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

Crocs Grå B01456U9NM Unisex Classic Clog Skifer Skifer Grå d131574 - 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
Crocs Grå B01456U9NM Unisex Classic Clog Skifer Skifer Grå d131574      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

Crocs Grå B01456U9NM Unisex Classic Clog Skifer Skifer Grå d131574

 

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

 

HELLE FASHION COMFORT KVINNER S GWEN COMFORT CLOG Svart , Calzuro Autoclavable Clog med øvre ventilasjon Blågrønn Steve Madden Womens Kaden Floral Multi Sika Fottøy Birchwood Comfort Work Clog , Birkenstock Arizona Unisex Lær Sandal Svart Oljert Lær Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Suede OLUKAI Nohea Mesh Shoe - Menns Bordeaux Suede , Franco Sarto Kvinners Firefly Heeled Sandal Taupe Snake Suede Sandgrens Swedish High Heel Wooden Clogs For Women | Lilla ask , XYD kvinner pekte tå kattunge muller lave hæler slippe på pumper skyve sandaler clog sko oransje , Klogs Klogs Klogs Black Tooled Easy Spirit ARORA Women s Slip On Lyseblå-krystall-suede Merrell Women s Encore Q2 Breeze Clog Blå himmel FSJ Kvinner Komfort Lav Hæl Mules Peep Toe Slide Sandaler Slip On Dress Pumps Sko Størrelse 4-15 US Rød-10cm , Alegria Women s debra Slip-On Svart Emboss Rose , Easy Street Women s Holly Mule Svart Crocs Women s Neria Pro Work Clog Hvit , Easy Street Women s Calm Mule Brune Burnish Stilig Pro XP Menn Mules & Clogs Sko, Elegant Fottøy, Mote, Brun Oljert, Størrelse - 42 Sort oljert , Sanita Jamie Wood Clogs Antikkbrun , OLUKAI Nohea Mesh Shoe - Menns Svart , Sanita Women s Professional Patent Clog Rødt patent David Tate Women s Gloria Slide platina Dansko Womens XP 2.0 Svart patent Alegria kvinners Keli profesjonell sko Goth Cutie FSJ Kvinner Mote Peket Toe Pumps Low Heels Casual Loafers Slip On Summer Shoes Størrelse 4-15 US Red-patent , OLUKAI Nohea Mesh Shoe - Menns Naturlig Snake , YDN Women Comfy Kitten Low Heel Mules Slip på Clog Sandals Open Toe Dress Pumps Slide Shoes Natural-5cm Bella Vita Women s Eden Mule Kamel Burnish ,

     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
Native Men s Apollo Chukka Mote Sneaker Regatta Blå / Shell Hvit / Shell Hvit Gummi , Dublin Ladies River Boots II Chocolate , VISCATA Women s Barceloneta Authentic & Original Spansk Made Espadrille Flats Blå Dr. Martens Women s 1460 Svart Easy Spirit ON CUE Women s Oxford Svart-black , Easy Spirit ON CUE Women s Oxford Svart-mørk Grå ED Ellen DeGeneres Womens Gillen Rent hvitt / rent hvitt krøllpatent / krøllepatent Stegmann Women s Wool Filt Clog Med PolyFlex Sole grafitt , Sanita Jamie Wood Clogs Svart L Artiste av Spring Step Women s Gipsy Flat Sandal Svart / Multi LifeStride Women s Dee Ballet Flat Svart 001 RIALTO Womens Vette Navy / E-print , FSJ Kvinner Komfort Lav Hæl Mules Peep Toe Slide Sandaler Slip On Dress Pumps Sko Størrelse 4-15 US Fuchsia-10cm Steve Madden Women s Granite Combat Boot Svart lær , DailyShoes Women s Bekjempe stil opp Ankel Bootie Quilted Militær Knit Credit Card Knife Money Lommebok lommebøker Quilted Black Pu Sanuk Women s Toasty Tails Short Boot Kull , RYKA Women s Luxury Mule Plum / Pink Birkenstock Women s Arizona Sandals Grasiøs lakris , Eastland Women s Tonie Twin Gore Slip On brun CLARKS Women s Muckers Trail Rain Sko Svart , Pendleton Heritage Kort Papago Park Boot Svart , Walking Cradles Kvinner Hartford Flat Sandal Hvit Accordion Perf SOREL Women s Joan of Arctic Wedge Mid Black / Quarry , Chooka Women s Wide Calf Memory Foam Rain Boot Mulberry Demonia Women s Damned-318 Knee High Boot Svart Vegan Leather NATURLIG SOUL Kvinners Mia Heeled Sandal taupe La Sportiva Women s TarantuLace Performance Rock Climbing Shoe Korall , Vionic Women s Walker Classic Sko Hvit / Rosa / Grå Carlos av Carlos Santana Women s Heron Flip Flop Platino , Adidas Adilette CF + sommer Kvinner Sandaler Svart / Mynt
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Crocs Grå B01456U9NM Unisex Classic Clog Skifer Skifer Grå d131574 - 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
Crocs Grå B01456U9NM Unisex Classic Clog Skifer Skifer Grå d131574           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 3850407
8536939
9222379
11281974
11858474
13127774
15458307
15656089
18033388
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