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 19996 Women s Final Exam Wedge Aerosoles Brun Final eksotisk 6cb7538 - 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 19996 Women s Final Exam Wedge Aerosoles Brun Final eksotisk 6cb7538      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 19996 Women s Final Exam Wedge Aerosoles Brun Final eksotisk 6cb7538

 

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

 

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 Drew Shoe Women s Delite Flat Hvit Kombinasjon , CLARKS Women Blanche West Flat Black Snake Print Leather FootThrills Women s Midtown Slip-on Sko Svart , Dansko Women s Edith Mary Jane Flat Svart lær TOMS Women s Classic Linen Rope-Sole Komfortabel og lett å bruke Slip-On Infinity Blue Heritage Canvas Crocs Unisex Classic Clog khaki Naturalizer Women s Stella Ballett Flat Beige Sirkus av Sam Edelman Kvinners Charlotte Ballett Flat Navy / White Denim Stripe Rieker 41385 Doris 85 Flat Schwarz / Leinen VANELi kvinners anemone sko Svart utskrift , Easy Street Women s Jules Flat Lysegrå / Hvit Metallisk YDN Kvinner Casual D Orsay Spisset Toe Ballett Flats Komfort Slip-On Walking Sko Med Ankel Rem Lett laks Mio Marino Komfortabel Flat Dress Sko til kvinner - Square Toe Satin Flats Lysegrå Walking Cradles Women s Heidi Pump Hvitt lær , Hot Chocolate Design Chocolaticas Frokost Kvinners Mary Jane Flat flerfarget , Auditions Crescent Women s Slip On Hvit , BOBS fra Skechers Women s Super Plush-Rhinestone Flat Svart , JA VIE Mote Designer Sko til kvinner Dress Sko Slip On Womens Flats for hver dag Wear Mom Tattoo ,

     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
Pleaser INFINITY-2020 9 \ Svart patent / blk , Twisted X Ladies Blue Slip-On Driving Mocs Oiled Saddle Kvinners Keds, Lori Ubal Lace Up Shoe Stein LFL av Lust for Life Women s Nimble Oxford Flat konjakk , Apex Women s A200WM095 Loafer Flat Svart Prop & eacute; t Women s Mabel Oxford Vin , Oboz Echo Hiking Sko - Kvinner Korall KEEN Women s Terradora Mid Waterproof Vandringssko Duck Green / Quiet Green Jack Rogers Women s Rebecca Suede Ballett Flat Svart Giesswein Unisex Acadia Grå (019 Anthrazit) Rockport Cobb Hill Collection Womens Cobb Hill Gratasha Zip Boot Mørkegrå Nubuck Crocs Menns Santa Cruz Loafer Espresso / Espresso SAS Women s Classic Slip on Mocha , OLUKAI Nohea Mesh Shoe - Menns Hvit , Sanita Women s Signature Dalton Oil Closed Clog Svart XYD Soft Suede Ankel Støvler Komfortabel Round Toe Block Heel Booties With Zipper For Women Svart , CLARKS Women s Sillian Sway Ankle Bootie Grå , Klogs Sko Kvinner Silverton Mary Jane Clog Svart prisme Birkenstock Unisex Boston Soft Footbed Leather Clog Habana Oljert lær Rockport Kvinners Kimly Ankle Bootie Svart lær Lugz Women s Drifter LX Chukka Boot Charcoal / Cream / Gum / Blåsvart Madden Girl Women s Prissley Riding Boot Grå stoff , LifeStride Women s Unity Harness Boot Svart , Kamik Women s Starling Snow Boot Sjokolade , Touch Ups Women s Jane Ankle-Strap Sandal Sølv Glitter Onex Women s Ariel Wedge Sandal Tan , Calvin Klein Kvinners Vable Kjole Sandal marinen La Sportiva Mythos Lace Up Climbing Shoe - Kvinner pistachio , NAOT Paris Metall Birkenstock Women s Gizeh Birkibuc Sandaler Svart / Flerfarget ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Aerosoles 19996 Women s Final Exam Wedge Aerosoles Brun Final eksotisk 6cb7538 - 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 19996 Women s Final Exam Wedge Aerosoles Brun Final eksotisk 6cb7538           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 274877
3305847
3395040
6253778
7370747
8767840
12237311
12406865
15984441
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