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

CLARKS Sneaker Women s Darleigh Cora Sneaker s Sage Darleigh Nubuck 66e740f - 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
CLARKS Sneaker Women s Darleigh Cora Sneaker s Sage Darleigh Nubuck 66e740f      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

CLARKS Sneaker Women s Darleigh Cora Sneaker s Sage Darleigh Nubuck 66e740f

 

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

 

Finn Comfort Women s Ikebukuro Oxford Svart Rodeobuk Easy Street Women s Lucky Oxford Brun Suede , Ros Hommerson Women s Nadia Oxfords Mørkbrun Kid Suede / Brown Croc Patent OTBT Womens Khora Sneakers Bein , Mona Flying Women s Lær perforert Lace-up Oxfords Sko til kvinner Wingtip Multicolor Brougue Shoes Blå-brun Converse Menns Street Tonal Canvas High Top Sneaker Cool Grå / Midnattsflåte Loeffler Randall Kvinners Callie Oxford Svart / natur Shellys London Women s Dilys Oxford Svart Easy Spirit Women s Devitt Oxford Flat Hvit Pikolinos Kvinners Royal Cap Toe Derby W4D-4565C1 Olmo FitFlop Women s Uberknit Ballerina Walking Slip-On Svart , Drew Shoe Women s Parade II Oxford Svart kalv , Wolverine Women s Oxford Blå / Grå OTBT kvinners kometsko Marine Vionic Womens Syra onyks Hush Valper Kvinners Chardon Oxford Isgrå Drew Amora Women s Oxford Brun-nubuck Timberland Women s Amherst Mote Sneaker Vintage Indigo Mephisto kvinners laserslip-up sko Chestnut Nubuck / Tinn Raptor , TOMS Kids 10009918 Alpargata-K Multi Akvarellhval Naturalizer Women s Morrison Mote Sneaker Gråt skinn / slange CLARKS Kvinne Janey Mae Oxford bivoks , SOREL Women s After Hours Lace Up Skinn og Suede Sandal havregrøt , Timberland Women s Amherst Mote Sneaker bjørk Skechers Sport Women s Empire Mote Sneaker Grå / Lyseblå Synergi , SoftWalk Womens Relax Svart / Grå Lycra / Microfiber , TOMS Kids 10009918 Alpargata-K Cognac Suede / Fringe , Drew Shoe Women s Fiesta Oxford Svart kalv , Cloud Footwear Women s Aika Fashion Sneaker Mørk grå ,

     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
Palladium menns Pampa Hi Originale Chukka Boot Sahara OLUKAI Malie Boot - Kvinner Rum Kelsi Dagger Brooklyn Women s Edison Oxford Flat Quartz Prop & eacute; t kvinners Cicely Flats Shoes Svart Taryn Rose Women s Annabella Ballett Flat Svart natt , Hush Valper Kvinners Heather Tassel Ballett Flat Tan , Teva Women s W Arrowood Swift Lace Vandringssko Hvit Dr. Comfort Vigor Women s Therapeutic Diabetic Extra Depth Vandring Boot Leather Lace Svart Dolce Vita Women s Saxon Oxford Hvit svart Minnetonka Women s Moosehide Fringed Kilty Moccasin Svart Naturalizer Women s Huntley Loafer Red Pepper Leather , svenske hasbeens kvinners Jill Clog Svart , Vionic Menns Orthaheel Technology White / Navy Walker Pink Orange , AVARCAS 101 Avarcas Kvinner Flat Leather Summer Sandals Hvit , CLARKS Kvinners Carleta Paris Boot Svart lær Easy Spirit Women s Tunein Mule Svart , Adtec Women s 8 \ To Tone , Marc Jacobs Kvinner Daisy Ballerina Ballett Flat Svart , FRYE Women s Ellen Short Western Boot Sand , ADRIENNE VITTADINI Fottøy Kvinner Senji Boot Svart myk kalv BZee Women s Terri Black / Polygon / Texture Material M Joules Women s Welly Print Rain Boot Grey Swan Geo , JSport av Jambu Women s Regatta Flat Charcoal / Peony , Bella Vita Women s Talina Ii Plus Harness Boot brun UGG Women s Classic Short II Boot Magenta Rose , Skechers Women s Keepsakes-Short 2 Button Winter Boot taupe Tommy Hilfiger Women s Gianna Western Boot Svart Suede Ariat Women s Heritage Western R Toe Western Cowboy Boot Distressed Brown , Dolce Vita Women s Rocko Heeled Sandal Metallic / Multi Leather FRYE Women s Roberta Ghillie Wedge Sandal Sand ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

CLARKS Sneaker Women s Darleigh Cora Sneaker s Sage Darleigh Nubuck 66e740f - 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
CLARKS Sneaker Women s Darleigh Cora Sneaker s Sage Darleigh Nubuck 66e740f           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

Oxfords

 

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 2197968
3675853
6137500
10306241
11775836
12668834
15373171
15777843
18015867
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