CALCULUS
OF ONE REAL VARIABLE

 

A TUTORIAL

 

 

By Pheng Kim Ving, BA&Sc, MSc
Email: pheng@phengkimving.com
Toronto
- Canada

View

 

If you're using Internet Explorer and if it doesn't display the view properly, such as misplaced or (partly-)missing tables or

graphics, please try Firefox or Chrome. If the problem still doesn't go away, please let me know and I'll email you copies of
the .docx and .pdf versions of the problematic pages.

 

Welcome To CALCULUS OF ONE REAL VARIABLE!!

 

This website posts a tutorial on the introductory calculus of one real variable, free!! It provides a complete treatment of the
introductory calculus of functions of one real variable. It's organized to accompany two one-semester first and second calculus
courses or one two-semester first calculus course.

 

Each chapter is divided into sections. Each section discusses the topics that are the subject of the section and provides
examples each followed by its complete solution. The presentation of each section is fairly comprehensive and detailed, almost
the same as in textbooks, not just a summary of the topics. Each section includes a set of problems with complete solutions.

The examples and problems supply drills on the basic techniques for the topics discussed in the section, and some are
theoretical and/or difficult.

 

If you have thoughts or comments about the site and you like to make them public, please don't hesitate to sign my guestbook.
The link to the guestbook is accessible under the heading “ Guess Who's The Guest!”, below.

 

 

Math Teacher/Tutor

 

If you need a math teacher/tutor and you live in (metro) Toronto then I'll be pleased to be your teacher/tutor. For more
information please click here.

 

 

 

Solutions To Problems

 

Do you need solutions to a set of problems you obtained from class or from a course? If you answer yes then please click here for more information.

 

 

Guess Who's The Guest!

 

You are the guest of this site!! If you like, please take the liberty to view or sign my guestbook. To go to the guestbook
please click here.

 

 

CONTENTS

 

Reference To A Function
Splitting Of The Topic Of The Applications Of The Derivative

Notations And Abbreviations

 

1. Limits And Continuity

 

     1.1 Limits
          1.1.1 Limits
          1.1.2 Properties Of Limits
          1.1.3 The Indeterminate Form Of Type 0/0

Wolky Comfort Lace Lace Up Up Sko Gobly Suede Metal Sofa Suede 03669e2 - awesome-b12.site

          1.1.4 One-Sided Limits
          1.1.5 Limits At Infinity And Infinite Limits
         
         

     1.2 Continuity
          1.2.1 Continuity
          1.2.2 Extrema
          1.2.3 The Intermediate-Value Theorem

 

2. The Derivative

 

     2.1 Tangent Lines And Their Slopes
Wolky Comfort Lace Lace Up Up Sko Gobly Suede Metal Sofa Suede 03669e2      2.2 Rates Of Change
     2.3 The Derivative
     2.4 Differentiability Vs Continuity

3. Rules Of Differentiation

 

     3.1 Differentiation Of Sums, Differences, And Polynomials
     3.2 Differentiation Of Products And Quotients
     3.3 Differentiation Of Compositions Of Functions – The Chain Rule
     3.4 Differentiation Of Inverse Functions

 

4. More On The Derivative


     4.1 Higher-Order Derivatives
     4.2 Implicit Differentiation
     4.3 The Differential

Wolky Comfort Lace Lace Up Up Sko Gobly Suede Metal Sofa Suede 03669e2

 

5. Applications Of The Derivative – Part 1

 

     5.1 The Mean-Value Theorem
     5.2 Critical Points And Extrema
     5.3 The First-Derivative Test
     5.4 Concavity And Inflection
     5.5 The Second-Derivative Test
     5.6 Sketching Graphs Of Functions
     5.7 Antiderivatives And Indefinite Integrals
     5.8 Motion

6. The Trigonometric Functions And Their Inverses

 

     6.1 The Trigonometric Functions
          6.1.1 The Trigonometric Functions

          6.1.2 Trigonometric Identities
          6.1.3 Limits Of Trigonometric Functions
          6.1.4 Differentiation Of Trigonometric Functions
          6.1.5 Graphs Of Trigonometric Functions
          6.1.6 The Projectile Motion
          6.1.7 The Simple Harmonic Motion

 

     6.2 The Inverse Trigonometric Functions

          6.2.1 The Inverse Trigonometric Functions
          6.2.2 Differentiation Of The Inverse Trigonometric Functions

 

Caterpillar Women s Maren Unconstructed Leather Lace Up Sko med Cutouts Oxford Rose Cole Haan Women s Tali Grand Lace Wedge 40 Pump Blazer Blue , Dr. Martens Unisex 1461 W PW 3-øye Gibson Leather, Fashion Oxfords Svart TOMS Kids 10009918 Alpargata-K Desert Taupe Suede Perforert , Caterpillar Kvinner Reegan Ii Wingtip Dress Oxford Svart Drew Shoe Women s Quest Komfort Slip On Casual Oxfords Svart lær / svart strekke , Easy Spirit AP1 Sport Walking Sko Vin T.U.K. rød Easy Spirit Women s Devitt Oxford Flat Svart Ros Hommerson Women s Nancy Oxfords Black Croc Patent / Tinnpiping Eastland Solstice MLB Women s Oxford Chicago Cubs-blå Fly London Women s Yaua699fly Oxford 000 Black Brito Skechers Women s Ghenter Srelt Work Shoe Svart Madden Women s M-Graves Oxford Navy Suede , Drew Sko Toscana kvinners terapeutisk diabetisk ekstra dybde skinn lær / mesh snøre opp Krem / Combo Arcopedico kvinners Ls Denim , ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Flats Studded Sneakers Sko rød Drew Shoe Women s Paradise II Velco Slip-on Hvit , Rockport Women s Ayva Oxford Lysegrå , Prop & eacute; t Women s Washable Walker Suede Oxford tinn Rockport Women s Total Motion Envelope Flat Svart lær Rockport Women s Truflex W Lace To Toe Sneaker Sand Nubuck Easy Spirit Women s Devitt Oxford Flat Gull Skechers Women s Ghenter Srelt Work Shoe Svart , Propet Women s Washable Walker Sneaker Sr Hvit Arcopedico kvinners Ls Gull Sparkle , MINIVOG Kvinners Perforerte Tasseler Wingtip Oxfords Tinn Skinn Brogue Sko tinn , Vionic Womens Hadley tinn , Cole Haan Women s Grand Crosscourt Ii Vasket Indigo Suede-hvit

     6.3 Transcendency
          6.3.1 Transcendental Functions
          6.3.2 Transcendency Of The Trigonometric Functions

 

7. The Exponential And Logarithmic Functions

     7.1 The Natural Exponential Function
     7.2 The Natural Logarithm Function

     7.3 General Exponential And Logarithmic Functions
     7.4 Logarithmic Differentiation
     7.5 Growth And Decay
     7.6 The Hyperbolic Functions
     7.7 The Inverse Hyperbolic Functions
     7.8 Transcendency Of The Exponential And Logarithmic Functions

 

8. Applications Of The Derivative – Part 2

 

     8.1 Optimization
     8.2 Related Rates
     8.3 Tangent-Line Approximations
     8.4 Approximations Of Errors In Measurement
     8.5 Approximations Of Roots Of Functions – Newton's Method
    
     8.7 More Indeterminate Forms

 

9. The Integral

 

     9.1 Summation Notation And Formulas
     9.2 Areas And Riemann Sums
     9.3 The Definite Integral
     9.4 The Fundamental Theorem Of Calculus

 

10. Techniques Of Integration

    
10.1 Integration By Inspection
    
10.2 The Method Of Substitution
    
10.3 Integration Of Trigonometric Functions
    
10.4 Integration Of Powers Of Trigonometric Functions
    
10.5 The Inverse Trigonometric Substitution
    
10.6 Other Substitutions
    
10.7 The Method Of Partial Fractions
    
10.8 The Method Of Integration By Parts

 

11. More On The Integral


    
11.1 Approximate Numerical Integration
SAS Women Free Time Mocha Rockport Women s Total Motion Cherene Boot Svart Kid Suede , Ellie Sko Kvinner Go-Go Boot Svart , ED Ellen DeGeneres Kvinner Zoila Black Coastal Calf / Tumbled Nubuck Cole Haan Kvinners Leah Skimmer Ballett Flat Svart lær JSport av Jambu Women s Regatta Oxford Flat Lysegrå / Stenblå , Merrell Women s Moab 2 Vent Vandringssko Røyk Merrell Women s Moab 2 GTX Vandringssko Sedona Sage , Crocs Women s Kadee Slingback Flat Svart Marc Fisher LTD Kvinner Zala 5 Blått stoff Marc Jacobs Kvinner Cleo Studded Ballerina Ballett Flat Svart Skechers BOBS Women s Plush Lite-Be Cool Flat Grå / Multi , Taos Fottøy Kvinner Marvey Leather Slip On Svart lær Softwalk Women s Meredith Clog Floral Medallion Leather OLUKAI Nohea Mesh Shoe - Menns Rock / Kano , FRYE Women s Demi Zip Bootie Boot Forest Soft Oiled Suede Franco Sarto Kvinners Richland Ankle Boot whisky , Birkenstock Women s Gizeh Birko-Flor (Tm) onyks , Naot Women s Vistoso Boot Vintage Grey Leather , Aerosoles Women s Effortless Boot Taupe Fabric Kmeioo Mules for Kvinner, Puff Pompom-Embellished Slip On Loafers Backless Pointed Tå Satin Mule lysbilder Blue-satin Ariat Women s Sherbourne H2O Country Fashion Boot Seal Brown SOREL Women s Glacy Elk 6 Tommers Stiletto Black Stetch Platform Lår Høy Støvler Dominatrix - Anne Klein Kvinner Langstyn Suede Snow Boot Mørk naturlig Columbia Women s Powder Summit Shorty Vanntette Støvler Isolerte Sko , Sam Edelman Women s Yaro Heeled Sandal Mykt naken lær FSJ Kvinner Sexy High Heel Stiletto Sandaler Ankel Rem Slingback Open Toe Aftesko Størrelse 4-15 US Svarte prikker , Blå av Betsey Johnson Women s Sb-Briel Dress Sandal Sølv Satin , Vionic Women s Tide Ii Gullkork
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Wolky Comfort Lace Lace Up Up Sko Gobly Suede Metal Sofa Suede 03669e2 - awesome-b12.site

 

12. Applications Of The Integral

 

     12.1   The Mean-Value Theorem For Integrals
     12.2   Areas Of Plane Regions
     12.3   Finding Volumes By Slicing
     12.4   Finding Volumes By Using Cylindrical Shells
     12.5   Distance And Displacement
     12.6   Arc Length
     12.7   Areas Of Surfaces Of Revolution
     12.8   Work
     12.9   Force Exerted By A Fluid
     12.10 Net Change

13. Plane Curves

 

     13.1 Parametric Curves
          13.1.1 Parametric Curves
          13.1.2 Tangent And Sketching Of Parametric Curves
          13.1.3 Arc Length And Area Of Surface Of Revolution Of Parametric Curves
Wolky Comfort Lace Lace Up Up Sko Gobly Suede Metal Sofa Suede 03669e2           13.1.4 Vector Study Of Motion In The Plane

     13.2 The Polar Coordinate System
          13.2.1 The Polar Coordinate System
          13.2.2 Sketching Polar Curves
          13.2.3 Area By Polar Curves
          13.2.4 Arc Length And Area Of Surface Of Revolution Of Polar Curves

Oxfords

 

14. Infinite Series

 

     14.1 Infinite Sequences
     14.2 Infinite Series
     14.3 The Comparison Tests
     14.4 The Root And Ratio Tests
     14.5 The Integral Test
     14.6 The Alternating-Series And Absolute-Convergence Tests
     14.7. Approximations Of Sums Of Series

 

15. Representations Of Functions By Power Series

 

     15.1 Power Series
     15.2 Derivatives And Integrals Of Power Series
     15.3 Taylor Series
     15.4 Applications Of Taylor Series
     15.5 Taylor Polynomials And Taylor Theorem
     15.6 The Binomial Series

 

16. Differential Equations

 

     16.1 First-Order Equations
          16.1.1 Introduction To Differential Equations
  16.1.2 Variables-Separable Equations
          16.1.3 First-Order Linear Equations

 

     16.2 Second-Order Linear Homogeneous Equations
          16.2.1 469396
956196
2056160
7044483
14228182
15093326
15881956
16237442
17080423
Equations With Constant Coefficients – Characteristic Equation
          16.2.2 Equations With Variable Coefficients – Reduction Of Order

     16.3 Second-Order Linear Non-Homogeneous Equations
          16.3.1 Equations With Constant Coefficients – Undetermined Coefficients
          16.3.2 Equations With Variable Coefficients – Variation Of Parameters

 

     16.4 Approximate Solutions

          16.4.1 Approximate Graphical Solutions – Direction Fields

          16.4.2 Approximate Numerical Solutions - Euler Method

 

 

Last Updated: 14 May 2015.

 

Return To Top Of Page