By Pheng Kim Ving, BA&Sc, MSc
- Canada



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.




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.





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

Rieker 41385 Doris 85 Flat Flat Schwarz Doris 19973/ Leinen 62aa4e0 -

          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
Rieker 41385 Doris 85 Flat Flat Schwarz Doris 19973/ Leinen 62aa4e0      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

Rieker 41385 Doris 85 Flat Flat Schwarz Doris 19973/ Leinen 62aa4e0


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


Easy Street Women s Jules Flat Lysegrå / Hvit Metallisk , YDN Kvinner Casual D Orsay Spisset Toe Ballett Flats Komfort Slip-On Walking Sko Med Ankel Rem Lett laks , Mio Marino Komfortabel Flat Dress Sko til kvinner - Square Toe Satin Flats Lysegrå , Walking Cradles Women s Heidi Pump Hvitt lær Hot Chocolate Design Chocolaticas Frokost Kvinners Mary Jane Flat flerfarget , Auditions Crescent Women s Slip On Hvit , BOBS fra Skechers Women s Super Plush-Rhinestone Flat Svart , JA VIE Mote Designer Sko til kvinner Dress Sko Slip On Womens Flats for hver dag Wear Mom Tattoo Dansko Women s Fawna Mary Jane Flat Navy Burnished Calf , Propet Women s Robin Shoe rød Trotters Women s Caroline Ballet Flat Mørkbrun Cognac , VANELi Kvinner Andie Ballett Flat Naturkork , Walking Cradles Kvinner Hartford Flat Sandal Hvit Accordion Perf , Tic-Tac-Toes kvinners Maryjane Dance Shoe Gull , Propet Women s TravelActiv Mary Jane Mote Sneaker Svart Apex Women s Comfort Tan MIA Women s Elsie Gladiator Sandal Naturlig Alegria Women s Paloma Flat Tinnpudder Dazzler , Trotters Women s Sizzle Signature Ballett Flat Gull Marc Joseph New York Women s Cypress Hill Loafer Navy Leaf Napa , UGG Womens Jonette Loafer Svart Jack Rogers Women s Anice Suede Ballett Flat Mørk grå , Skechers BOBS Women s Plush Lite-Be Cool Flat Grå / Multi , Birkenstock Women s Gizeh Birko-Flor (Tm) Bright White Patent Nine Seven Genuine Leather Women s Pointy Toe Comfort Håndlagde Flats Sko Aprikos Propet Women s W0089 Pedic Walker Sandal Dusty Taupe Nubuck , Camper Women s Right Nina 21595 Ballett Flat Svart Sella Negro Alegria Women s Pesca Shoe Tawny TOMS Women s Classic Linen Rope-Sole Komfortabel og lett å bruke Slip-On marinen ,

     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
Dualyz Unisex Kush Classic Slipemaskin med flyttbar Sole Mørkegrå / Gul TOMS Kids 10009918 Alpargata-K Desert Taupe Striped Slub Chambray Sorel - Kvinner Emelie Lace Premium Non Shell Boot Cordovan / Major ED Ellen DeGeneres Womens Gillen Svart / Sort Crinkle Patent / Crinkle Patent , Crocs Women s Malindi Flat Sukkerspinn , SoftWalk Women s Napa Ballett Flat Tan Multi , Easy Spirit Nine West Women s Giovedi Suede Ballett Flat Elektrisk Blå Aerosoles Kvinners Ballett Flat Black Snake FLY London Women s Yip Boot Oxblood Touch Ariat Women s Wexford H2O arbeidsstøvle, Taupe, 7,5 B US Java , Hush Valper Kvinners Cyra Catelyn Ankel Boot Vintage Indigo Pelle Moda Women s Wager Ankle Boot Blush Patent Easy Street Women s Columbus Mule Svart Cole Haan Women s Abbot Ankle Boot Morel Marc Jacobs Kvinner Rita Pointy Toe Ballerina Ballett Flat Svart , Athena Alexander Kvinners Tallye Ballett Flat Svart Leopard Rockport Kvinners Cobb Hill Abigail Kjolepumpe brun , Easy Street Women s Bryson Boot Tan Burnish / Gore Jessica Simpson Women s Zamia Ankle Boot Rouge Noir Hunter Womens Original Short Wellington Syntetisk Rain Boots Militær Rød , Jileon Extra Wide Kalv Gummi Regn Støvler for kvinner-bredeste Pass opp støvler i US-opp til 21 tommer Calves-Wide in Foten og ankelen Navy With White Polka Dots , Via Spiga Kvinner Forte Block Heel Heeled Sandal Sky Patent Tommy Hilfiger Kvinner Kitty Dress Sandal Hvit Ll , Tommy Hilfiger Kvinners Katz Heeled Sandal Svart , FSJ Kvinner Allsidig Open Toe Mule Sko Feminin Slingback Stiletto Sandaler Størrelse 4-15 US Rosa Easy Street Women s Anita Heeled Sandal marinen FSJ Kvinner Casual Peep Toe Mule Sandaler Stiletto High Heels Party Evening Shoes Størrelse 4-15 US turkis Nine West Womens Provein Block Heel Sandal Lett naturlig syntetisk Pleaser Women s Adore-709 Ankle-Strap Sandal Svart Patent / Svart , La Sportiva Miura Klatringsko - Kvinner Hvit / Jade Grønn ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Rieker 41385 Doris 85 Flat Flat Schwarz Doris 19973/ Leinen 62aa4e0 -


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
Rieker 41385 Doris 85 Flat Flat Schwarz Doris 19973/ Leinen 62aa4e0           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



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 235980
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