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

Seychellene Kvinner Mountain Monk Mountain Strap Flat B00F2O7K5M Eksotisk Gull Eksotisk fe4293a - 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
Seychellene Kvinner Mountain Monk Mountain Strap Flat B00F2O7K5M Eksotisk Gull Eksotisk fe4293a      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

Seychellene Kvinner Mountain Monk Mountain Strap Flat B00F2O7K5M Eksotisk Gull Eksotisk fe4293a

 

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

 

Coolway Women s Topaz Ballet Flat Cue Aerosoles Women s Spin Class Ballett Flat Pink Multi FRYE Women s Sienna Micro Stud Ballett Flat Sortpolert myk fullkorn Sam Edelman Kvinner Flynt Ballett Flat Hot Pink Suede , ECCO Damas Damara Crisscross Gladiator Sandal Rose Dust / Coral Earth Bijou Women s Pump Burgundy Trykt Suede LifeStride Kvinner Eclipse Dress Sandal Inky Navy , CLARKS Women s Gracelin Lea Ballett Flat Svart lær , Walking Cradles Women s Heidi Pump Navy Leather , Skoleproblem Prodigy 5100 Mary Jane Uniform Shoe (Toddler / Little Kid / Big Kid) Navy Nvc TOMS Women s Zebra Printed Calf Hair Avalon 10006320 , ILSE JACOBSEN Kvinner Tulip 3475 Slip-On Laurel Green Haflinger Women s Energy Flat Kapteiner Blue TOMS Classic Floral Embroidery Striped Damessko Multi hekle , Walking Cradles Kvinner Hartford Flat Sandal Rød Nubuck , Therafit Shoe Women s Rio Ballett Flat marinen YDN Women s Soft Casual Round Toe Flats Slip On Suede Ballet Driving Shoes Wide Width brun , Bernie Fru Kvinner Catniss Hvit Shimmer Propet Women s Ladybug Walking Shoe Cayenne , Rieker Mirjam 11 kvinners pumpe Hvitt lær CLARKS Women s Neenah Vine Ballet Flat Tinnlær Badgley Mischka Women s Davis Pointed Toe Flat Platino Brocade , Sanuk Women s Funky Fiona Flat Slate Blue Zig Zag Crocs Women s Busy Day Stretch Mesh Lace-up Svart Naot Women s Kumara Flat Vintage Smoke, Black Raven , Amalfi av Rangoni Kvinner Oslo Almond Toe Slide Flats hav , Rockport Women s Cobb Hill Laila Mary Jane Flat Svart , Trotters Women s Sante Ballett Flat Svart Suede , Dansko Womens Lisanne Ballett Flat Blå Milled Nubuck ,

     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
TOMS Kids 10009918 Alpargata-K Rose Cld Cht Emb / T G , Allrounder av Mephisto Women s Dakona Oxford Elektrisk blå luftmesh , Katy Perry Women s Harra Ballet Flat Spanskrød , Crocs Women s Swiftwater Sandal Paradis rosa / hvitt Mona Flying Women s Lær perforert Lace-up Oxfords Sko til kvinner Wingtip Multicolor Brougue Shoes brun KEEN Kvinners Hush Strikk-W Vandringssko Svart / Gargoyle , Bernie Fru Kvinner Flettet Catwalk Svart / Tinn , Sanita Kvinners Koi Kupid Clog Rose Sanita Women s Original Pro. fuchsia ara Women s Leather Clogs & Mules Grå Birkenstock Unisex Boston Soft Footbed Leather Clog Taupe Suede Naturalizer Women s Cassandra Ankle Bootie Svart , Drew Shoe Women s Joy Sky Blue Suede / Mesh , Jack Rogers Kvinner Adair Dress Sandal Hvit / karibisk blå Coach Women s Carmen Ankle-High Lizard Boot Mink / Mink Semi Matte Calf / Suede , David Tate Womens Glow brun Dr. Martens Women s askins Mary Jane Flat Black Temperley + Sport Spacer Mesh , Naturalizer Women s Jaye Flat taupe Muck Boot Unisex Muckmaster Hei Wellington Støvler Svart Kelsi Dagger Brooklyn Women s Ashton Ankel Boot ingefær Aerosoles Women s Mythology Boot Lys Tan Suede , Top Moda Women s COCO 1 knestrømkjøring Tan Gold , Ikke vurdert Kvinners G-Funk Chelsea Boot taupe Kamik Women s Heidi Rain Boot Oliven , SOREL Women s Conquest Wedge Booties Svart , Merrell Women s Chateau Mid Lace Polar Vanntett Snow Boot Merrell Oak Jessica Simpson Women s Emagine Heeled Sandal Aztec Blue , Alegria Women s debra Slip-On Hvit Tumble Leather , Easy Spirit Women s Cindie Dress Pump Naturlig Nubuck Skechers Cali Women s Rumblers-Sosial Butterfly Wedge Sandal Tinnestreng
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Seychellene Kvinner Mountain Monk Mountain Strap Flat B00F2O7K5M Eksotisk Gull Eksotisk fe4293a - 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
Seychellene Kvinner Mountain Monk Mountain Strap Flat B00F2O7K5M Eksotisk Gull Eksotisk fe4293a           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 1891682
3204183
7724591
8108454
9626632
12153724
13408339
14669941
17720422
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