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

Sesto Meucci Zingato Kvinner Platino Evonne Kvinner Flat Platino Zingato f81ccb9 - 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
Sesto Meucci Zingato Kvinner Platino Evonne Kvinner Flat Platino Zingato f81ccb9      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

Sesto Meucci Zingato Kvinner Platino Evonne Kvinner Flat Platino Zingato f81ccb9

 

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

 

David Tate Womens Glow Bone Croc Print Nine West Women s Antonecia Suede Slip-On Loafer Grønn Suede STEVEN av Steve Madden Women s Saunders Flat Sandal Hvitt lær Skechers BOBS Kvinners BOBS Plysj-Sit Flat Svart , Dansko Women s Cindy Flat Sandal Svart Nubuck Sanuk Women s Donna Hemp Flat Oliven Grå L Artiste av Spring Step Women s Copa Flat Sandal Blå / Multi YDN Women Pointed Toe Slip på Flats Skjulte Low Heels Pumps Comfort Shoes med stropper marinen Easy Street Women s Garrett Flat Sandal Svart , Trotters Women s Liz Flat Svart , Mephisto Women s Elettra Ballett Flat Grå magisk / mørkegrå Ceylan , Drew Shoe Women s Geneva Lace-Up Esprsso BOBS fra Skechers Women s Highlights-Jewel Rock Ballet Flat Lyserosa , Michael Michael Kors Women s Fulton Moccasin Svart lær , STEVEN av Steve Madden Women s Cocoa Flat Sandal Tan Suede BeautiFeel Kvinner Amada Ballett Flat Gold Pixel Print Suede Kombinasjon Kinesisk Klesvask Kristin Cavallari Kvinners Capri Mule Svart lær , KEEN Kvinner Kona Flip-W Flat Sandal Stål Grå / Sort , Indigo Rd. Blå LifeStride Women s I-Loyal Flat Mørkegrå 2 , Aetrex Women s Corinne Slip On Svart , Jord Origins Harley Svart SoftWalk Women s Hadley Mary Jane Flat Svart , SCHUTZ Kvinner Sasha Ballett Flat Svart , STEVEN av Steve Madden Women s Pace s Flat Oliven Satin , Finn Comfort Women s Hanoi Oxford Svart Nappa / Nubuck , Trotters Women s Sizzle Flat Blekrosa Perforert Dr. Scholl s Shoes kvinners leder ballett flat Blå Mikrofiber , Arcopedico Kvinner Shawna Flats Sko Svart ,

     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
Cobb Hill Rockport Women s FitSpa Flat Korall , Vaneli Womens Jabot Svart Suede / Matchende Elastisk , Mona Flying Women s Lær perforert Lace-up Oxfords Sko til kvinner Wingtip Multicolor Brougue Shoes Kaffe-grå Allrounder av Mephisto Womens Funny Trend Ivy Suede N InterestPrint Damestøvler Unikdesignede Comfort Lace Up Boots Multi 10 , FRYE Women s Addie Hiker Ankel Bootie Svart , Trotters Women s Sizzle Signature Ballett Flat Sølv Cobb Hill Women s Abilene Boot Blå / Teal , Mountrek Women s Brook Path Mid Boot Svart , Vasque Men & Fnof; _Ts Breeze III GTX Fottøy, Black Olive Brun Oliven / Bungee Cord , SAS Kvinner Duo Open Toe Casual Sports Sandaler Svart Alegria kvinners Keli profesjonell sko Chestnut Luster , Myk stil av Hush valper Kvinners pleats være med deg kle pumpe Svart L Artiste av Spring Step Women s Bande Mule Brun / Multi , BO Brun (Bagasje) Via Spiga Kvinners Sophia Backless Bootie Ankle Boot Marina Blue Suede , XOXO Women s Karol Boot Svart Women Round Toe Platform Pumps Stiletto High Heels Dress Sko Leopard , 206 kollektive kvinner Ballard Chelsea Ankel Boot Chocolate Brown Leather Franco Sarto Kvinners Edaline Ankel Bootie Mogano T.U.K. Svart , Tommy Hilfiger Women s Kandess Heeled Sandal rød Lucky Brand Women s Jakeela Ankel Boot travertin De Blossom Collection Kvinners kjole Party Satin Round Toe Dress Ballet Flat med Rhinestone Dekorert spenne marinen MuckBoots Kvinner Breezy Casual All Purpose Ankle Boot Svart Gingham , Ikke rangert Kvinner Gemini Ankel Bootie taupe , Kinesisk Klesvask Women s Spring Street Boot Svart , UGG Women s Aysel Winter Boot Svart Dolce Vita kvinners Lorna Heeled Sandal Av hvitt lær Calvin Klein Women s EFA Wedge Sandal Sand
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Sesto Meucci Zingato Kvinner Platino Evonne Kvinner Flat Platino Zingato f81ccb9 - 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
Sesto Meucci Zingato Kvinner Platino Evonne Kvinner Flat Platino Zingato f81ccb9           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 6008103
8944241
9961380
10618463
11421210
16028410
16521295
16692966
17413659
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