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

SoftWalk s Topeka Women s Topeka Flat Flat Mørkebrun 98bb4b4 - 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
SoftWalk s Topeka Women s Topeka Flat Flat Mørkebrun 98bb4b4      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

SoftWalk s Topeka Women s Topeka Flat Flat Mørkebrun 98bb4b4

 

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

 

Me Too Women s Hailey Ballet Flat Svart Kid Leather , Calvin Klein Women s Gredel Ballett Flat Svart , Sykepleier Mates Women s London Slip On Safirblå Soft Style Hush Valper Damer Darlene Flat Bright Multi Butterfly Grosgrain Skechers Cali Women s Meditation Slingback Yoga Flip-Flop Blå / Multi Nine West Women s Huebert Naturlær Marc Fisher kvinners Alen Mary Jane Flat Svart Soft Style Hush Valper Kvinners Rory Flat Naturlig Sparkle Linen , Sam Edelman Women s Gilly Flat Sandal Sal Ros Hommerson Kvinner Rebecca Cap Toe Flat Navy Leather, Patent FitFlop Kvinner F-Sporty Elastisk Mary Jane Flat Midnight Navy Patent Easy Street Women s Giddy II Flat Tan FSJ kvinner komfortable Low Heels Flats Pekte tåpumper Spenne Slip On Office Dress Sko Størrelse 4-15 US Gul Walking Cradles Women s Heidi Pump Hvitt lær , Kate Spade New York Kvinner Farrah Flat Sandal Sølv , ECCO Women s Bluma Mary Jane Flat Wild Dove / Betong Romika Women s Cassie 07 Flat burgunder Soft Style Women s Angel II Pump Bone Elegance Propet Women s Chloe Mary Jane Flat Mørkerød Alegria Women s Paloma Flat renessanse LifeStride Women s Farrow Wedge Pump Sort 2 ECCO Fottøy Kvinner Bouillon II Gladiator Kjole Sandal Varm Grå , Earth Women s Dione Shoe Alpaca , Gabor Womens 74.163 Svart Aravon Women s Bromly Ghillie Flat Sandal Svart Naturalizer Women s Marianne Svart , LifeStride Women s Aliza Flat Svart Naot Women s Laticia Flat Bagasje Brun Skinn / Kanel Skinn / Hawaiian Brown Leather Drew Shoe Women s Bloom II Bone Stretch ,

     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
SOREL Women s After Hours Booties Tobakk , U-lite Women s Perforated Lace-Up Wingtip Skinn Flat Oxfords Vintage Oxford Sko Brogues burgunder Easy Spirit AP1 Sport Walking Sko Natural-microfiber Alegria kvinners Keli profesjonell sko Hickory , Capezio Stretch Jazz Ankel Sko Tan BOBS fra Skechers Women s Pureflex Supastar Flat marinen Fransk Sole Women s Zeppa Slip On Shoes Black Wave , MIA Women s Abba Clog-Inspired Sandal Bagasje , Myk stil av Hush valper Kvinners pleats være med deg kle pumpe Sjøfisk Kid / Patent , NAOT kvinners Rongo Mary Jane Flat Quartz Leather / Dusty Silver Leather , Nine West Women s Schull Suede Ballett Flat Lyserød / Rosa Prop & eacute; t Women s Ridgewalker Low Boot Svart Aravon Women s Peggy-AR Boot Svart , SOREL Women s Snow Angel Lace Boot Svart Birkenstock Thongs Sølv , Apex Women s A200WM095 Loafer Flat Svart Sanuk Women s Elle V. Åtte Flat burgunder , Sbicca Women s Annex Engineer Boot Lysebrun Charles David Women s Prism Ankle Boot Svart , Kvinners pytter Regn og snøboot Multi Color Mid Calf Knee High Waterproof Rainboots G-floral Propet Women s W0001 Breeze Walker Sandal Hvit korn , Modemoven Women s Round Toe Lase-Up Ankel Støvletter Ladies Leather Combat Booties Fashion Martens Boots khaki Fergalicious Women s Tootsie Knee High Boot Doe , Bogs Women s Classic Butterflies Snow Boot Navy Multi Ivanka Trump Women s Sarena Boot Mørk grå Sam Edelman Women s Bronte Combat Boots Svart / Mørk Multi Naturalizer Women s Irena Heeled Sandal Svart , Walking Cradles Women s Sky 3 Wedge Sandal Hvete Nappa Pleaser Women s Delight-601 Platform Slide Klar / Sølv Chrome Sanuk Women s Yoga Sling 2 Print Vintage The Ranch Aqua ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

SoftWalk s Topeka Women s Topeka Flat Flat Mørkebrun 98bb4b4 - 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
SoftWalk s Topeka Women s Topeka Flat Flat Mørkebrun 98bb4b4           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 777284
7057244
8319330
8874462
13049901
14317301
14789623
16239617
16631974
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