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 http://math1.skku.ac.kr/ 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.

CONTENTS


Preface


Part I  Single Variable Calculus

 


Part II  Multivariate Calculus

 

References

Index

Copyright

 

  : Calculus with Sage Lectures (Youtube Movies)

 

1.1 History of Calculus :  http://youtu.be/ODfMaHgIhAc

 

Calculus with Sage Week 1 how to manage our class Review :  http://youtu.be/XWEQFlv4jKc

 

Chapter 1. Functions  http://youtu.be/cl8GqIWIRD0  ( )

 

1.1 Functions and its graph      Ǯ  http://youtu.be/rQ2CB8EvkoE  

1.2 Symmetry    Ǯ http://youtu.be/BNKUzSohiD8

1.3 Common Functions   Ǯ http://youtu.be/x0E0ZMxZ3Og

1.4 Translation, Stretching and Rotation of Functions Ǯ http://youtu.be/vx7GCWY68Zw

 

Chapter 2. Limits and Continuity

 

   2.1 Limits of functions         :     http://youtu.be/VBCeAllP1M0

   2.2 Continuity                   :    http://youtu.be/zGxx3PUCTnM

 


Chapter 3. Theory of Differentiation

 

    3.1 Definition of Derivatives, Differentiation          : http://youtu.be/A-vDsF9ulTs

    3.2 Derivatives of Polynomials, Exponential Functions, ... , The product rule: http://youtu.be/XXMnCESesfQ

    3.3 The Chain Rule and Inverse Functions              : http://youtu.be/HfScHEsPfKI

    3.4 Approximation and Related Rates :  http://youtu.be/ViRwEJ0Wfkw

 

Chapter 4. Applications of Differentiation

 

    4.1 Extreme values of a function :  http://youtu.be/mXVU8OqIHJY

    4.2 The Shape of a Graph :   http://youtu.be/cZrAF_77On4

    4.3 The Limit of Indeterminate Forms and LHospitals Rule :  http://youtu.be/vp-gck5-gKE  

    4.4 Optimization Problems : http://youtu.be/k0NtkmZFnh8

    4.5 Newtons Method : http://youtu.be/VxCfl2JzMYU


Chapter 5. Integrals


    5.1 Areas and Distances : http://youtu.be/mT_oxlD6RSA


    5.2 The Definite Integral : http://youtu.be/GIm3Oz58Ti8  

    5.3 The Fundamental Theorem of Calculus :  http://youtu.be/Zf1HT2H2fbA

    5.4 Indefinite Integrals and the Net Change Theorem : http://youtu.be/E6I3EDzAVuU

    5.5 The Substitution Rule  : http://youtu.be/h7tmvmNOliU  

    5.6 The Logarithm Defined as an Integral : http://youtu.be/kD0Z9PqetsA  


Chapter 6. Applications of Integration

 

     with Sage Sec-6-1 Areas between Curves, SKKU  http://youtu.be/o53phm5cqJE

     with Sage Sec-6-2 Volumes, SKKU http://youtu.be/4-ChOAFbJAs

     with Sage Sec-6-3 Volumes by Cylindrical Shells, SKKU  http://youtu.be/qM1izf8qeX8

     with Sage Sec-6-4 Work by SKKU : http://youtu.be/u3ZaJWhKy6k

     with Sage Sec-6-5 Average Value of a Function  by SKKU http://youtu.be/zmEeGmwQTB0

 

Chapter 7. Techniques of Integration

 

with Sage Sec-7-1 Integration by Parts  by SKKU : http://youtu.be/WX-6C9tCneE

with Sage Sec-7-2 Trigonometric Integrals  by SKKU : http://youtu.be/sIR0zNGQbus

with Sage Sec-7-3 Trigonometric Substitution  by SKKU : http://youtu.be/avTqiEUi8u8

with Sage Sec-7-4 Integration of Rational Functions by the Method of Partial Fractions  by SKKU : http://youtu.be/KLTHp_7G4cI

with Sage Sec-7-5 Guidelines for Integration  by SKKU : http://youtu.be/Fgn8U4We60o

with Sage Sec-7-6 Integration Using Tables   by SKKU : http://youtu.be/tn9jLkgTMp8

with Sage Sec-7-7 Approximate Integration  by SKKU : http://youtu.be/hg2pw1n1cZI

with Sage Sec-7-8 Improper Integrals  by SKKU : http://youtu.be/rquxbYrC0Yc


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  http://youtu.be/rQ2CB8EvkoE

with Sage Sec-1-2  Symmetry, Problem, Ǯ by JHKwak  http://youtu.be/BNKUzSohiD8

with Sage Sec-1-3 Common Functions, Problem, Ǯ by CYJang http://youtu.be/x0E0ZMxZ3Og

with Sage Sec-1-4 Translation, Stretching and Rotation of Functions, Problem, Ǯ by HJIm http://youtu.be/vx7GCWY68Zw


with Sage Sec-2-1 Limits of functions, Problem, Ǯ by Jang-Lee http://youtu.be/LZSmRPAAXME

with Sage Sec-2-2 Continuity, Problem, Ǯ by Lee http://youtu.be/azrkT1RP4-c

with Sage Sec-2-2 Continuity, Epsilon-Delta Proof, by ICHwang  http://youtu.be/hj8d-j_DGf4


with Sage Sec-3-1 Definition of Derivatives, Differentiation, Problem, Ǯ by DHKim http://youtu.be/7wTBWuk2CzU

with Sage Sec-3-2 Derivatives of Polynomials, Exponential Functions, Trigonometric, Problem, Ǯ by Cho http://youtu.be/Ei5KGW9vZhE

with Sage Sec-3-3 The Chain Rule and Inverse Functions, Problem, Ǯ by Yoo http://youtu.be/aSKm12922FE

with Sage Sec-3-4 Approximation and Related Rates, Problem, Ǯ http://youtu.be/JmBOv6_D6qA


with Sage Sec-4-1 Extreme values of a function, Problem, Ǯ by TYKim http://youtu.be/_V4MryNEzWY

with Sage Sec-4-2 The Shape of a Graph, Problem, Ǯ by TYKim http://youtu.be/SVOWADHlzV8

with Sage Sec-4-3 Indeterminate Forms and L'Hospital's Rule, Problem, Ǯ by Shin http://youtu.be/gR2luDDPsMY

with Sage Sec-4-4 Optimization, Problem, Ǯ by Lee http://youtu.be/AELEV2ElaeQ

with Sage Sec-4-5 Newton's Method, Problem, Ǯ by Lee  http://youtu.be/fdBHQ46g9RE


with Sage Sec-5-1 Area and Distance, Problem, Ǯ by THNam  http://youtu.be/Y_nCn76RPmY

with Sage Sec-5-2 Definite Integral, Problem, Ǯ by THNam   http://youtu.be/iUsf1h_hTAE

with Sage Sec-5-3 and 5-4 Fun Theorem of Calculus Net Change Theorem, Problem, Ǯ by Jung & Kim http://youtu.be/Pa4Z38KkDVY

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

with Sage Sec-5-5 Substitution, Problem, Ǯ by HWLee http://youtu.be/0TMbpCPO4Uc

with Sage Sec-5-6 Log and Exponential, Problem, Ǯ by HWLee  http://youtu.be/ymDImdIQ90c

 

 

Internet resources :


Sage-Reference: http://matrix.skku.ac.kr/2009-Sage/Sage-Reference.html  

Sage Tutorial: http://www.youtube.com/watch?v=GJcym7gMKrg&feature=results_main&playnext=1&list=PL9168C6B83FE306CE  

2011-How to use Sage 1: http://matrix.skku.ac.kr/2011-Album/Sage-02.html

2011-How to use Sage 2: http://matrix.skku.ac.kr/2010-Album/Math-talk-Sage.html

2011-How to use Sage 3: http://matrix.skku.ac.kr/2011-Album/Sage-01.html

William Stein demos sage math: http://www.youtube.com/watch?v=kIQZU_uZGlc

2011-Mobile Math with Sage: http://matrix.skku.ac.kr/2011-Sage/2011-Mobile-Math/MobileMath.html

Sage Interact / ODE and Mandelbrot: http://www.youtube.com/watch?v=_258y4kMYyQ  

Sage Multivariable Calculus (1 of 2) by Travis: http://www.youtube.com/watch?v=rqACCzGYOm8  

Sage Multivariable Calculus (2 of 2) by Travis: http://www.youtube.com/watch?v=SwgFWKK0oCg

http://bkmath.skku.ac.kr/bk21/index.html

http://matrix.skku.ac.kr/sglee



http://matrix.skku.ac.kr/2013-Calculus-Sage/Cal-lab-0-3/cal-lab-ch0to3.htm

http://matrix.skku.ac.kr/2013-Calculus-Sage/Web-Cover/CH-0-Cover.pdf


* http://matrix.skku.ac.kr/cal-lab/Math-CAS.htm