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

Uregelmessige Turkis valgkvinners Ja Du Peli Kan Kan Turkis Flat Ballerinas Ballerinas Sko Størrelse 0e4f2ff - 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
Uregelmessige Turkis valgkvinners Ja Du Peli Kan Kan Turkis Flat Ballerinas Ballerinas Sko Størrelse 0e4f2ff      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

Uregelmessige Turkis valgkvinners Ja Du Peli Kan Kan Turkis Flat Ballerinas Ballerinas Sko Størrelse 0e4f2ff

 

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

 

Munro American Women s Jerrie Shoes Lilac Nubuck , VANELi Kvinner FC-313 Flat Svart Suede / Brown Coryl Print , Gabor Women 74.161 Bronse / Svart Metallisk Sam Edelman Kvinners Rodney Ballett Flat Svart lær , TOMS menns klassiske lærreds Slip-On Universitetet Mørkegrå Eastland Women s Hazel Dress Sandal Svart Walking Cradles Kvinner Ante Flat Svart stoff , Dr. Scholls kvinnelige vennlig flat Rose Gold / Metallic Jack Rogers Women s Rebecca Ballet Flat Eik / Platinum , Merrell Womens Barrado Svart , Alegria Women s Paloma Flat Pretty Swirl Alegria Women s Paloma Flat Hasselnøtt , Trotters Women s Sizzle Flat Rosa , Anne Klein Women s Zarenna Slip Ballet Flat Silver Natural UGG Women s Royale Slide Sandal Ribbon Rød , ALDO Kvinners Wiliwiel Ballett Flat Lyserosa , Lucky Brand Women s Lucky Emmie Ballett Flat travertin Prop & eacute; t Women s Onalee Mary Jane Flat Rød, Sølv , Softwalk Women s Haddley Mary Jane Flat Svart / Mørk Grå Ønsket \ Hvit Propet Women s TravelFit Flex Ballett Flat Svart hvit , Trotters Women s Estee Ballet Flat Svart Madden Girl Kvinner Bryceee Toe Ring Sandal Rose gull , Drew Shoe Women s Orchid Hook and Loop Brun Nubuck Naturalizer Women s Eliza Flat Sandal Rosa , CLARKS Kvinners Keesha Rosa Kjolepumpe Gold / Metallic Splendid Women s Renee II Ballett Flat Leopard Aerosoles Women s Spin Off Ballett Flat Svart stoff , Sam Edelman Kvinner Felicia Ballett Flat Sal

     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
Georgia GB00164 Mid Calf Boot Mørkebrun Soft Style Hush Valper Women s Valda Oxford Black Snake , Cole Haan Kvinners Leah Skimmer Ballett Flat Svart lær , VANELi Kvinner FC-313 Flat Svart-black-patent , Kvinner Thorogood SR EVA Slip-On Classic Comfort Skinn Oxford Sko Sko Aravon Women s Farah Rødbrun , Hush Valper Kvinner Ekko Gwen Oxford Black Cracked Suede , Keds Women s Chillax Vasket Laceless Slip-On Sneaker taupe Franco Sarto Kvinner Shirley Ballett Flat Blå David Tate Maple kvinners pumpe brun , FSJ Kvinner Sommer Ankel Rem Sandaler Åpen Tå Chunky Low Heel Komfortabel Walking Sko Størrelse 4-15 US Red Suede , Klogs Women s Mission Clog Mustang Brown Gentle Souls Kvinner Brynn Mary Jane Flat marinen , Oka-B Kvinner Taylor Flat Rødme , Cobb Hill Rockport Kvinner Pandora Boot Svart , Rockport Cobb Hill samling Kvinner Cobb Hill Janna Cross Strap metallic Sanita Women s O2 Life Classic Mule Svart CLIFFS BY WHITE MOUNTAIN Cliffs KUDROW Women s Bootie Hvete , PUMA Women s Vega Mesh Ballet Flat Grå Violett-Puma Sølv , Rockport Women s Cobb Hill Janet Wedge Pump Stormy Grey Cougar Women s Dash Casual Loafer Shoe Svart David Tate Women s Dallas Sandal Bagasje , Bella Vita Kvinners Nicky II Ankel Bootie Burgund Super Suede Western Chief Women s Vanntett Trykt Tall Rain Boot Leopardess Trotters Women s Blast III Boot Svart boks , Journee Collection Women Regular og Wide Calf Faux Suede Over-The-Knie Boots Svart Bogs Women s Berkley Footprints Rain Boot Turkis / Multi , Gracosy High Top Sneaker, Women Vinter varm krok Loop Snø Sko Fur Fôr Casual Støvler Ankel Bootie rød , SCARPA Origin Climbing Shoe Iron Grey Propet Women s Aurora Wedge Sandal Svart ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Uregelmessige Turkis valgkvinners Ja Du Peli Kan Kan Turkis Flat Ballerinas Ballerinas Sko Størrelse 0e4f2ff - 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
Uregelmessige Turkis valgkvinners Ja Du Peli Kan Kan Turkis Flat Ballerinas Ballerinas Sko Størrelse 0e4f2ff           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 2185560
6809115
7528026
9209434
10076174
11260623
13498521
15332121
17204767
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