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

Aerosoles Women Women s Suede Fable Ballett 19153 Flat Navy Suede b290642 - 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
Aerosoles Women Women s Suede Fable Ballett 19153 Flat Navy Suede b290642      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

Aerosoles Women Women s Suede Fable Ballett 19153 Flat Navy Suede b290642

 

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

 

CLARKS Women s Cheyn Web Mary Jane Flat Black Tumbled , NAOT Kvinners LUGA Lær, Suede, Kork, Latex, Mary Janes Grå Iguana, Tinn Grå, Sterling Splendid Women s Brett Sandal Naken Naturalizer Women s Samantha Pointed Toe Flat Svart lær , Jack Rogers Women s Hamptons Wide Dress Sandal Gull Marc Jacobs Women s Heart Ballerina Ballett Flat Mørk Lilla / Multi Trotters Women s Caroline Ballet Flat Mørkbrun Cognac Naturalizer Women s Brittany Ballet Flat Lady Blue Suede , Sirkus av Sam Edelman Kvinners Cayenne Ballett Flat Cranberry Multi Faded Houndstooth , CLARKS Women s Candra Light Flat Dusty Pink Patent , YDN Women Classic Pointy Toe Flats Suede Casual Shoes Low Cut Slip On Soft Purple , Bernie Fru Kvinner, Casual TWO09 Slip On Sko Taupe taupe CLARKS Kvinner Leisa Claytin Flat Sandal Hvitt lær Bella Vita Women s Briar II Flat Svart velvet Alegria Women s Paloma Flat Gothic Steel Skechers BOBS fra Women s Chill Luxe Flat Natural Flex Aubrey | Svart Suede , Badgley Mischka Kvinner Morgen II Mary Jane Flat Sølv Drew Shoe Women s Joy Hvitt lær / Mesh Miz Mooz Kvinners Carissa Kjole Pumpe whisky Sperry Top-Sider Women s Seabrook Surf Breton Stripe Flat Sandal Tan , Gabor Epworth Women s Modern Ballerina Shoes River Flower Stamp Crocs Unisex Classic Clog marinen Jambu Women s Cherry Blossom Mary Jane Flat Taupe / Peach Nectar Naturalizer Women s Bayberry Ballet Flat Bronse Aravon Women s REVsolace Flat Svart Glatt , Aerosoles Women s High Bet Ballett Flat Svart lær TOMS Classic Floral Embroidery Striped Damessko Teal Heritage Canvas Walking Cradles Kvinner Ante Flat Svart Stretch Fabric

     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
Ariat Mens Spitfire Rugged West Bar Room Brown , Propet Women s Travelwalker Lo Pro Sko Hvit , TOMS Kids 10009918 Alpargata-K Imperial Blue Dot Chambray / Bow Under Armour Women s Post Canyon Mid Vanntett Apollo Grey (962) / Overcast Grey , CLARKS Women s Everlay Kennon Mary Jane Flat Brunt lær Vionic Womens oppreist Rory Ankle Boot taupe , JOE N JOYCE Rio SynSoft Soft-Footbed Sandales Hammertone Bordeaux JOE N JOYCE Paris SynSoft Soft-Footbed Sandaler Floriblue Klogs Fottøy Kvinner Portland Clog Turf Apache , FSJ Women Stilig Peep Toe Slouch Ankel Støvler Chunky Stacked Heels Party Evening Shoes Størrelse 4-15 US brun , Birkenstock Unisex Boston Soft Footbed Leather Clog Mocha Suede OLUKAI Nohea Mesh Shoe - Menns Black Shearling Suede Sanita Women s Original Olivia Clog Svart , SOREL Women s Lea Wedge Booties Elk , Lucky Brand Women s Belia Ankle Bootie cashew BEARPAW Women s Bethany Snow Boot Svart / Grå ECCO Women s Jab Toggle Sandal Hvit Blomstertrykk / Hvit / Pulver , Hunters Boots Women s Original Raffinerte Gloss Boots Scarlett Calvin Klein Women s Enith Flat Crimson Red , Durango Kvinners DRD0192 Western Boot Brun / Royal Blue , CORRAL Kvinners hvite glitter og krystaller Cowgirl Boot Square Toe - A3397 Hvit Pleaser Kvinner Fab2023 / b Boot Svart Str Patent , RF FASHION Emma Women s Shearling Fur Fôret Med Quilt Mønster Winter Eskimo Støvler Svart Su Muck Boot Women s Reign Supreme Snow Brun / Bison Easy Street Women s Fantasia Heeled Sandal Hvit LifeStride Women s Mexico Wedge Sandal Bein Spring Step L artiste av Women s Flourish GLADIATOR Sandal Naturlig , Naturalizer Women s Caitlyn Dress Sandal Bronse Fly London Women s Yasi682fly Wedge Pump Nikotinmousse , FitFlop Women s Crystal Flip Flop tinn
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Aerosoles Women Women s Suede Fable Ballett 19153 Flat Navy Suede b290642 - 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
Aerosoles Women Women s Suede Fable Ballett 19153 Flat Navy Suede b290642           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 1554296
3247081
6012912
12101401
12932572
14230498
14273487
14747207
18677533
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