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

Trotters Women 11128 s Women Jake Flat/ Svart/ Svart 2ff70f8 - 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
Trotters Women 11128 s Women Jake Flat/ Svart/ Svart 2ff70f8      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

Trotters Women 11128 s Women Jake Flat/ Svart/ Svart 2ff70f8

 

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

 

Bella Vita Women s Briar II Flat Stenkrokodille Skechers Sport Kvinner EZ Flex Flighty Majesty Slip-on Flat Svart , Peerage Marina Kvinner bred bredde Round Toe Casual Dress Skimmer Flats (Størrelse og måling Diagram tilgjengelig) Beige JA VIE Walking Sko for kvinner Ballet Flats stil for hver dag, bruk kjøring Sand Stripe , Drew Shoe Women s Bloom II Svart Nubuck , Walking Cradles Women s Heidi Pump Leopard Patent Lær , Jessica Simpson Kvinner Mandalaye Ballett Flat Kaffe Uregelmessig Choice Nick Of Time Womens Shoes rød CLARKS Women s Medora Nina Flat Svart Nubuck , ECCO Damas Damara Crisscross Gladiator Sandal Rose Dust / Coral , Dr. Comfort Lu Lu Kvinners terapeutisk diabetisk ekstra dybde skinnlær velcro kastanje , CLARKS Women s Medora Nina Flat Svart lær Kaitlyn Pan Pusset Toe Studded Ballerina Leather Flats Pastell Rosa Matte / Naken Trim / Gullpinner Dr. Scholls kvinnelige vennlig ballettflate Svart patent , Naturalizer Women s Neina Huarache Sandal Gingersnap Meg for kvinners Halle Ballet Flat Drivved Nappa , JBU av Jambu Women s Blossom Vegan Mary Jane Flat Sement , Alegria Women s Paloma Flat hieroglyph Easy Street Women s Garrett Flat Sandal marinen , Melissa Women s Puzzle Peep Toe Flats Klar , CLARKS Women s Gracelin Lea Ballett Flat Rødt lær Bella Vita Women s Switch II Flat Black Snake Aravon Women s Francesca Flat Svart Alegria Women s Paloma Flat Yeehaw , Earth Shoes Allegro Svart , Lucky Brand Women s Emmie Ballett Flat Sesam Leopard Arcopedico Women s Vegas Svart , CLARKS Kvinners Keesha Rosa Kjolepumpe Gold / Metallic Sesto Meucci Kvinner Evonne Flat Platino Zingato

     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
Dr. Comfort Vigor Women s Therapeutic Diabetic Extra Depth Vandring Boot Leather Lace Svart , Josef Seibel Women s Ida Flat Svart , Pleaser Demonia Av Kvinners Dank-101 Lace Up Sort polyuretan CLARKS Kvinner Jocolin Vista Båtsko Sort perforert tekstil Salewa Women s Wildfire S GTX Teknisk Tilnærming Sko Cypress / River Blue SAS Women Free Time Bein , Alegria Women s Paloma Flat Jazzy Black OLUKAI Ohana Sandaler - Menn Husk / Clay Alegria Women s Paloma Flat Hibiscus & Co , Dafna av Naot Women s Amy Rain Boot brun , Easy Spirit Women s Billian Boot Svart lær Crocs Women s Lodge Point Lace Snow Boot Svart YDN Kvinner Stacked Low Heel Mules Slide On Closed Toe Clogs Block Casual Slippers Shoes rød , Vionic Menns Orthaheel Technology White / Navy Walker Plomme Vince Camuto kvinners Talorini motorsykkel støvel Granitt Peak , SoftWalk Women s Norwich Ballett Flat tinn Nine West Womens Starship Fabric pekte tå Ballet Flats Navy Multi Fabric UGG Women s Bailey Bow Grå , FRYE Kvinner Veronica Back-Zip Boot Svart mykt vintage lær-79707 , FitFlop Women s DueBoot Biker Winter Boot Sjokoladebrun CAPE ROBBIN Kvinner Satin Lår High Stiletto Boot - Kjoler, Fest, Kostyme - GF71 Grå Woolrich Women s Doe Creek II Vinterboot Sjokolade / Kendall Creek NCAA Texas A & M Aggies Kvinner 13-tommers Gameday Boots Messing , SOREL Women s Emelie Tall Premium Vanntette Støvler, Svart 7 - , Sam Edelman Kvinner Patti Dress Sandal Klassisk Naken Lær , ECCO Women s Women s Shape 35 Block Slingback Heeled Sandal Svart Bandolino Women s Armory Heeled Sandal Sølv , Crocs Women s Leigh Wedge Sandal Havregryn / Mushroom Yellow Box Women s Sawyer Wedge Sandal Svart UGG Women s Tawney Flip-Flop Svart ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Trotters Women 11128 s Women Jake Flat/ Svart/ Svart 2ff70f8 - 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
Trotters Women 11128 s Women Jake Flat/ Svart/ Svart 2ff70f8           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

Flats

 

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 1001119
1429477
2081705
5366867
7539350
9180646
10817026
13910708
16509818
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