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

Walking Cradles Valerie Women T-Strap s Valerie T-Strap 19141 Sandal Svart 7b5c57d - 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
Walking Cradles Valerie Women T-Strap s Valerie T-Strap 19141 Sandal Svart 7b5c57d      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

Walking Cradles Valerie Women T-Strap s Valerie T-Strap 19141 Sandal Svart 7b5c57d

 

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

 

CLARKS Women s Reedly Variel Wedge Sandal Sand Nubuck BC Sko Kvinner Snack Bar Wedge Sandal taupe , LifeStride Kvinner Marcela Wedge Sandal Tan / Hvit Crocs Women s Adrina III Mini Wedge Svart / Svart Koolaburra av UGG Women s W Briona Wedge Sandal Insignia Blue , Gentle Souls Kvinner Morrie Platform Ankel Strap Wedge Sandal Palm Multi , Badgley Mischka Women s Winter Wedge Sandal elfenbein , Touch Ups Women s Jazz Wedge Sandal Gull Aravon Women s Candace Sandal brun , BC Sko Kvinner Snack Bar Wedge Sandal Rose gull Glimrende kvinner Fenwick Wedge Sandal Svart Toms Women s Avery Syntetisk Suede Bootie Smidd jern Naturalizer Women s Cinda Wedge Sandal Hvit Cole Haan Women s Adderly Wedge Sandal Naken Franco Sarto Kvinners Pinot Espadrille Wedge Sandal Svart , Vince Camuto Kvinners Leera Espadrille Wedge Sandal Støvete Mink , NAOT Fottøy Dames Treasure Wedge Sandal Fishnet Leather / Black Patent Leather Dr. Scholls Shoes Women s Barton Wedge Sandal Green Snake Print , Eric Michael Women s Lily Sandal Blå , Bernie Fru Kvinner Comfi Flat Korall , GUESS Women s HONORA2 Pump Svart WHITE MOUNTAIN Women s Blue Jay Beige / Multi , Blowfish Women s Bungalow Wedge Sandal Birch Dyecut Pu Sam Edelman Women s Darlene Wedge Sandal Svart Naot Women s Reserve Wedge Sandal Metall L / Tinn L / Metall L CLARKS Women s Maritsa Nila Wedge Sandal Khaki Leather , ECCO Women s Shape Plateau Wedge Sandal Rose Dust , Miz Mooz Women s Shay Wedge Sandal Hasselnøtt Kenneth Cole REACTION Kvinner Calm Water Platform Sandal Marigold

     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
SOREL Women s Major Low Leather Boot Steinbrudd / hai Merrell Women s Moab 2 Mid GTX Hiking Boot Sedona Sage LifeStride Women s Farrow Wedge Pump Tan , Taos Fottøy Kvinners karusell 2 skinnsandal Kanel Multi Cole Haan Women s Tali Bow Ballet Flat Sølv / Gunmetal Glitter , Tamaris Women s Nao 28140 Oxford Flat konjakk Klogs USA Women s Mission Clog Pin Drop Patent , Drew Shoe Women s Bloom II Taupe Nubuck , Crocs Unisex Classic Clog Hvit , Rockport Womens Cobb Hill Penfield Svart Sandal - 10 M - , Skittent Klesvask av Kinesisk Klesvask Kvinners Tessa Ankel Boot Camel Split Suede Mephisto Women s Filipina Bootie Svart Adrianna Papell Kvinner Trala Mary Jane Flat Navy Attalie Lace , Peerage FIC Daphne Kvinner bred bredde Peep Toe Pleated Øvre Slingback med spenne (størrelse / mål Guide) brun , XYD Kvinner Retro Strappy Gladiator Sandaler Block Heel Open Toe Mule Slip On Slippers Shoes rød Alegria Women s Debra Exclusive Professional Shoe Hvit verktøyet Sam Edelman Women s Tinsley Rain Boot Moss Green / Black Cole Haan Women s Findra II Flat Sandal Tahitian Tide Suede , Dr. Scholls kvinnelige vennlig flat Tapioca Daydreamer Canvas , Birkenstock Unisex Boston Soft Footbed Leather Clog Taupe Suede , Hunters Boots Women s Original Raffinerte Gloss Boots Mørk Oliven Roper Women s Spade Western Boot Tan , FSJ Women Comfy Åpen tå sommer Sandaler Ankelrem Kattunge Mid Heels Sko Lakkert Størrelse 4-15 US Blå , Franco Sarto Kvinners Haneli Heeled Sandal Melk , Naturalizer Women s Yolanda Slip-On Loafer Turtle Dove Leather , CLARKS Women s Petrina Gail Platform Rød Nubuck SCARPA Women s Dam V Climbing Shoe turkis , FitFlop Women s Tiararama Flip Flop Svart Teva Womens W Azure Flip Sandal Raya Teal , Vionic Women Elevated Becca Wedge Lace-Up Lys Tan
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Walking Cradles Valerie Women T-Strap s Valerie T-Strap 19141 Sandal Svart 7b5c57d - 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
Walking Cradles Valerie Women T-Strap s Valerie T-Strap 19141 Sandal Svart 7b5c57d           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

Plattformer og kiler

 

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 746635
9164887
9455224
10375392
12738098
13401887
13466286
13890766
17340521
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