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

Mephisto Women B071DWP7SX s Svart Cross Walking Shoe Cross Svart Bucksoft/ Mote c8cf85e -

          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
Mephisto Women B071DWP7SX s Svart Cross Walking Shoe Cross Svart Bucksoft/ Mote c8cf85e      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

Mephisto Women B071DWP7SX s Svart Cross Walking Shoe Cross Svart Bucksoft/ Mote c8cf85e


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


Native Men s Apollo Chukka Mote Sneaker Cavalier Rød / Cavalier Rød , Miz Mooz Harmony Women s Oxford Loafer Naken , ECCO Fottøy Womens Fara Tie Oxford, Black, 38 EU / 7-7.5 M US - Earth Shoes serval Lysrød , Bed Stu Kvinner Lita Oxford sko Navy Rustic , TOMS Women s Deia Boot Svart Suede / Glimmer , Cole Haan Women s Original Grand Wingtip Oxford Navy Printed Floral-elfenben Cole Haan Women s Grandpro Hi Grå Pinstripe Oljert Vachetta-Grå Flann , Synes at! Stahl , Propet Women s Washable Walker Sneaker Sr Royal Blå / Hvit FitFlop Women s Uberknit Ballerina Walking Slip-On Neon Blush / White Earthies Santana Svart pensel av lær Clarks Women s Un.Hinton Oxfords Shoes Black Cow Nubuck , Pedro Garcia Womens Cristina Perle Satin , CLARKS Women s Wave Andes Walking Shoe Sage Nubuck , Cole Haan Women s Grand Crosscourt Ii Svart Suede-hvit , Drew Shoe Women s Paradise II Velco Slip-on Hvit Rose Petals Kvinner Caliente Dress Pump Svart mikrofiber Alegria Women s Essence Sneaker Forever Yours , CLARKS Womens Un Haven Lace Sand Nubuck Spring Step Womens Cinzano Oxford Svart Sesto Meucci Womens Dirce Black Old West , Allrounder av Mephisto Womens Lets Go Lace Up Shoes Cloudburst Mesh / Cemento Suede , Arcopedico kvinners Ls Red Starry Night K-sveitsiske kvinner Belmont SO Fashion Sneaker Fersk laks / hvit Jessica Simpson Womens Nalicia Sneaker Opal / Peach , Dr. Martens Women s 1461 Oxford , Native Unisex Apollo Moc Mote Sneaker. Fakkel Rød / Shell Hvit / Naturlig Gummi Cobb Hill Rockport Kvinners Isabella-CH Casual Sko taupe

     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
ALDO Women s Harber Oxford Sølv Sammitop Kvinners 8 cm Stiletto Heel Ankel Støvler Zip Up Vinter Kort Støvler Med Fur Patent Black , Giesswein Unisex Acadia Lamm , Oboz Women s Phoenix Mid Waterproof Boot Mineralblå , TOMS Women s Classic Linen Rope-Sole Komfortabel og lett å bruke Slip-On Pink Icing , Bloch Women s Aspiration Pointe Comfort Ballett Flats Rosa Ros Hommerson Friste Kvinner Casual Sko Lær Spenne Sølv / Glitter , Aquatalia av Marvin K. Kvinner Stella Mule Platinum Metallic Tumbled Kate Spade New York Women s Mariel Mule Gammelt gull OLUKAI Nohea Mesh Shoe - Menns Mink Shearling Suede Chooka Damemote Rain Bootie Wedge Black Vionic med Orthaheel Serena Women s Boot Svart Camper Women s Peu Cami Mote Sneaker Lys Pastellgrå Bernardo Women s Heather Fashion Boot Røyk Suede Aravon Women s Faith Flat Svart , svensk hasbeens kvinners Jill Plateau Clog Natur The Fix Women s Jaeda Open Weave Mule Shoetie Ankel Bootie Matte Sølv / Metallisk Sesto Meucci Kvinner Evonne Flat Svart Nabuk Propet Women s TravelActiv Mary Jane Mote Sneaker Sølv , Ivanka Trump Women s Evia Mule Middels naturlig suede Prop & eacute; t kvinners Lumi Ankle Zip Snow Boot Grå Women Round Toe Platform Pumps Stiletto High Heels Dress Sko Mørke blå , Bella Vita Kvinner Baxter Flat Svart Kid Suede Soft Style Women s Angel II Pump Bone Elegance Hunters Boots Women s Original Raffinerte Gloss Boots Svart LH Unisex Neoprene Gardening Boots og Rain Boots brun ALDO kvinners Cilalla Ankel Boot Brown Nabuck , Nine West Women s mcglynn Leather Heeled Sandal Svart lær , Steve Madden Women s Stecy Dress Sandal Naturlær LifeStride Women s Groovy Wedge Pump Svart 3
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Mephisto Women B071DWP7SX s Svart Cross Walking Shoe Cross Svart Bucksoft/ Mote c8cf85e -


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
Mephisto Women B071DWP7SX s Svart Cross Walking Shoe Cross Svart Bucksoft/ Mote c8cf85e           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 7116078
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