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

UGG Women s Svart Nolita Women Oxford Oxford Svart 6ada4b2 - 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
UGG Women s Svart Nolita Women Oxford Oxford Svart 6ada4b2      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

UGG Women s Svart Nolita Women Oxford Oxford Svart 6ada4b2

 

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

 

Marc Joseph New York Womens Bryant Park Closed Toe Loafers Ny Leopard , Drew Amora Women s Oxford Brun-nubuck Finn Comfort Women s Ikebukuro Oxford Oliven , Amalfi Orazio Women s Slip On marinen Easy Spirit Women s Solana Mote Sneakers Mørk Grå Multi Nubuck Dr. Martens Women s Clarissa Sandal Tyggegummi , L amour Des Pieds Womens Boccoo Oxford Svart , Shellys London Women s Frankie Oxford Flat Svart Blomstret ECCO Footwear Womens CS 14 Stropp Oxford Svart TOMS Kids 10009918 Alpargata-K Svart Lace Glimmer Finn Comfort Women s Ikebukuro Oxford Plata / Schwartz Mephisto Women s Tomasia Oxford Silver Magic / Python; , Caterpillar Women s Windup Leather Lace Up Fashion Oxford rød , FitFlop Women s F Pop Leather Oxford All Black Leather IVY KIRZHNER Womens Lunar Svart skog , UGG Women s Marji Snake Oxford Svart Vionic Women s Walker Classic Sko Hvit / Blå / Grå , Prop & eacute; t Women s Mabel Oxford Vin , Rieker Women s Sina 78 Pump Grå , Taos Women s Ideal Oxf Shoes Black Leather Svart FRYE 79981 Alexis Low Lace Sneaker Hvit svart , Ros Hommerson Women s Nancy Oxfords Black Croc Patent / Tinnpiping , Propet Women s Washable Walker Sneaker Bone / Wht , ECCO Fottøy Dam Damara II Tie Oxford Dusty Purple , Mephisto Women s Poppy Smoking Oxford Kamel Silencio / Hasselnøtt Mano Cole Haan Womens Zerogrand Wingtip Oxford Walking Shoe Roccia / Pavement Mordenmiss Women s Leather Short Boots Nye Sko Stil 1-svart , Skoleproblem Upper Class 7300 Saddle Shoe (Toddler / Little Kid / Big Kid) Svart hvit Rockport Women s Trustride Derby Trainer Mote Sneaker Stone Multi Leather

     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
Bernie Fru Kvinner Tidløse Svart ED Ellen DeGeneres Womens Aviana Stone / Stone Oiled Veg Leather / Bucato Suede , Cordani Womens Bree Svart lær , Dingo Women s Floral Tooled Knotted Strap Ankel Boot Round Toe - Di8951 Tan , Marc Jacobs Women s Brittany Lace Up Oxford Antikk Silver Cole Haan Women s Rodeo Tassel Driver Loafer Malbec Suede , Earth Origins Women s Reid Slip On Svart , Skechers Women s Flex Appeal 2.0 - Hourglass Ankel-High Slip-On Sko Svart hvit Skechers Cali Women s Meditation Slingback Yoga Flip-Flop Grå , LifeStride Women s Doit Flat Lux Navy Ros Hommerson Women s Eloise Lightweight Casual Flats Tinnestrek , Trotters Women s Josie Mary Jane Flat Svart , Sanita Women s Professional Patent Clog Svart , Easy Street Women s Origin Mule Svart Lamey / Svart / Gore , Soft Style Hush Valper Dame Prema Flat Sandal tinn , Fransk Sole FS / NY Kvinner Passportr Ballett Flat Grå , Skechers Sport Women s Beathe Easy Vår Song Fashion Sneaker Sølv , TOMS Womens Classics Slip On Alpargata Flat Shoe (Blonder) Sort , Trotters Women s Jenn Loafer marinen Klogs USA Women s Harley Mule Skifer Glatt , Merrell Women s Aurora 6 Ice Plus vanntett snøbrett Aske LifeStride Women s Jamie Ankle Boot Svart Sensible Soles Women s Dover Rain Boot Svart Crocs Unisex Classic Clog khaki , Aravon Women s Andrea-AR Ballett Flat Svart FRYE Dame Reina Bootie Dusty Rose Pajar Women s Grip Boot Svart / Svart Laredo Kvinner Eva Vintage Stockman Boot Square Toe Blå Merrell Women s Chateau Tall Lace Polar Vanntett Snøbrett Svart , Teva Women s Arrabelle Leather Sandal brun
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

UGG Women s Svart Nolita Women Oxford Oxford Svart 6ada4b2 - 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
UGG Women s Svart Nolita Women Oxford Oxford Svart 6ada4b2           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 170457
1406465
15136082
15289028
15532704
15919399
15995067
18088059
18832162
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