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

Vionic Womens Rory oppreist Womens Rory Boot Ankle Boot taupe 54ffe10 - 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
Vionic Womens Rory oppreist Womens Rory Boot Ankle Boot taupe 54ffe10      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

Vionic Womens Rory oppreist Womens Rory Boot Ankle Boot taupe 54ffe10

 

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

 

YDN Women Comfy Kitten Low Heel Mules Slip på Clog Sandals Open Toe Dress Pumps Slide Shoes Natural-5cm , SoftWalk Women s Tilton Slip-On Loafer Svart / Svart , YDN Women Block Low Heels Mules Slip On Clogs Runde Toe Pumps Slide Casual Shoes Blå , Alegria Women s Classic Clog Chained Black , Birkenstock Women s Mayari Birko-Flor Sandal Ocean Blue Birko-flor , Klogs Kvinners Charleston Clog Leather Buckle Svart Tintoretto , Crocs Unisex Classic Clog marinen , Mavirs Loafers For Women, Womens Loafers Velvet Backless Slip On Loafers Broderi Mule Tøfler Svart pels Easy Street Campus Women s Slip On Lt. Blått Skinn / Stoff Birkenstock Unisex Boston Soft Footbed Leather Clog Mocha Suede , Sanita Women s Professional Greta Leather Clogs Blå , OLUKAI Nohea Mesh Shoe - Menns Naturlig Snake Birkenstock Unisex Boston Clog, Habana Oljert Lær, 38 M EU - Walking Cradles Kvinners Adobe Clog Tobakk YDN Kvinner Stacked Low Heel Mules Slide On Closed Toe Clogs Block Casual Slippers Shoes brun Birkenstock Milano Unisex Lær Sandal Lysebrun , SoftWalk Women s Mason Clog salvie , Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Amalfi Leather , OLUKAI Nohea Mesh Shoe - Menns Trench Blue / Dive Camo , Crocs Women s Sarah Lined Clog Mule Svart , Alegria Women s Debra Exclusive Professional Shoe Hvit verktøyet Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Suede OLUKAI Nohea Mesh Shoe - Menns Naturlig Snake Columbia Dorado CVO PFG Shoe Womens Damp / Golfstrøm OLUKAI Nohea Mesh Shoe - Menns Mokka-mocha Suede , El Naturalista Women s Yggdrasil N178 Mule Black-grafito Rieker Womens 46369 Daisy 69 Elefant / is / jeans Avec Les Filles Kvinners Marie Slide Sandal Sølv / Metallisk Nappa , PIC / PAY Dacia - Kvinners lærespenne mule - Ryggløs Slip-On Flat Leather Loafer Tan / Black Leopard 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
SoftMoc Women s Debra II Hi Back Zip Fringe Moccasin Kobber , Aerosoles Women s Sidecar Oxford Taupe Metallic Demonia Poison 101 Womens Boots Blk Vegan Leather Anne Klein AK Sport Kvinner Ulisa Fabric Ballet Flat Tinn Multi Fabric Vionic Women Elevated Becca Wedge Lace-Up Lys Tan Kinesisk Klesvask Kvinners Cecilia Oxford Svart patent , FRYE Women s Terri Perf Oxford Flat Hvit , Ahnu Women s W Sukkerpute Vanntett Vandringssko Dark Skifer , Ted Baker Women s Naoi Monk Strap Flat Svart Alegria Women s Paloma Flat konjakk Easy Spirit Women s Travelport26 Mule Hvit-blå-multi Klogs USA Women s Cardiff Mule Svart Wigwam Født - Kvinner - Toby II Blå , OLUKAI Nohea Mesh Shoe - Menns Mørk Java / Rojo , Mavirs Loafers For Women, Womens Loafers Velvet Backless Slip On Loafers Broderi Mule Tøfler Svarte blomster Stacy Adams Stabler menns støvel Svart Lucky Brand Women s Jillah Mule glasert , Alegria Women s Khloe Slip-On Svart Nappa YDN Women Pointed Toe Low Heels Muller Slip On Block Sandals Casual Slide Shoes Purple John Deere Women s JD3694 Ankel Boot brun D2C Beauty Women s Studded Platform Chunky High Heel Lace Up Ankel Booties Hvit , Merrell Women s Murren Strap Vanntett-W Snow Boot Bracken Australia Luxe Collective Women s Reneade Boot Svart Steve Madden Women s Carlita Heeled Sandal Blush Leather Blå av Betsey Johnson Women s Sb-Adley Dress Pump Rød stoff , Tahari Kvinner Ta-Favor Wedge Sandal Rødme , A | X Armani Exchange Women s Flip Flops Wedge Sandal papaya , Bernie Fru Kvinner Comfi Flat Sort Multi , Amalfi av Rangoni Dame Doris Wedge Sandal Naturlig Talco Marc Fisher Kvinners Amina Sandal Rødme
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Vionic Womens Rory oppreist Womens Rory Boot Ankle Boot taupe 54ffe10 - 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
Vionic Womens Rory oppreist Womens Rory Boot Ankle Boot taupe 54ffe10           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 559397
1234953
1368181
2173457
5348172
9254826
9568099
13648428
16036082
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