. [Applications of the Derivative](https://github.com/mikeroyal/Calculus-Guide#applications-of-the-derivative)

- Linear Approximation
- Maximum and Minimum Problems
- Second Derivatives: Minimum vs. Maximum
- Graphs
- Ellipses, Parabolas, and Hyperbolas
- Iterations x,+ ,= F(x,)
- Newton's Method and Chaos
- Mean Value Theorem and L 'Hôpital's Rule

5. [The Chain Rule](https://github.com/mikeroyal/Calculus-Guide#the-chain-rule)

- Derivatives by the Charin Rule
- Implicit Differentiation and Related Rates
- Inverse Functions and Their Derivatives
- Inverses of Trigonometric Functions

6. [Integrals](https://github.com/mikeroyal/Calculus-Guide#integrals)

- What is an Integral
- Antiderivatives
- Summation vs. Integration
- Indefinite Integrals and Substitutions
- The Definite Integral
- Properties of the Integral and the Average Value
- The Fundamental Theorem and Its Consequences
- Numerical Integration

7. [Exponentials and Logarithms](https://github.com/mikeroyal/Calculus-Guide#exponentials-and-logarithms)

- The Exponential e^x
- Growth and Decay in Science and Economics
- Logarithms
- Separable Equations Including the Logistic Equation
- Powers Instead of Exponentials
- Hyperbolic Functions

8. [Techniques of Integration](https://github.com/mikeroyal/Calculus-Guide#techniques-of-integration)

- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitutions
- Partial Fractions
- Improper Integrals

9. [Applications of the Integral](https://github.com/mikeroyal/Calculus-Guide#applications-of-the-integral)

- Areas and Volumes by Slices
- Length of a Plane Curve
- Area of a Surface of Revolution
- Probability and Calculus
- Masses and Moments
- Force, Work, and Energy

10. [Polar Coordinates and Complex Numbers](https://github.com/mikeroyal/Calculus-Guide#polar-coordinates-and-complex-numbers)

- Polar Coordinates
- Polar Equations and Graphs
- Slope, Length, and Area for Polar Curves
- Complex Numbers

11. [Infinite Series](https://github.com/mikeroyal/Calculus-Guide#infinite-series)

- The Geometric Series
- Convergence Tests: Positive Series
- Convergence Tests: All Series
- The Taylor Series for e^x, sin x, and cos x
- Power Series

12. [Vectors and Matrices](https://github.com/mikeroyal/Calculus-Guide#vectors-and-matrices)

- Vectors and Dot Products
- Planes and Projections
- Cross Products and Determinants
- Matrices and Linear Equations
- Linear Algebra in Three Dimensions

13. [Motion along a Curve](https://github.com/mikeroyal/Calculus-Guide#motion-along-a-curve)

- The Position Vector
- Plane Motion: Projectiles and Cycloids
- Tangent Vector and Normal Vector
- Polar Coordinates and Planetary Motion

14. [Partial Derivatives](https://github.com/mikeroyal/Calculus-Guide#partial-derivatives)

- Surface and Level Curves
- Partial Derivatives
- Tangent Planes and Linear Approximations
- Directional Derivatives and Gradients
- The Chain Rule
- Maxima, Minima, and Saddle Points
- Constraints and Lagrange Multipliers

15. [Multiple Integrals](https://github.com/mikeroyal/Calculus-Guide#multiple-integrals)

- Double Integrals
- Changing to Better Coordinates
- Triple Integrals
- Cylindrical and Spherical Coordinates

16. [Vector Calculus](https://github.com/mikeroyal/Calculus-Guide#vector-calculus)

- Vector Fields
- Line Integrals
- Green's Theorem
- Surface Integrals
- The Divergence Theorem
- Stokes' Theorem and the Curl of F

# Calculus Learning Resources
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

[Calulus](https://en.wikipedia.org/wiki/Calculus) is the mathematical study of continuous change where there is two major branches, [Differential calculus](https://en.wikipedia.org/wiki/Differential_calculus) and [Integral calculus](https://en.wikipedia.org/wiki/Integral).

[Wolfram Course Assistant Apps: Calculus Step-by-Step](http://products.wolframalpha.com/courseassistants/calculus.html)

[Wolfram Course Assistant Apps: multivariable Step-by-Step](https://products.wolframalpha.com/courseassistants/mvcalculus.html)

[Calculus - Math is Fun](https://www.mathsisfun.com/calculus/)

[Calculus 1 | Khan Academy](https://www.khanacademy.org/math/calculus-1)

[Differential Calculus | Khan Academy](https://www.khanacademy.org/math/differential-calculus)

[Top Calculus Courses Online | Coursera](https://www.coursera.org/courses?query=calculus)

[Introduction to Calculus | Coursera](https://www.coursera.org/learn/introduction-to-calculus)

[Mathematics for Machine Learning: Multivariate Calculus | Coursera](https://www.coursera.org/learn/multivariate-calculus-machine-learning)

[Calculus - Mathematics for Data Science - Machine Learning](https://www.udemy.com/course/deep-learning-calculus-data-science-machine-learning-ai/)

[Calculus and Optimization for Machine Learning Course | Coursera](https://www.coursera.org/learn/calculus-and-optimization-for-machine-learning)

[Learn Calculus 1 and Become a Calculus Master Course | Udemy](https://www.udemy.com/course/calculus1/)

[Become a Calculus 2 Master Course | Udemy](https://www.udemy.com/course/calculus-2/)

[Become a Calculus 3 Master Course | Udemy](https://www.udemy.com/course/calculus-3/)

[Calculus 3 (multivariable calculus) | Udemy](https://www.udemy.com/course/calculus-3-multivariable-calculus-part-1-of-2/)

[Learn Calculus with Online Courses | edX](https://www.edx.org/learn/calculus)

[Online Calculus Courses | Harvard University](https://online-learning.harvard.edu/subject/calculus)

[Calculus 1 | Harvard University - Harvard Online Courses](https://online-learning.harvard.edu/course/calculus-1)

[Mathematics | MIT OpenCourseWare | Free Online Courses](https://ocw.mit.edu/courses/mathematics/)

[Single Variable Calculus | Mathematics | MIT OpenCourseWare](https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/)

[The Organic Chemistry Tutor - Calculus Video Playlist | YouTube](https://www.youtube.com/playlist?list=PL0o_zxa4K1BWYThyV4T2Allw6zY0jEumv)

[Calculus I - Pauls Online Math Notes](https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx)

[Calculus II - Pauls Online Math Notes](https://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx)

[Calculus III - Pauls Online Math Notes](https://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx)

# Introduction to Calculus
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Velocity and Distance

### Calculus Without Limits

### The Velocity at an Instant

### Circular Motion

### Trigonometry Review

### Thousand Points of Light

### Computing in Calculus





Distance and Velocity Formulas.

# Derivatives
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### The Derivative of a Function

### Powers and Polynomials

### The Slope and the Tangent Line

### Derivative of the Sine and Cosine

### The Product and Quotient and Power Rules

### Limits

### Continuous Functions





Derivative Rules. Source: [onlinemathlearning](https://www.onlinemathlearning.com/derivative-rules.html)

# Applications of the Derivative
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Linear Approximation

### Maximum and Minimum Problems

### Second Derivatives: Minimum vs. Maximum

### Graphs

### Ellipses, Parabolas, and Hyperbolas

### Iterations x,+ ,= F(x,)

### Newton's Method and Chaos

### The Mean Value Theorem and L'Hôpital's Rule





Derivatives and Limits. Source: [EEWeb](https://www.eeweb.com/tools/online-scientific-calculator/)

# The Chain Rule
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Derivatives by the Charin Rule

### Implicit Differentiation and Related Rates

### Inverse Functions and Their Derivatives

### Inverses of Trigonometric Functions





Chain Rule. Source: [onlinemathlearning](https://www.onlinemathlearning.com/chain-rule.html)

# Integrals
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### What is an Integral

### Antiderivatives

### Summation vs. Integration

### Indefinite Integrals and Substitutions

### The Definite Integral

### Properties of the Integral and the Average Value

### The Fundamental Theorem and Its Consequences

### Numerical Integration





Properties of Integrals & Integration Rules. Source: [OSU.edu](https://mslc.osu.edu/sites/mslc.osu.edu/files/Integral%20Calculus%20Formula%20Sheet_0.pdf)





Fund. Theorem of Calulus, Riemann Sums, and Net Change. Source: [OSU.edu](https://mslc.osu.edu/sites/mslc.osu.edu/files/Integral%20Calculus%20Formula%20Sheet_0.pdf)

# Exponentials and Logarithms
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### The Exponential e^x

### Growth and Decay in Science and Economics

### Logarithms

### Separable Equations Including the Logistic Equation

### Powers Instead of Exponentials

### Hyperbolic Functions





Exponents and Logarithms. Source: [onlinemathlearning](https://www.onlinemathlearning.com/logarithms-exponents.html)

# Techniques of Integration
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Integration by Parts

### Trigonometric Integrals

### Trigonometric Substitutions

### Partial Fractions

### Improper Integrals





Techniques of Integration. Source: [OSU.edu](https://mslc.osu.edu/sites/mslc.osu.edu/files/Integral%20Calculus%20Formula%20Sheet_0.pdf)





Techniques of Integration Part 2. Source: [OSU.edu](https://mslc.osu.edu/sites/mslc.osu.edu/files/Integral%20Calculus%20Formula%20Sheet_0.pdf)

# Applications of the Integral
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Areas and Volumes by Slices

### Length of a Plane Curve

### Area of a Surface of Revolution

### Probability and Calculus

### Masses and Moments

### Force, Work, and Energy





Surface Area of Revolution of Parametric Equations. Source: [YouTube](https://www.youtube.com/watch?v=USiKDtILhmc)





Areas and Volumes. Source: [OSU.edu](https://mslc.osu.edu/sites/mslc.osu.edu/files/Integral%20Calculus%20Formula%20Sheet_0.pdf)





Physical Applications. Source: [OSU.edu](https://mslc.osu.edu/sites/mslc.osu.edu/files/Integral%20Calculus%20Formula%20Sheet_0.pdf)

# Polar Coordinates and Complex Numbers
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Polar Coordinates

### Polar Equations and Graphs

### Slope, Length, and Area for Polar Curves

### Complex Numbers





Polar Coordinates - Rectangular. Source: [getcalc](https://getcalc.com/geometry-rectangle-polar-calculator.htm)





Area in Polar Coordinates. Source: [Calcworkshop](https://calcworkshop.com/polar-functions/area-polar-coordinates/)

# Infinite Series
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### The Geometric Series

### Convergence Tests: Positive Series

### Convergence Tests: All Series

### The Taylor Series for e^x, sin x, and cos x

### Power Series





Geometric Series. Source: [onlinemathlearning](https://www.onlinemathlearning.com/geometric-series.html)





Taylor and MacLaurin Series. Source: [onlinemathlearning](https://www.onlinemathlearning.com/taylor-maclaurin-series.html)





Power Series. Source: [onlinemathlearning](https://www.onlinemathlearning.com)

# Vectors and Matrices
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Vectors and Dot Products

### Planes and Projections

### Cross Products and Determinants

### Matrices and Linear Equations

### Linear Algebra in Three Dimensions





Scalar, Vector, Matrix. Source: [mathisfun](http://www.mathsisfun.com/algebra/scalar-vector-matrix.html)

# Motion along a Curve
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### The Position Vector

### Plane Motion: Projectiles and Cycloids

### Tangent Vector and Normal Vector

### Polar Coordinates and Planetary Motion





Position Vectors. Source [YouTube](http://www.youtube.com/watch?v=v3uhB6LhXis)

# Partial Derivatives
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Surface and Level Curves

### Partial Derivatives

### Tangent Planes and Linear Approximations

### Directional Derivatives and Gradients

### The Chain Rule

### Maxima, Minima, and Saddle Points

### Constraints and Lagrange Multipliers





Chain Rule With Partial Derivatives. Source: [YouTube](https://www.youtube.com/watch?v=XipB_uEexF0)

# Multiple Integrals
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Double Integrals

### Changing to Better Coordinates

### Triple Integrals

### Cylindrical and Spherical Coordinates





Double Integrals. Source: [YouTube](https://www.youtube.com/watch?v=BJ_0FURo9RE)





Triple Integrals. Source: [YouTube](https://www.youtube.com/watch?v=7iy83x8bv6o)





Cylindrical and Spherical Coordinates.

# Vector Calculus
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

### Vector Fields

### Line Integrals

### Green's Theorem

### Surface Integrals

### The Divergence Theorem

### Stokes' Theorem and the Curl of F





Vector Calculus Forumlas. Source: [vectorified](https://vectorified.com/vector-calculus-formulas)

## Contribute

- [x] If would you like to contribute to this guide simply make a [Pull Request](https://github.com/mikeroyal/Calculus-Guide/pulls).

## License
[Back to the Top](https://github.com/mikeroyal/Calculus-Guide#table-of-contents)

Distributed under the [Creative Commons Attribution 4.0 International (CC BY 4.0) Public License](https://creativecommons.org/licenses/by/4.0/).