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

FitFlop Women Gitter s Leather Gitter 19993 Surfa Floral Flip-Flop FitFlop Svart 757099f - 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
FitFlop Women Gitter s Leather Gitter 19993 Surfa Floral Flip-Flop FitFlop Svart 757099f      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

FitFlop Women Gitter s Leather Gitter 19993 Surfa Floral Flip-Flop FitFlop Svart 757099f

 

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

 

FRYE Women s Ally Western Flip-Flop Hvit Sanuk Women s Yoga Chakra Flip-Flop Svart Mephisto Kvinner Helen Thong Sandaler Copacabana / Camel UGG Women s Maddie Flip Flop Marino Birkenstock Gizeh Unisex Lær Sandaler Tobakk brun lær , Vince Camuto Women s Chania Flip-Flop Sunset Orange FitFlop Kvinners Lulu Thong Sandal Urban White CLARKS Women s Wave Coast Flip Flop Svart Sam Edelman Women s Gracie Flip-Flop Ny naken leopard , Splendid Women s Madrid Flip Flop Lysegrå , Yellow Box Women s Gateway Wedge Flip Flop Svart tinn Sam Edelman Women s Gracie Flip-Flop Candy Red Vionic Women s Tide Ii Bronse Metallisk , Sanuk Women s Yoga Sling 2 Print Vintage Rapture Rose Ditsy Floral , Vionic Women s Tide Ii Lyse blå , Mephisto Kvinner Helen Thong Sandaler rød , Birkenstock Unisex Gizeh Fringe Grasiøs karamell Birko-flor ECCO Women s Intrinsic Toffel Thong Sandal Hvit Vionic med Orthaheel Tide Sequins Gullkork , FitFlop Kvinners Cha Cha Sandal Bronse , NIKE Women s Ultra Celso Thong Flip-Flop Svart hvit , Sanuk Women s Yoga Sling 2 Sp17 Aqua Havaianas Women s Slim Flip Flop Grønn te Vionic Women s Tide Ii Svart NIKE Women s Solarsoft Thong 2 Athletic Sandal Hvit Yellow Box Women s Jello Sandal Hvit , Havaianas Women s Slim Flip Flop petunia Sanuk Women s Yoga Mat Wander Flip-Flop brun , FitFlop Kvinners Lulu Shimmer Print Thong Sandal Gold Shimmer Print ,

     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
Wolky Jacquerie Rød Dessin Suede W / Black Collar , Native Unisex Apollo Moc Mote Sneaker. Snapper Rød / Cabo Blå / Shell Hvit Dr. Scholl s Shoes Women s Knoll Boot Svart mikrofiber , Mona Flying Women s Lær perforert Lace-up Oxfords Sko til kvinner Wingtip Multicolor Brougue Shoes brun Tommy Hilfiger Women s Frank Sneaker Navy / Navy The North Face Women s Storm Low Top Fabric Vandringssko Sepia Brown / Spicy Orange , Discovery Expedition Women s Boot Gun Metal , The North Face Nuptse Purna Shorty Boot Women s Desert Palm Brown / Storm Blue Blå av Betsey Johnson Women s Sb-Willa Pointed Toe Flat Elfenben Satin , Jord Origins Tova Svart , Drew Shoe Women s Rose Mary Jane marinen Birkenstock Unisex Boston Soft Footbed Leather Clog Brun Amalfi Leather Eastland Women s Newbury Penny Loafer brun , Corso Como Women s Retreat Flat Grå Nubuck Caterpillar Women s Charade Pull Bootoe med V Shape Cutout Ankel Boot Puritan , Drew Shoe Women s Joy Olive Combo Alegria Women s Ella Mary Jane Pump Kjæreste , Crocs Sarah Mule Clog Svart / Black / Black Easy Street Women s Columbus Mule Tan Qupid Women s Olee-32 Boot Light Taupe Sperry Women s Saltwater Boot Tan / Navy Blondo Kvinner Victoria Vanntett Regnsko Sopp suede , Womens Hunter Original Justerbar Back Gloss Wellington Winter Rain Boot marinen , Baffin Women s Shannon Snow Boot Kull Jessica Simpson Women s Grandie Winter Boot Svart BareTraps Women s Yoshie Snow Boot taupe , Roma Boots Women s EMMA Mid Rain Boots Blågrønn Kinesisk Klesvask Kvinners Savana Heeled Sandal Basil Leather , Jessica Simpson Kvinners Kariss Kjole Sandal Svart , Cole Haan Women s Adderly Wedge Sandal Naken ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

FitFlop Women Gitter s Leather Gitter 19993 Surfa Floral Flip-Flop FitFlop Svart 757099f - 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
FitFlop Women Gitter s Leather Gitter 19993 Surfa Floral Flip-Flop FitFlop Svart 757099f           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

Flip-Flops

 

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 341754
1087939
2298725
3251552
4809316
12190410
15939360
16220528
17422386
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