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

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/).