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

OLUKAI Nohea Nohea Mesh Hvit Shoe - Menns Menns Hvit b1fb17e -

          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
OLUKAI Nohea Nohea Mesh Hvit Shoe - Menns Menns Hvit b1fb17e      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

OLUKAI Nohea Nohea Mesh Hvit Shoe - Menns Menns Hvit b1fb17e


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


Rachel Comey Women s Wald Mule Eyes Needlepoint Broderi , Antelope Women s 826 Leather Bib-Front Wedge taupe , Lucky Brand Women s LARSSON2 Clog Svart , Linea Paolo Axel | Cognac / Black Leopard Hair Calf Leather , Daniel Green Women s Glamour Slipper Svart Kidskin , Dansko Professional Clog Blå Scrunch SoftWalk Women s Alcon Mule Bright Multi / Dark Brown , OLUKAI Nohea Mesh Shoe - Menns Hvit , OLUKAI Nohea Mesh Shoe - Menns Carbon / Deep Red Eastland Women s Mae Clog brun FRYE Women s Dani Shield Sling Mule Khaki Suede Alegria Women s Paloma Limited Edition profesjonell sko Sukker skaller , CLARKS Women s Addey Trust Clog brun , Daniel Green Women s Meg Slipper Svart Easy Spirit Women s Tunein Mule Svart CLARKS Womens Patty Tayna Mule M Navy Leather , XYD Women Fashion Pointed Toe Slingback Pumps High Heel Slip On Dress Sko med buer Hot Pink , OLUKAI Nohea Mesh Shoe - Menns Svart Lotta Fra Stockholm Highwood High Heel Clogs Tan Leather G.H. Rose Nubuck Xelero Madera Women s Comfort Therapeutic Ekstra Dyb Casual Sko Leather Lace-up Svart Sam Edelman Kvinners Kerry Mule Bright White Leather , FSJ Kvinner Mote Peket Toe Pumps Low Heels Casual Loafers Slip On Summer Shoes Størrelse 4-15 US Green-suede Sanita Women s Professional Patent Clog Purple Shimmer Easy Spirit Women s Gessica Flat Medium Grå Combo Syntetisk , Rieker Womens 46369 Daisy 69 Elefant / is / jeans , Sol Sana Womens Charli Mule Rust Suede Stud Lotta fra Stockholm Low Wood Clogs i Brown Oiled Nubuck - Walking Cradles Women s Caden Clog Svart Nappa ,

     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
Allrounder av Mephisto Kvinner Kalibra Sport Sko Tåke Nubuck / Suede Sperry Top-Sider Women s Coil Ivy Perf Boat Sko Svart , SOREL Women s Dacie Lace Booties karri Belleville Hot Weather Desert Flight & Combat Vehicle (Tanker) Boot, Menn, Sko Tan , Lane Boots Women s Cody Boot (Brown) brun , Hush Valper Damer Chaste Ballet Flat Svart Stud , Easy Spirit AP1 Sport Walking Sko Svart HUMTTO Vandringssko Menn og kvinner Utendørs Walking Walking Støvler 6907 Lilla , Dansko Kvinner Vera Flat Sandal Tinn Nappa , CAMSSOO Kvinners Classic Rivets Pointy Toe Slip på Comfort Flats Dress Pumps Shoes Naken Patient Pu Ivanka Trump Women s Tropica Ballett Flat Rosa , Merrell Women s Siren Sport Merrell Stone / Cordovan Lauren av Ralph Lauren Women s Betha-Sn-CSL Flat Sandal Blå Mist Sanita Women s Yara Square Flex Platform Sandal Brun - Braun (Antikkbrun 78) Trotters Women s Sizzle Ballet Flat marinen , Dr. Martens 1460 Originals Åtte-øye snøreoppstart Svart glatt lær Apex Women s A200WM095 Loafer Flat Brunt lær , Spring Step Women s Cosmic Mary Jane Flat rød Cole Haan kvinners Alice Bow skimmer pekte tå Flat Roccia Snake Print , Easy Spirit Women s Traveltim280 Mule Navy / Navy Multi Suede , Spring Step Women s Streetwise Sandaler Svart lær Miz Mooz Women s Fritz Loafer tinn Linea Paolo Frieda | Svart lær Palladium Women s Pallabosse Chelsea L Chukka Boot Svart Alegria Women s Cami Boot Bølgete marinen , L & rsquo; Artiste av Spring Step Women s Simonetta Boot Svart Vince Women s Benetta Heeled Sandal Svart , CLARKS Women s Leisa Vine Platform Hvitt lær The Original MuckBoots Voksen Våtland Boot brun VISCATA Tossa 2,5 \ Svart Noir
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

OLUKAI Nohea Nohea Mesh Hvit Shoe - Menns Menns Hvit b1fb17e -


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
OLUKAI Nohea Nohea Mesh Hvit Shoe - Menns Menns Hvit b1fb17e           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 228250
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