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

Easy Street Women Women s Garrett Easy Flat Sandal 19912 Svart f78410f - 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
Easy Street Women Women s Garrett Easy Flat Sandal 19912 Svart f78410f      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

Easy Street Women Women s Garrett Easy Flat Sandal 19912 Svart f78410f

 

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

 

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 , Sanuk Women s Donna Flat Rosa Poncho , Easy Street Women s Zone Flat Sandal Dark Tan TOMS Classic Floral Embroidery Striped Damessko Navy Washed Canvas Tile Sole , OLUKAI Kvinner Pehuea Powwow Closed Toe Mules Teal / Legion Blue , Lauren Ralph Lauren Kvinners Belen Flat Kunstnere Crème , Walking Cradles Women s Hayden Loafer Svart lær , CAMSSOO Kvinners Classic Rivets Pointy Toe Slip på Comfort Flats Dress Pumps Shoes Rose Patant Pu , Kinesisk Klesvask Kristin Cavallari Kvinners Knock Out Flat Sandal Svart lær Kenneth Cole New York Women s Brigid Espadrille med Stud Ballet Flat Naken , FRYE Women s Sienna Suede Ballet Flat Cashew Suede ,

     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
Naot Fottøy Womens Groovy Desert Suede / Vintage Grey Leather / Mulberry Suede , Nine West Women s Bonnelyn Fabric Slipper Svart SCARPA Women s Women s Terra GORE-TEX Vandring Boot brun , Garmont Pordoi Nubuck Mid GTX Vandring Boot - Kvinner Olive Green / Light Green Propet Women s Washable Walker Sneaker Sr Bone / White Easy Spirit Women s Motion Sport Lace-Up Sneaker Tornado KEEN Women s Versatrail vanntett sko Zinfandel / Magnet FitFlop Women s Superbendy Mirror-Toe Ballerinas Ballett Flat Rødme Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Amalfi Leather Kmeioo Mules for Kvinner, Puff Pompom-Embellished Slip On Loafers Backless Pointed Tå Satin Mule lysbilder Hot Pink-satin Dr. Martens Women s Pascal Leather Combat Boot Kirsebærrød Virginia Badgley Mischka Kvinner Bianca Ballett Flat elfenbein LifeStride Women s Motorway 2 Flat Sopp OLUKAI Nohea Mesh Shoe - Menns Onyx / Onyx , Jack Rogers Women s Captiva Flat Sandal Orange / Gold SOREL Women s Joan Tall Gloss Rain Boot Svart, havsalt , Womens Hunter Original Tour Gloss Wellies Wellington Snow Rainboots Mørk Oliven FRYE Kvinner Sabrina 6G Lace Up Suede Boot Tre Merrell Women s Aurora 6 Ice Plus vanntett snøbrett Merrell Tan Hunters Boots Dames Originale Tour Short Boots Militær Rød , The North Face Womens Shellista II Mid Split Rock Brown & Dove Grey Brinley Co Womens Blitz Faux Suede Vanlig og bred kalv Over-the-kne Lace-up Brogue Boots Grå Columbia Women s Minx Slip Ii Omni-Heat Snow Boot Skifer, Dark Raspberry InterestPrint Dames Snow Boots Unikdesignede Comfort Winter Boots Multi 24 , SOREL Kvinner Joan of Arctic Wedge Mid x Feiring Svart / natur , Ikke vurdert Fottøy Kvinner Santana Laser Cut Bootie Light Taupe , CLARKS Women s Corsio Calm Flat Sandal Svart lær , Loeffler Randall Kvinner Vera Kjole Sandal Svart , Vionic Women s Naples Arch Støtte Thong Wedge Sandal Champagne , Kenneth Cole REACTION Women s Great Stretch Low Wedge Sandal Rose gull ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Easy Street Women Women s Garrett Easy Flat Sandal 19912 Svart f78410f - 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
Easy Street Women Women s Garrett Easy Flat Sandal 19912 Svart f78410f           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 281307
605299
3268710
4286383
6903337
6967422
10265605
11629175
14054829
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