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

Nine Women West Women s Garroy Nubuck Oxford Nubuck Flat Off Off White Nubuck 049bb65 -

          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
Nine Women West Women s Garroy Nubuck Oxford Nubuck Flat Off Off White Nubuck 049bb65      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

Nine Women West Women s Garroy Nubuck Oxford Nubuck Flat Off Off White Nubuck 049bb65


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


Cobb Hill Women s Abilene Boot Bordeaux , Tahari Kvinner Ta-Randi Høyrød , Tamaris Women s Vivi 28325 Oxford Flat konjakk , Mordenmiss Kvinners etniske unike Emobossed Rose Sweet Ankle Bootie Street Lace Flat Fleece Green , Franco Sarto Women s Imagine Oxford Svart G.H. Brun / Multi , Lauren av Ralph Lauren Womens Barrett-SH Svart / Sort Sigburncalf / Kid Suede Native Unisex Apollo Moc Mote Sneaker. Torch Rød / Shell Hvit , Ønskede Womens Beekman Oxford Sølv K-Swiss Women s Novo Demi Fashion Sneaker Hvit / Off White , Aerosoles Women s Sidecar Oxford Vinlær Mark Nason Los Angeles Women s Debbie Sneaker Rosa Propet Women s Alice Oxford Svart , Propet Women s Alice Oxford Svart Finn Comfort Women s Ikebukuro Oxford Sølv Propet Women s Washable Walker Sneaker Hvit , Drew Shoe Women s Fiesta Oxford Svart kalv Vince Camuto Women s Lesta Oxford Flat Fransk Taupe Propet Women s Washable Walker Sneaker Hvit , SOREL Women s After Hours Lace Up Skinn og Suede Sandal havregrøt , Amalfi Orazio Women s Slip On tinn Hush Valper Kvinners Chardon Oxford Svart , Mona Flying Women s Lær perforert Lace-up Oxfords Sko For Kvinner Low Heels Wingtip Brougue Shoes Grå , ED Ellen DeGeneres Womens Lilliane2 Pinot Avalon Nappa / Patent Pu , Dr. Martens Unisex 1461 W PW 3-øye Gibson Leather, Fashion Oxfords Svart , Cole Haan Women s Jagger Grand Weave Oxford tinn CLARKS Women s Candra Blush Flat Aubergine , Timberland Women s Eivissa Sea Oxford Navy Canvas , Fly London Women s Hira128fly Oxford Svart teppe

     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
CLARKS Women s Caddell Hop Boot brun Hell La Victoire Women s Indy Oxford vanilje Cole Haan Women s Hayes Flat Ankel Bootie Java Suede , Jord Origins Harley Coco , Walking Cradles Kvinner Hartford Flat Sandal Hvit Accordion Perf Walking Cradles Kvinner Ante Flat Svart Softee Prop & eacute; t Propet Women s Delaney Ankle Bootie marinen DailyShoes Women s Militær snøre seg opp Buckle Combat Boots Ankel Høy Eksklusiv Quilted Credit Card Pocket Bootie Gullplate Svart , Cobb Hill Women s Oliana Panel Ankel Boot Svart trekk opp lthr , Trotters Women s Jenn Loafer tinn , Klogs USA Women s DUSTY Clog Blå fugl , Crocs Women s Meleen Twist Sandal Navy / Storm , Drew Shoe Dames Kool Slitesterkt Lær Casual Støvler Svart Lær / Svart Suede Drew Shoe Women s Orchid Hook and Loop Antikk kobber marmor , YDN Women Block Low Heel Mules Slip On Pumps Åpne Tå Slide Sandaler Slipesko rød Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Suede Sandgrens Swedish Wooden Low Heel Clog Sandals for Women | Svart (mørk base) , Josef Seibel Women s Britney 02 Ankle Boot anthrazit SOREL Women s Tivoli Ii Snow Boot Svart / Havsalt Australia Luxe Collective Women s Foxy Boot Beva Bogs Women s North Hampton Plaid Vanntett Isolert Boot Rød / Multi , Salomon Women s Toundra Pro CSWP Vinter støvler i Superlight Leather og Airgel Isolasjon Phantom / Black / Amparo Blue Katy Perry Women s Kenzie Heeled Sandal Sølv , CL av Chinese Laundry Women s Tasty Wedge Sandal Gunmetal / Black , Alegria Women s Lara Wedge Sandal shmancy , Katy Perry Women s Annalie Heeled Sandal Blush Naken , Bella Vita Women s Kimmy Heeled Sandal Champagne , Spring Step Women s Abigail Wedge Sandal Hvit TKEES Women s Liners Flip Flop Sable , Soludos Kvinner Menorca Høy Platform Espadrille Wedge Sandal Naken ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

Nine Women West Women s Garroy Nubuck Oxford Nubuck Flat Off Off White Nubuck 049bb65 -


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
Nine Women West Women s Garroy Nubuck Oxford Nubuck Flat Off Off White Nubuck 049bb65           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 2117460
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