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

OLUKAI OLUKAI Nohea Mesh Shoe - Shoe Menns Black Shearling B01ITVJ8U4 Suede fd49839 - 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
OLUKAI OLUKAI Nohea Mesh Shoe - Shoe Menns Black Shearling B01ITVJ8U4 Suede fd49839      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

OLUKAI OLUKAI Nohea Mesh Shoe - Shoe Menns Black Shearling B01ITVJ8U4 Suede fd49839

 

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

 

Birkenstock Unisex Boston Soft Footbed Leather Clog Habana Oljert lær Easy Street Women s Sportster Clog Brun Croco CLARKS Women s Wave.Run Svart lær Easy Spirit Women s Travelport26 Mule White-lilla-multi Skechers for Work Women s Slip Resistant Clog Svart patent OLUKAI Nohea Mesh Shoe - Menns Mokka-mocha Suede , Alegira Alegira Kvinner Freesia Skinn Clog Limited Edition Jet Luster Alegria Women s Carina Wedge Sandal Bronzemosaic , SoftWalk Women s Abby Clog Svart , OLUKAI Nohea Mesh Shoe - Menns Naturlig Snake Klogs Klogs Klogs Sea Whims Patent Sanita Morse Oiled Justerbar Fleksibel Danske Tøfler (Art: 1200038) Antikkbrun , OLUKAI Nohea Mesh Shoe - Menns Rock / Kano , Easy Spirit Women s Dolly Skreddersydd Casual Sko tinn Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Suede FSJ Women Basic Open Toe Sandaler Med Block Heels Faux Suede Ankel Rem Pumps Size 4-15 US rød OLUKAI Nohea Mesh Shoe - Menns Naturlig Snake YDN Kvinner Lace Up Suede Flat Mules Low Heel Peep Toe Tøfler Komfortable Walking Shoes Mørk grå Dansko Professional Clog Faded Stripe Patent OLUKAI Nohea Mesh Shoe - Menns Mørk Java / Rojo XYD Kvinner Åpne Tå Strappy Low Block Heel Sandal Pumps Ankel Strap Wedding Dress Shoes Purple Calzuro Autoclavable Clog med øvre ventilasjon Blågrønn , Klogs USA Unisex Boca Mule Hvit , Alegria kvinners Keli profesjonell sko Swirly Godhet FRYE Women s Gwen Perf Slide Mule taupe Walking Cradles Women s Caden Clog Svart verktøyet lær , FSJ Kvinner Komfort Lav Hæl Mules Peep Toe Slide Sandaler Slip On Dress Pumps Sko Størrelse 4-15 US Royal Blue-6cm XYD Women Low Heel D Orsay Pumps Peep Toe Slip On Dress Sandaler Sko med Bowknot Svart , Calaier Womens Cafresh Closed Toe 4CM Block Heel Slip-on Mule Sko Svart

     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
Nike Golf Women s Tech Essentials Classic Long Short Lysegrå / Seillys Bone-sunset Tint Skechers BOBS fra Women s Chill Luxe Flat Natural Flex Propet Dame W3910 Vista Walker Comfort Sko Svart Glatt , Adrianna Papell Kvinner Trala Mary Jane Flat Svart Attalie Lace , Bos. Hvete / Beige Nubuck / Kokerull , Boreal Climbing Boots Women Lightweight Super Latok Orange Black 47404 Oransje Svart Beast Fashion Carter-05 Women D Orsay Slip On Pointy Cap Toe Extreme Cut Out Ankel Flat Bootie Grey Taryn Rose Women s Rosalyn Ballet Flat Black Sheep Nappa , VANELi Kvinner FC-313 Flat marinen , Softwalk Women s Meredith Clog Svart olje , Easy Street Women s Origin Mule Svart Glatt SoftWalk Women s Mason Clog Svart , Trotters Women s Liz Ballett Flat Svart , Calzuro Autoclavable Clog uten øvre ventilasjon Lyse blå Født Kvinner - Toby Duo Svart Romika Kvinner Mokassetta 279 Mule Svart Corkys Elite Women s Heidi Shoe, Amber , ECCO Damas Damara Crisscross Gladiator Sandal Svart / Svart Aigle Womens Aiglentine Synthetic Boots Navy Marine , Rampage Kvinners Ivelia Mote Knee High Casual Riding Boot (Tilgjengelig I Bred Kalv) Sort / Cognac Merrell Women s City Leaf Tall Snow Boot Merrell Oak Eileen Fisher Women s High Knot High Boot Tan Columbia Women s Minx Mid II Omni-Heat Wool Snow Boot Svart / brent Henna , Katy Perry Women s Lizette Heeled Sandal Ivory / Blush Sam Edelman Women s Yaria Heeled Sandal Inky Navy Kid Suede Leather , Spring Step L artiste av Women s Flourish GLADIATOR Sandal Svart , Sanita Women s Yara Square Flex Platform Sandal Grå , Dansko Women s Jacinda Taupe Snake Wedge Sandal Taupe Snake , Alegria Women s debra Slip-On urteaktig Sanuk Women s Yoga Sling 2 Print Vintage Tan / Black Geo Stripes ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

OLUKAI OLUKAI Nohea Mesh Shoe - Shoe Menns Black Shearling B01ITVJ8U4 Suede fd49839 - 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
OLUKAI OLUKAI Nohea Mesh Shoe - Shoe Menns Black Shearling B01ITVJ8U4 Suede fd49839           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 204872
834514
2338584
6498944
6935624
7040804
15020323
16833466
17498039
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