Main Author : Sang-Gu Lee

Co-Authors : Eung-Ki Kim, Yoonmee Ham, Ajit Kumar, Robert A. Beezer, Phong Vu, S.-G. Hwang, J.-D. Sim, B.-S. Jang, Lois S. Simon, ... (not  more than 10)

Reviewers : R. Sakthivel, K. Das, I. Hwang, J. Lee, ... (more than 10)




  Calculus is the mathematical foundation for much of university mathematics, science, and engineering curriculum. For the mathematics student, it is a first exposure to rigorous mathematics. For the engineer, it is an introduction to the modeling and approximation techniques used throughout an engineering curriculum. And for the future scientist, it is the mathematical language that will be used to express many of the most important scientific concepts.


  In the first semester, that is for the beginners of calculus, we start with differential and integral calculus on functions of single variable and then study L'Hospital's theorem, concavity, convexity, inflection points, optimization problems, ordinary differential equations as applications of differential and integral calculus, parameter equations, polar coordinates, infinite sequences and infinite series accordingly. In the second semester of calculus, we cover vector calculus that includes vectors, coordinate space, partial derivatives and multiple integrals. Concepts, definitions, terminology, and interpretation in calculus should be as current as possible. This book has many problems presenting calculus as the foundation of modern mathematics, science and engineering.


  Many recent calculus textbooks are using Computer Algebra System (CAS) including a variety of visual tools in it. But in most cases its use by students is limited. Therefore, for this book, we have adapted a wonderful free and open-source program, Sage. With the new learning environment of universities, students will take a full advantage of 21st century, state of the art technology to learn calculus easily and be better prepared for future careers. We can use Sage easily on popular web browsers such as Firefox or Chrome. The system language for Sage is Python, a powerful mainstream computer programming language.


  More content and related materials will be added to be viewed on the web. When you see a CAS or web mark in the book, this means you will be able to find relevant information by clicking on the address. This will save you a lot of work.


  Finally, the book also combines technology, reform, and tradition in a way that offers a wider view to students. Most importantly, we appreciate everyone who has contributed to the project of writing this book.

                                 2013. 2. 1.



Part I  Single Variable Calculus


Part II  Multivariate Calculus






  : Calculus with Sage Lectures (Youtube Movies)


1.1 History of Calculus :


Calculus with Sage Week 1 how to manage our class Review :


Chapter 1. Functions  ( )


1.1 Functions and its graph      Ǯ  

1.2 Symmetry    Ǯ

1.3 Common Functions   Ǯ

1.4 Translation, Stretching and Rotation of Functions Ǯ


Chapter 2. Limits and Continuity


   2.1 Limits of functions         :

   2.2 Continuity                   :


Chapter 3. Theory of Differentiation


    3.1 Definition of Derivatives, Differentiation          :

    3.2 Derivatives of Polynomials, Exponential Functions, ... , The product rule:

    3.3 The Chain Rule and Inverse Functions              :

    3.4 Approximation and Related Rates :


Chapter 4. Applications of Differentiation


    4.1 Extreme values of a function :

    4.2 The Shape of a Graph :

    4.3 The Limit of Indeterminate Forms and LHospitals Rule :  

    4.4 Optimization Problems :

    4.5 Newtons Method :

Chapter 5. Integrals

    5.1 Areas and Distances :

    5.2 The Definite Integral :  

    5.3 The Fundamental Theorem of Calculus :

    5.4 Indefinite Integrals and the Net Change Theorem :

    5.5 The Substitution Rule  :  

    5.6 The Logarithm Defined as an Integral :  

Chapter 6. Applications of Integration


     with Sage Sec-6-1 Areas between Curves, SKKU

     with Sage Sec-6-2 Volumes, SKKU

     with Sage Sec-6-3 Volumes by Cylindrical Shells, SKKU

     with Sage Sec-6-4 Work by SKKU :

     with Sage Sec-6-5 Average Value of a Function  by SKKU


Chapter 7. Techniques of Integration


with Sage Sec-7-1 Integration by Parts  by SKKU :

with Sage Sec-7-2 Trigonometric Integrals  by SKKU :

with Sage Sec-7-3 Trigonometric Substitution  by SKKU :

with Sage Sec-7-4 Integration of Rational Functions by the Method of Partial Fractions  by SKKU :

with Sage Sec-7-5 Guidelines for Integration  by SKKU :

with Sage Sec-7-6 Integration Using Tables   by SKKU :

with Sage Sec-7-7 Approximate Integration  by SKKU :

with Sage Sec-7-8 Improper Integrals  by SKKU :

Chapter 8. Further Applications of Integration


    8.1 Arc Length

    8.2 Area of a Surface of Revolution

    8.3 Applications of Integral Calculus

    8.4 Differential equations


Chapter 9. Parametric Equations and Polar Coordinates


    9.1 Parametric Equations

    9.2 Calculus with Parametric Curves

    9.3 Polar Coordinates

    9.4 Areas and Lengths in Polar Coordinates

    9.5 Conic Section


Chapter 10. Infinite Sequences and Infinite Series

     10.1 Sequences and Series

     10.2 Tests for convergence of series with positive terms

     10.3 Alternating Series and Absolute Convergence

     10.4 Power Series

 Syudent Activity (л Ǯ ):

with Sage Sec-1-1 Functions and Graph, Problem, Ǯ by ICHwang

with Sage Sec-1-2  Symmetry, Problem, Ǯ by JHKwak

with Sage Sec-1-3 Common Functions, Problem, Ǯ by CYJang

with Sage Sec-1-4 Translation, Stretching and Rotation of Functions, Problem, Ǯ by HJIm

with Sage Sec-2-1 Limits of functions, Problem, Ǯ by Jang-Lee

with Sage Sec-2-2 Continuity, Problem, Ǯ by Lee

with Sage Sec-2-2 Continuity, Epsilon-Delta Proof, by ICHwang

with Sage Sec-3-1 Definition of Derivatives, Differentiation, Problem, Ǯ by DHKim

with Sage Sec-3-2 Derivatives of Polynomials, Exponential Functions, Trigonometric, Problem, Ǯ by Cho

with Sage Sec-3-3 The Chain Rule and Inverse Functions, Problem, Ǯ by Yoo

with Sage Sec-3-4 Approximation and Related Rates, Problem, Ǯ

with Sage Sec-4-1 Extreme values of a function, Problem, Ǯ by TYKim

with Sage Sec-4-2 The Shape of a Graph, Problem, Ǯ by TYKim

with Sage Sec-4-3 Indeterminate Forms and L'Hospital's Rule, Problem, Ǯ by Shin

with Sage Sec-4-4 Optimization, Problem, Ǯ by Lee

with Sage Sec-4-5 Newton's Method, Problem, Ǯ by Lee

with Sage Sec-5-1 Area and Distance, Problem, Ǯ by THNam

with Sage Sec-5-2 Definite Integral, Problem, Ǯ by THNam

with Sage Sec-5-3 and 5-4 Fun Theorem of Calculus Net Change Theorem, Problem, Ǯ by Jung & Kim

( with Sage Sec-5-4  Net Change Theorem, Problem, Ǯ by Kim)

with Sage Sec-5-5 Substitution, Problem, Ǯ by HWLee

with Sage Sec-5-6 Log and Exponential, Problem, Ǯ by HWLee



Internet resources :


Sage Tutorial:  

2011-How to use Sage 1:

2011-How to use Sage 2:

2011-How to use Sage 3:

William Stein demos sage math:

2011-Mobile Math with Sage:

Sage Interact / ODE and Mandelbrot:  

Sage Multivariable Calculus (1 of 2) by Travis:  

Sage Multivariable Calculus (2 of 2) by Travis: