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

Soft Style Women s Angel II II Pump Women Bone s Elegance 5dc4522 - 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
Soft Style Women s Angel II II Pump Women Bone s Elegance 5dc4522      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

Soft Style Women s Angel II II Pump Women Bone s Elegance 5dc4522

 

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

 

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 , Easy Street Women s Gnist Flat Sandal Tan , Sam Edelman Kvinners Rodney Ballett Flat Blå skygge Dupioni Ros Hommerson Kvinner Rebecca Cap Toe Flat Tinnlær , LifeStride Women s Influx Ballett Flat brun Haflinger Women s Atb Slip-On Loafer Navy Speckle , Hot Chocolate Design Chocolaticas Habana Carmine Women s Mary Jane Flat flerfarget , Aerosoles Women s Final Exam Wedge Brun eksotisk , Trotters Women s Sizzle Signature Ballett Flat Svart Easy Street Women s Zone Flat Sandal Dark Tan , Sebago Women s Plaza II Penny Loafer Cordo Leather , Dr. Comfort Women s Merry Jane Lycra Stretchable Diabetisk Mary Jane Shoes lycra CAMSSOO Kvinners Classic Rivets Pointy Toe Slip på Comfort Flats Dress Pumps Shoes Red Soft Pu , Crocs Women s Lina Suede Ballett Flat Flamme Seychellene Women s Prospect Pointed Toe Flat Gull , Naot Women s Arataki Dress Sandal Rumba Leather , Freebird Women s Echo Loafer Flat Naturlig Multi , CLARKS Women s Kinzie Light Loafer Flat Black Tweed Trotters Women s DEA Ballett Flat Svart Seychellene Women s Prospect Pointed Toe Flat rød

     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
Belleville 390 Hot Weather Boot Tan , Dr. Martens Women s 1461 W / Zip Oxford Hvit , TuffRider kvinners barokke kjole støvler Svart , Minnetonka Women s El Paso II Moccasin taupe Hi-Tec Women s Bandera II Low Waterproof Hiking Shoe Charcoal / Cool Grey / Lichen , Hi-Tec Women s Altitude Lite I Vanntett Fottøy Charcoal / Cool Grey / Lichen Easy Street Women s Garrett Flat Sandal Bein Skechers BOBS fra Women s Bobs Plush-Peace and Love Hvit Easy Spirit Women s Motion Sport Lace-Up Sneaker Tinnlær , Hush Valper Damer Chaste Ballet Flat Svart lær , FRYE Women s Melissa Chelsea Boot Redwood SoftWalk Women s Napa Ballett Flat Metallisk leopard , Finn Comfort Women s Gomera - 82562 Sandal Taupe / Bronze , American Rag Womens Desyre Closed Toe Ankel Mote Støvler Svart JBU av Jambu Women s Yasmin Gladiator Sandal Saddle / Sage Vionic Womens oppreist Sterling Ankel Boot Sal , SoftWalk Women s Tilton Slip-On Loafer Svart Lucky Brand Women s Yasamin Fashion Boot tapenade , Splendid Women s Rina Ankle Boot Bordeau , Dr. Scholls kvinners Brilliance Riding Boot whisky , Demonia Women s Scene-107 Knee High Boot Svart Vegan Leather RF-ROM FUNKSJON Kvinners snøre opp Ankel Midkalve Artic varm pels Fôrbestandig Vannbestandig Eskimo Snow Boots Ny Tan-1 , Franco Sarto Kvinners Crimson Wide Calf Over Knee Boot Cocco Ellie 511-ALLY Womens Sexy Komfortabel 5 \ Blk Kingshow Women s Globalwin Vanntette Vinterstøvler Grå Sperry Top-Sider Mens Gold Cup Lug Lace Boot Tan , David Tate Women s Rosette Evening Sandal Sølv Satin , CL av Chinese Laundry Women s Tasty Wedge Sandal Hvit / blek naken , Ivanka Trump Kvinner Dalyna Wedge Sandal Mørke blå , Sam Edelman Kvinners Gigi Thong Sandal Saddle Leather ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Soft Style Women s Angel II II Pump Women Bone s Elegance 5dc4522 - 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
Soft Style Women s Angel II II Pump Women Bone s Elegance 5dc4522           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 821323
7088491
7161122
13190896
13378226
15387120
15757286
17808760
18071691
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