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

Alegria kvinners 15123 Keli profesjonell sko Tile B01NCB38ZI Alegria Me More Black 302326b - 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
Alegria kvinners 15123 Keli profesjonell sko Tile B01NCB38ZI Alegria Me More Black 302326b      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

Alegria kvinners 15123 Keli profesjonell sko Tile B01NCB38ZI Alegria Me More Black 302326b

 

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 Pesca Fisherman Sandal Midnight Burnish OLUKAI Nohea Mesh Shoe - Menns Trench Blue / Dive Camo , David Tate Doll Women s Slip On Naturlig nakenkalve , MIA Women s Greta Mule Svart OLUKAI Nohea Mesh Shoe - Menns Mørk Java / Rojo , Birkenstock Unisex Boston Soft Footbed Leather Clog Mocha Suede OLUKAI Nohea Mesh Shoe - Menns Carbon / Deep Red Crocs Women s Carlie Cutout Clog Coral / Oyster , Merrell Women s Encore Kassie Moc Clog Rødbete , Ivanka Trump Women s Tissa Mule Svart , YDN Women Peep Toe Chunky High Heel Loafers Mules Slip On Sandaler Dress Slide Shoes rød , Wolky Comfort Lin Beige Nubuck , Wolky Women s Jewel Sandal Orange Vegi Leather , Arcopedico Womens Leina Mary Jane Hvit bluss FSJ Damemote Mid High Heels Sandaler Suede Slip On Open Toe Mules Sexy Slide Shoes Størrelse 4-15 US Rosa-10 cm YDN Kvinner Stacked Low Heel Mules Slide On Closed Toe Clogs Block Casual Slippers Shoes rød , OLUKAI Nohea Mesh Shoe - Menns Black Shearling Suede Klogs USA Women s Austin Clog Taupe Tapestry , RYKA Women s Luxury Mule Bjørnebær Easy Spirit Women s Dolly Skreddersydd Casual Sko Svart OLUKAI Nohea Mesh Shoe - Menns Black Shearling Suede , Easy Spirit Women s Pierson Mule Svart / Multi Lær 4EurSole RH014 kvinners helsetjenester Clog Plaid Patent Plaid Patent , Troentorp Bastad Clogs Kvinner Davinci Original Svart Birkenstock Boston Unisex-Adult Clogs Tabacco Brown Sanita Arla Åpne Trykt Suede Bordeaux FSJ Women Pointed Toe Flats Low Heels Med Studs Casual Slip On Mules For Daily Wearing Size 4-15 US Blå , Birkenstock Unisex Boston Soft Footbed Leather Clog Habana Oljert lær , Stegmann Edelweiss Ull Clog, Kirsebær - ,

     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
Eastland Solstice MLB Women s Oxford Baltimore Orioles-svart , KEEN Kvinners Fremont Zip Vanntett Sko Cordoba , Freja | Hvit / tynt skinn CLARKS Women s Wave Andes Walking Shoe Sage Nubuck Dansko Professional Clog Blå Damask Patent L & rsquo; Artiste av Spring Step Women s Aneria Mule Svart , Katy Perry Women s Tatum Flat Sandal Sølv , Birkenstock esra Sandal Silver Splashes Mud , Crocs Women s Literide Clog Navy / Hvit , Stuart Weitzman Women s Mulearky Mule Hvit G.H. Sjokolade / Navy , Dolce Vita Women s BAZ Moccasin Denim Linen XYD Elegant High Heel Stiletto Ankel Støvler Faux Suede Peep Toe Pumps Booties For Women marinen Kate Spade New York kvinners Liverpool Ankle Boot Svart Easy Street Women s Origin Mule Blå blomsterverktøy , Koolaburra av UGG Women s Kenz Fashion Boot Sable Western Chief Women Vanntett Fire-Eye Lace Up Duck Up marinen Nina Womens Kiyrah Svart Luster Satin , Womens Hunter Original Tour Gloss Galoshes Snow Wellington Rain Boots Lilla Urchin , Sirkus av Sam Edelman Women s Sibley Knee High Boot Stone Blue , Kamik Women s Olivia Rain Boot Grønn , Smoky Mountain Women s Hayden Pulll On Holes Kvadrat Toe Brown Oil Distress / Brown Boots , Nautica Women s Stringer Rain Boot Storm Grå , Dansko Women s Marta Heeled Sandal Red Oiled Badgley Mischka Jewel Kvinners Kjole Sandal Hot Pink , Bella Vita Women s Seraphina II Espadrille Wedge Sandal Hvit , NAOT Women s Sophia Kanel Skinn / Biscuit Leather / Champagne Leather Propet Women s Aurora Wedge Sandal Svart SCARPA Women s Techno X Klatringsko Sølv / Begonia Vionic med Orthaheel Tide Sport Women s Sandal Svart ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Alegria kvinners 15123 Keli profesjonell sko Tile B01NCB38ZI Alegria Me More Black 302326b - 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
Alegria kvinners 15123 Keli profesjonell sko Tile B01NCB38ZI Alegria Me More Black 302326b           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 2093252
4033286
4137428
6586443
9397539
15390856
15446357
18041807
18969487
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