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

Birkenstock 17997 Unisex Footbed Boston Svart Soft Footbed Leather Clog Svart Suede 3faaa70 - 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
Birkenstock 17997 Unisex Footbed Boston Svart Soft Footbed Leather Clog Svart Suede 3faaa70      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

Birkenstock 17997 Unisex Footbed Boston Svart Soft Footbed Leather Clog Svart Suede 3faaa70

 

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

 

OOFOS Unisex OOcloog Sport Clog Mule brun , Qupid Women s Barnes-01a Ankel Bootie Taupe Suede , Easy Spirit Women s Gessica Flat Sort Multi , NAOT Aster Floral Women Clogs og Mules Shoes Vintage Grey Leather Crocs Women s Neria Pro Work Clog Svart OLUKAI Nohea Mesh Shoe - Menns Bordeaux Suede , FSJ kvinner Basic Mid Chunky Heels Ankel Rem Sandaler Open Toe Komfort Uformelt Sko Størrelse 4-15 US Rosa OLUKAI Nohea Mesh Shoe - Menns Mocha / Mocha , XYD Kvinner Retro Strappy Gladiator Sandaler Block Heel Open Toe Mule Slip On Slippers Shoes rød , YDN Kvinner Casual Flip Flops Sandaler Slip On Sandals Low Heels Thong Sko Med Spenne Oliven , SR Max Geneva Women s Slip Resistant Clog marinen Easy Street Campus Women s Slip On marinen YDN Women Kitten Low Heels Loafers Slip On Mules Pointed Toe Komfort Sko turkis RIALTO Womens Vette Brun / E-print Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Suede FRYE Women s Joan Campus Mule Dusty Rose Soft Oiled Suede David Tate Women s Dallas Sandal Dallas Bone , OLUKAI Nohea Mesh Shoe - Menns Mørk skygge / mørk skygge Sanita Women s Signature Pro. Blå Skechers BOBS fra Women s Keepsakes Ice Angel Slipper Sjokolade Softwalk Women s Meredith Clog Cirrus Leather , Alegria Women s Venice Sandal Impressionista Cloud Nine Sheepskin Women s Sunrise Clog Slipper kastanje , DAWGS Women s Premium Working Hvit svart Crocs Kvinner Capri Shimmer Cross-Band Sandal Oyster / Cobblestone , Easy Spirit Women s Traveltime Mule rød , NAOT Women s Sophia Brushed Black / Black Crinkle / Metallic Rd Sanita Women s Smart Step Profesjonell Sophia Mule Grønn / Multi , Kinesisk Klesvask Kristin Cavallari Kvinners Capri Spisset Tå Flat Gull lær

     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
Themost Womens Pointed Toe Sexy Trykt Stretch Stiletto High Heels Over Knee Låret High Boots Blå , Eric Michael Laguna Womens Boots rød CLARKS Women s Maypearl Alice Ankle Bootie Svart lær KEEN Kvinners Terradora Lær WP-w Fottøy Sko Svart / Stål Grå , Bernie Fru Kvinne Zig Zag Ankel Bootie jeans , CLARKS kvinners Leisa Sadie Mule Dark Tan Leather Sandgrens Swedish High Heel Clog Sandals for Women | Oliven (Dark Base) , Timberland Women s Teddy Fleece Fold-Down Waterproof Boot Svart , Allrounder av Mephisto Women s Nigata Tex Winter Boot Sort Gummi / Mid Rød Suede , Gx av Gwen Stefani Women s Akira Bootie Hvit / Grå / Rosa Badgley Mischka Jewel Kvinner Thalia Flat Sandal Svart , Sam Edelman Women s Francis Ballet Flat Smeltet gull Franco Sarto Kvinners Richland Ankle Boot Svart OLUKAI Nohea Mesh Shoe - Menns Hvit , London Fog Womens Aldgate Vevd Stretch Casual Sko Grå Hunters Boots Women s Original Raffinerte Gloss Boots Dark Skifer Hunters Boots Women s Original Raffinerte Gloss Boots Dark Skifer , Onlymaker Women s Sexy Rivets High Heel Round Toe Slip On Stiletto Pumps Shoes Blå SoftWalk Women s Norwich Ballett Flat Svart , KAMIK KVINNES VANDPROOF JENNIFER RAIN BOOTS Kull , Adidas Outdoor Women s CW Choleah High CP Lær Snow Boot Damp Grå / Sort / Teknisk Jord , Timberland Women s Wheelwright Tall Lace Boot WP Boots Mørkebrun Lucky Brand Women s Xaylah Heeled Sandal Tapenade 1 Kinesisk Klesvask Kvinners Foxie Heeled Sandal Dark Nude Suede , YDN Women Comfy Kitten Low Heel Mules Slip på Clog Sandals Open Toe Dress Pumps Slide Shoes Pink-5cm , Carlos av Carlos Santana kvinners tropiske sandal Azalea Aerosoles Women s Poppy Plush Wedge Sandal Korall , Sanuk Women s Yoga Sling 2 Print Vintage Hvit Multi Icicle FitFlop Kvinner Halo Toe Thong Sandal Svart , Helly Hansen Women s Ten-Below HT-isolerte Winter Sneaker Svart / Charcoal / Glacier ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Birkenstock 17997 Unisex Footbed Boston Svart Soft Footbed Leather Clog Svart Suede 3faaa70 - 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
Birkenstock 17997 Unisex Footbed Boston Svart Soft Footbed Leather Clog Svart Suede 3faaa70           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

Mules & Clogs

 

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 3861879
6135960
6174930
7023875
7486681
13853979
14475455
18463935
18844310
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