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

Dansko s Women s Professional Mule Mule Cordovan trekke B071L8J8F6 opp 296670e - 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
Dansko s Women s Professional Mule Mule Cordovan trekke B071L8J8F6 opp 296670e      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

Dansko s Women s Professional Mule Mule Cordovan trekke B071L8J8F6 opp 296670e

 

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

 

Drew Women s Milan II Slide Slate Blue Leather XYD Kvinner Åpne Tå Strappy Low Block Heel Sandal Pumps Ankel Strap Wedding Dress Shoes Svart Sanita Women s Textured Oil, Pro Mule Oljert lær Antikbrunt , Softwalk Women s Murietta Clog Svart , Drew Amora Women s Oxford Black-nubuck Alegria kvinners Keli profesjonell sko Birdland Alegria Women s Dayna Professional Shoe sprinkles Haflinger unisex-voksen GZ Classic Grizzly Clog marinen Sanita Women s Professional Patent Clog Purple Shimmer OLUKAI Nohea Mesh Shoe - Menns brun Nedal Kvinner Casual Velvet Mule Slip On Loafers Gold-blomst Vionic Womens oppreist Rory Ankle Boot Mørkebrun FRYE Women s Flynn Mules Cognac Tumbled Buffalo , Sam Edelman Women s Lewellyn Mule Moss Green Suede , SoftWalk Women s Mason Clog Svart , Klogs USA Unisex Boca Mule Drue FSJ Damemote Mid High Heels Sandaler Suede Slip On Open Toe Mules Sexy Slide Shoes Størrelse 4-15 US Rosa-10 cm YDN Women Comfy Kitten Low Heel Mules Slip på Clog Sandals Open Toe Dress Pumps Slide Shoes Blå Suede-10cm , Birkenstock kvinner fra Polyuretan Synthetic-Clogs Hvit Vionic Womens oppreist Rory Ankle Boot taupe , Kensie Women s Helina Mule taupe , Spring Step Women s Chino Grå Dansko Women s Sonja Patent Leather Clog Svart Sam Edelman Women s Navya Mule Sølv / Metallisk Skinn , Birkenstock Unisex Boston Soft Footbed Leather Clog Taupe Suede , Birkenstock Unisex Boston Soft Footbed Leather Clog Taupe Suede , FSJ Kvinner Low Chunky Block Heel Mules Åpne Toe Single Strap Sandals Summer Dress Shoes Størrelse 4-15 US Svart Easy Spirit Women s Traveltim278 Mule Svart / Sort Multi Suede Klogs USA Women s Naples Mule Svart Glatt

     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
Vionic Women s Walker Classic Sko Svart SOREL Women s Major Low Leather Boot Svart / vannkoker , Skechers Women s Cleo Sincere Wide Ballet Flat Svart Mordenmiss Women s Summer Håndlaget Ekte Lær Sandal Stil 1-mørk grå , LifeStride Kvinner Emilia Flat Sandal tinn Gentle Souls av Kenneth Cole Women s Noa Flat Svart OLUKAI Nohea Mesh Shoe - Menns Mustang / Lime Peel , FSJ Kvinner Mote Faux Suede Mules Sandaler Pebbende Toe Casual Slip On Shoes Low Heels Størrelse 4-15 US Svart Matte , Ikke vurdert Kvinner Glitzy Trio Harness Boot Svart , XYD Elegant High Heel Stiletto Ankel Støvler Faux Suede Peep Toe Pumps Booties For Women rød Vionic Women s Snug Adeline Mule marinen , SoftWalk Women s Rocklin Chelsea Boot konjakk Seychelles Women s So Blue Ankle Boot Hvit Aerosoles Kvinners Plakat Jente Ballett Flat Blå Snake , Bernie Mev Kvinners koselige Mary Jane Flat Flerfarget , Reef Women s Rose LX Mary Jane Flat Svart hvit Rieker Melgar Womens Casual Shoes 32 Rose Cotswold Sandringham Buckle-Up Womens Wellington Støvler Purple Trotters Women s Sizzle Signature Ballett Flat Bordeaux Vasket Metallic Microfiber Suede FRYE Women s Paige Tall-Apu Riding Boot Slate Journee Collection Women s Over-the-knee Faux Suede Boots Svart Rocket Dog Women s Slope Calf Fur Vinter Støvler kastanje UGG Women s Suvi Snow Boot Svart Pleaser Women s SED3000 / S / PU Boot Svart , InterestPrint Dames Snow Boots Unikdesignede Comfort Winter Boots Multi 25 Qupid Women s Sochi-127 Ankel Boot Mørk Taupe , Littleboutique Utsmykket Cutout High Heel Bootie Rhinestone Studded Sandal Heels Dress Sandal Black2 Bella Vita Women s Seraphina II Espadrille Wedge Sandal Pastellblomster Merrell Women s Lidia Thong Sandal Mahogany Calvin Klein Kvinners Vable Kjole Sandal Mørk Sølv ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Dansko s Women s Professional Mule Mule Cordovan trekke B071L8J8F6 opp 296670e - 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
Dansko s Women s Professional Mule Mule Cordovan trekke B071L8J8F6 opp 296670e           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

Mules & Clogs

 

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 3785058
4639652
6231564
8369682
12048911
12478250
12673930
14360315
17224832
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