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

Blowfish Women s/ B0776BKQHB Granola Fisherman Sandal Sandal Birk/ Naken/ Grå Dyecut be8be1a - 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
Blowfish Women s/ B0776BKQHB Granola Fisherman Sandal Sandal Birk/ Naken/ Grå Dyecut be8be1a      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

Blowfish Women s/ B0776BKQHB Granola Fisherman Sandal Sandal Birk/ Naken/ Grå Dyecut be8be1a

 

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

 

Blowfish Women s Granola Fisherman Sandal Sage Dyecut Pu Franco Sarto Kvinner Alex Ballett Flat Svart , SoftWalk Women s Adora Flat Svart LifeStride Women s Cora Flat Sandal Svart , Easy Spirit Women s Gessica Ballet Flat Grå , Trotters Women s Sizzle Signature Ballett Flat Mørkbrun Suded , Franco Sarto Women s Rye Flat Sandal Sølv Finn Comfort Women s Gomera - 82562 Sandal multi , TOMS Classic Floral Embroidery Striped Damessko Rose Gold / Natural Canvas Dots TOMS Classic Floral Embroidery Striped Damessko Multi hekle Alegria Women s Paloma Flat midt i blinken , Minnetonka Silvie Kvinner N / S Open-Toe Lerret Slingback Sandal Multi-korall TOMS Classic Floral Embroidery Striped Damessko Teal Heritage Canvas Earth Topaz Svart Jack Rogers Women s Hamptons Sandal Sølv J.Renee Women s Valeria Ballet Flat rød , JBU av Jambu Women s Wildflower Encore Mary Jane Flat Kull Verocara Women s Flat Heel Round Tå med bue-slips ekte lær Casual Balleria Pumpe til bruden Parti marinen , Minnetonka Silvie Kvinner N / S Open-Toe Lerret Slingback Sandal Multi-korall Propet Women s TravelFit Flex Ballett Flat Navy / Hvit , Propet Women s TravelFit Flex Ballett Flat Taupe / Grå Alegria Women s Paloma Flat Hibiscus & Co Calvin Klein Women s Ghita Ballet Flat Sheer Satin Easy Street Women s Laurel Flat Navy Leather / Suede Leather , BeautiFeel Kvinner Jolie Ballett Flat Champ Sh Sc S , Katy Perry Women s Shannon Ballet Flat Baby Rosa Corso Como Women s Corrinee Ballett Flat Naken , Dr. Scholls Kvinner Frankie Ballet Flat Svart Savory Tory Burch Women s Miller Svart Blomstret

     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
Shellys London Women s Tommy Oxford Naken , Ariat Menns arv Stockman Western Boot, Native Muskat, 7 D US Svart , FRYE Women s Natalie Short Lug Engineer Boot Cognac fullkornsbørste av Palladium menns Pampa Hi Originale Chukka Boot Butternut / Tarmac Vasque Women s Talus Vanntett Vandringssko Bungee Cord / Purple Plumeria Forsake Range Low & ndash; Sort / Oliven , Ahnu Women s Sugarpine Hiking Boot Mørk grå , ECCO Fottøy Kvinner Babett Lace Oxford Svart Mephisto Mens Cooper Lysegrå Suede Corso Como Women s Corrinee Ballett Flat Naken Soludos Kvinner Bees Beaded Mule Svart Easy Street Campus Women s Slip On Svart Sugar Women s Tikki Ankle Boot Svart Bella Vita Kvinner Everest Ii Ankle Bootie Black Croco , Penny Loves Kenny Women s Knot Fl Ballett Flat Svart blonder , Easy Spirit Hartwell Women s Sandal Black Snake CLARKS Women s Medora Nina Flat Svart lær , Arcopedico Kvinner ES Slip On Loafers Shoes Plomme , Easy Street Womens Conca Fabric Open Toe Casual Slide Sandaler Naturlig Aquatalia Women s Brann Perforated Suede Ankel Boot Sand Capezio Women s 700 T-Strap Character Shoe Svart Badgley Mischka Women s Davis Pointed Toe Flat Mørk rosa FRYE Kvinner Samantha Zip Tall Motorcycle Boot Forest Soft Oiled Suede SOREL Women s Whitney Lace Vanntett Vinter Boot Quarry , ECCO Women s Trace Tie Snow Boot Bjørk / bjørke , Dingo Women s Nelly Western Boot brun SOREL Women s 1964 Premium CVS Snow Boot Karamell, nektar Columbia Women s Himmelske Omni-Heat Lace Up Boot Quarry / Dove , Walking Cradles Mark Lemp av Kvinners Leash Dress Sandal Sølvfår Vionic med Orthaheel Tide Sport Women s Sandal Svart ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Blowfish Women s/ B0776BKQHB Granola Fisherman Sandal Sandal Birk/ Naken/ Grå Dyecut be8be1a - 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
Blowfish Women s/ B0776BKQHB Granola Fisherman Sandal Sandal Birk/ Naken/ Grå Dyecut be8be1a           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 1009529
3412797
4084839
5958322
7339845
11538654
12206533
15908383
18506905
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