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

CLARKS Women s Women Amberlee Crest Oxford Crest 12561 marinen a815bf3 - 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
CLARKS Women s Women Amberlee Crest Oxford Crest 12561 marinen a815bf3      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

CLARKS Women s Women Amberlee Crest Oxford Crest 12561 marinen a815bf3

 

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

 

CLARKS Women s Funny Dream Oxford Svart lær BC Fottøy Kvinner Sidekick Båtsko marinen Arcopedico Women s Francesca Oxford Grå , Nine Seven Ekte Lær Kvinner Square Toe Chunky Heel Casual Work Håndlaget Business Dress Pumps Svart Dr. Martens Cavendish Temperley Leather Mono Sneaker Svart , CLARKS Women s Cheyn Ava Oxford Navy Leather ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Flats Studded Sneakers Sko Svart patent Skoleproblem Upper Class 7300 Saddle Shoe (Toddler / Little Kid / Big Kid) Svart ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Flats Studded Sneakers Sko Hvit , Mephisto kvinners laser walking sko Indigo Bucksoft / Blue Cobb Hill Women s Abilene Boot Mandel , Adrianna Papell Women s Pam Slip-on Loafer Naturlig Rocket Dog Women s Destin Orchard Hanger Bomull Oxford Oliven PIC / PAY Hazel | Red Suede , Jambu Women s Tangerine Oxford Flat taupe Mephisto Women s Valentina Oxford Grå Bucksoft , Kenneth Cole New York Kvinner Annie Menswear Styled Oxford Flat Blå , Allrounder av Mephisto Women s Nigata TEX Winter Boot Sort Gummi / Mid Rød Suede , Shellys London Women s Tommy Oxford burgunder Geox Women s Kookean 3 Oxford marinen Prop & eacute; t Kvinner Tour Walker Strap Oxford taupe Massimo Matteo Womens Oxford Wing Tips Bordo Onitsuka Tiger Mexico 66 Klassisk Blå / Hvit , Rockport Menns Symon Wingtip Dress Oxford konjakk Aravon Women s Revsmart Oxford Svart Propet Women s Washable Walker Sneaker Sr Indigo , MICHAEL Michael Kors Kvinners Sunny Lace-up Svart , Native Unisex Apollo Moc Mote Sneaker. Dublin Grå / Dublin Grå , TOMS Women s Del Rey Sneaker Sort / Hvit Plaid ,

     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
Hush Valper Damer Abney Chrissie Lo Gladiator Sandal Lett Taupe Leather , Drew Shoe Women s Joy Brown Combo , Verocara Women s Flat Heel Round Tå med bue-slips ekte lær Casual Balleria Pumpe til bruden Parti marinen , Mishmi Takin Kameng Mid Event Waterproof Fottøy Grå Walking Cradles Kvinner Hartford Flat Sandal Svart / Svart Steve Madden Women s Henly Toe Ring Sandal Svart lær Finn Comfort Women s Orb Svart Nappa , KEEN Women s Elsa Vanntett Vinter Boot Svart ull Kate Spade New York Women s Monteclair Mule Flerfarget , SoftWalk Women s Rancho Boot Red Distress Sam Edelman Kvinner Felicia Ballett Flat Hot Coral Silk Dupioni Marc Jacobs Women s Aurora Mule Rosa Bogs Womens Auburn Slip-On taupe , SOREL Women s Danica Short Booties Quarry / Black , SoftWalk Women s Napa Mj Mary Jane Flat Tan / Oppbevaring , Walking Cradles Women s Logan Wedge Sandal Tobakk Taos Fottøy Kvinner Star Fashion Sneaker Chocolate Wash Canvas , Jordoliven Vintage lær / stein , Rockport Kvinders Total Motion Kalila Perf Shootie Fashion Boot Svart Nubuck , Hush Valper Women s Melodi Langdon Boot Navy Suede , Tretorn Women s Storm Rain Boot Grønn Kamik Women s Jennifer Rain Boot burgunder , FRYE Women s 12R Harness Boot Cognac Vasket Oiled Vintage-77298 , Nine Seven Genuine Leather Women s Pointed Toe Chunky Heel Ankel Rem Handmade Ankle Boots Beige , Madden Girl Women s Jagg Fashion Boot Svart stoff Eileen Fisher Women s Sport Sandal Fløtekaramell / Cream Nubuck , JBU av Jambu Women s Sugar Encore Platform Sandal marinen , CLARKS Kvinner Lexi Marigold Q Tinnlær Therafit Claire kvinners lærgladiator justerbar sandal - for plantarfasciitt / fotpine Hvit , Bernardo Women s Lilly Flat Sandal rød
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

CLARKS Women s Women Amberlee Crest Oxford Crest 12561 marinen a815bf3 - 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
CLARKS Women s Women Amberlee Crest Oxford Crest 12561 marinen a815bf3           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 1227167
2449696
3193362
4317163
6760035
8217535
11791712
15275272
16515744
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