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

Bogs Women Women B076QMHKTM s Berkley Rain Rain Boot Svart 0e5dfb3 - 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
Bogs Women Women B076QMHKTM s Berkley Rain Rain Boot Svart 0e5dfb3      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

Bogs Women Women B076QMHKTM s Berkley Rain Rain Boot Svart 0e5dfb3

 

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

 

Stuart Weitzman Women s Spartan Mid Calf Boot Svart Dr. Martens Kvinner Vonda Lace Up Boot, Svart, 7 UK / 9 M US - La Canadienne Women s Blanche Boot Svart patent The North Face Womens Shellista II Mid Duck Green / Smijern Ariat Women s Round opp Ryder Western Boot Himmelsblå , BEARPAW Women s Leslie Snow Boot taupe UGG Women s Classic Short Vanntett Snow Boot Sand Caterpillar Kvinner Alexi Combat Boot Kvinner Svart Badgley Mischka Women s Martine Ankle Boot Midnatt , TOMS Women s Majorca Peep Toe Mid Calf Boot Black Suede Perforated Leaf , NCAA SMU Mustangs Dame 13-tommers Gameday Støvler Svart Steve Madden Women s Troopa Lace-Up Boot Lilla lær Demonia DEMONIAW Womens CLASH-206 / BVL-CA støvler , Woolrich Women s Fw Tundracat Snow Boot Svart , Adidas Outdoor Women s CW Choleah High CP Lær Snow Boot Damp Grå / Sort / Teknisk Jord , Toe Warmers Women Boots Michelle Mørkebrun , Lucchese Bootmaker Kvinner Sierra Western Boot Svart , Baffin Women s Hannah Snow Boot Sjokolade , Trotters Women s Blizzard III Snow Boot konjakk , SOREL Women Conquest Carly II Snow Boot Svart, Sølv Sage Sugar Women s Linjal Ankel Bootie brun SOREL Women s Joan Short Rain Boot Brent Henna , Sugar Women s Tegan Lace-Up Mid-Calf Combat Boot Svart , Durango Women s Crush 12 \ Mørk sjokolade Hush Valper Kvinners Aydin Catelyn Perf Boot Pulverblå Suede , KAMIK KVINNES VANDPROOF JENNIFER RAIN BOOTS periwinkle Baffin Women s Shannon Snow Boot Kull MuckBoots Women s Tack II Mid Riding Work Boot Svart / Hot Pink Naturalizer Women s Rebecca Boot Brandy

     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
Gabor Womens Gabor 72.722 Svart Dr. Martens Women s 1461 Mono Smooth Oxford Hvit 5.11 Menns Pursuit Lace Up Shoe Mørkebrun , MIA Amore Women s Ellen Flat Sandal Rose gull SoftWalk Women s Hollis Flat Bagasje Wolverine 1883 av Women s Jude Oxford Tan Leather Dr. Comfort Vigor Women s Therapeutic Diabetic Extra Depth Vandring Boot Leather Lace kastanje , Naturalizer Women s Macie Platform Pump Grå NIUERTE Kvinners ekte lærflate rund tå med bue-slips Casual Formal Parties Sko Naken Patent , Steve Madden Women s Mesa Sandal Denim Multi XYD kvinner pekte tå kattunge muller lave hæler slippe på pumper skyve sandaler clog sko Grå Nine West Women s Argyle Ankle Boot Mørk naturlig suede , J.Renee Womens Francie Sølv Satin / Rhinestones , Easy Spirit Women s Traveltime Mule Svart hvit Crocs Unisex Classic Blitzen III Clog Mule Espresso / Espresso , Merrell Women s City Leaf Chelsea Snow Boot Merrell Oak , Softwalk Women s Hanover Boot Cognac Soft Nappa Leather , Vince Women s Dryden Fashion Boot kyst~~POS=TRUNC , ECCO Women s Women s Touch Ballerina Bow Ballet Flat Stone / Sand , Propet Women s TravelFit Flex Ballett Flat Taupe / Grå BEARPAW Women s Aretha Snow Boot Hickory , Blundstone M Menns BL1316 Winter Boot Svart, rød Nine West Women s Sweuce Fabric Mid Calf Boot Svart / svart stoff , Alegria Kvinner Sitka Støvler Snowflake Choco , BEARPAW Quinevere Tall vanntett støvel for kvinner Svart Ii , Baffin Women s Snogoose Winter Boot Kull , Taos Fottøy Kvinner Star Fashion Sneaker Sky Blue Wash Canvas Kamik Women s Starling Snow Boot khaki Kinesisk Klesvask Kvinners Ryktet Heeled Sandal Dark Nude Suede FitFlop Flip Women s Sandal Helt svart
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Bogs Women Women B076QMHKTM s Berkley Rain Rain Boot Svart 0e5dfb3 - 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
Bogs Women Women B076QMHKTM s Berkley Rain Rain Boot Svart 0e5dfb3           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

Midt på leggen

 

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 9229547
9568771
10758483
14001669
15974249
16930684
17279922
17603959
18426225
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