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

Hush 19992 Valper Valper Burlesque Burlesque Mørkerød cec48ce - 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
Hush 19992 Valper Valper Burlesque Burlesque Mørkerød cec48ce      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

Hush 19992 Valper Valper Burlesque Burlesque Mørkerød cec48ce

 

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

 

Women Round Toe Platform Pumps Stiletto High Heels Dress Sko rød , PUMA Kvinner Vega SL Ballett Flat Rapture Rose-love Potion Drew Shoe Women s Bloom II Taupe Nubuck , Naturalizer Women s Samantha Pointed Toe Flat Hvit perle TOMS Classic Floral Embroidery Striped Damessko Svart Suede / Rope , LifeStride Women s Relay 2 Ballett Flat Blek rosa Walking Cradles Women s Flynn Loafer Flat Navy Leather Cole Haan Women s Tali Bow Ballet Flat Gull / Sølv Glitter 206 kollektive kvinner Madison Ballett Flat Sølv Metallisk Skinn Allrounder av Mephisto Women s NIRO Cemento Suede / Open Mesh Earth Women s Bay Gladiator Sandal Safir Multi Soft Leather Stuart Weitzman Women s Strapdown Ballett Flat Svart , Amalfi av Rangoni Kvinner Oslo Almond Toe Slide Flats Navy-krokodille , Hot Chocolate Design Chocolaticas Sitron Kvinner Mary Jane Flat flerfarget , Trotters Women s Sizzle Ballet Flat marinen LifeStride Women s Diverse Flat Blå , Aibarbie Kvinner Galala Patent-lær Point-toe Flats Kontor Off-duty Flats Sko Grosgrain Bow Sko Sjokolade Vionic Women s Gnist Minna Ballett Flat Berry Snake , YDN Women s Classic Pointed Toe Lace Up Ballett Flats Ankel High D Orsay Dance Shoes Svart , San Antonio sko Kvinner Avslappet Lær Åpen Tå Avslappet Slingback Sandaler Rav , Hush Valper Kvinner Analyse Smart Flat Kamel SoftWalk Women s Napa Mj Mary Jane Flat Tan / Oppbevaring , Bernie Fru Kvinne Zig Zag Ankel Bootie jeans Wolky Comfort Nimes Sienna Italian Softy , LifeStride Women s Affirm Flat Svart Aerosoles Women s Spin Class Ballett Flat Sort kombinasjon Cobb Hill Rockport Women s Fiona Sandal taupe , ALDO Kvinners Ybalelia Mule Gull Sam Edelman Women s Rupert Ballet Flat Svart / hvit Gingham Print ,

     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
Twisted X Boots Mens Lace Up Vanntett Old Brown Hiker brun , Rieker Womens L7121 Eike 21 Svart / Svart Skechers BOBS fra Women s Plush Fashion Slip-On Flat Svart / Sort Adorbs Rockport Cobb Hill Samling Natashya Slouch Mid Women s Boot Svart Nubuck , Alegria kvinners Keli profesjonell sko Shard Oh Yay , Aerosoles A2 Women s Best Roll Boot Svart Alegria Women s Paloma Flat Fandamonium Softwalk Women s Meredith Clog Hvit , SoftWalk Women s Waverly Mary Jane Flat Bagasje , Trotters Women s Lauren Dress Wedge Naken Suede FLY London Women s Yip Boot Svart Oljert Sanita Women s Signature Professional Clog turkis Nine West Women s Yorada Leather Dress Sandal Svart Skechers Women s Bikers-Fan Club-Sporty Slip-on Mesh-Bungee Avslappet Fit Mule Grå , Kate Spade New York Women s Matilda Mule rød , Anne Klein Women s Chelsey Suede Western Boot Cognac Leather Marc Joseph New York Women s Cypress Hill Loafer Multi Sapphire Blue Skechers Women s Cleo Razz Dazz Ballett Flat Taupe / Gold , Crocs Menns Santa Cruz Loafer Svart / Khaki , OLUKAI Nohea Mesh Shoe - Menns Mørk skygge / mørk skygge , Sanuk Women s Donna Tribal Flat Olive / Multi Tribal Stripe , Joules Women s Fieldwelly Gloss Rain Boot lime , Dan Post Kvinner Maria Western Boot Svart , UGG Women s Cecile Winter Boot Svart , Blundstone Unisex Dress Series Rustikk brun Sam Edelman Women s Yaro Heeled Sandal Svart lær , Sam Edelman Kvinner Patti Dress Sandal Candy Red , NAOT Kvinners Kayla Wedge Sandal Buffalo Leather , Stuart Weitzman Women s Sunny Heeled Sandal Platinum Noir , Camper Women s Misia K200568 Wedge Sandal Hvit ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Hush 19992 Valper Valper Burlesque Burlesque Mørkerød cec48ce - 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
Hush 19992 Valper Valper Burlesque Burlesque Mørkerød cec48ce           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 9134307
9804831
10608771
11931305
13451664
14195613
16258102
17630375
18697823
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