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

DAWGS Women 14411 s Premium Working Hvit Working Hvit svart ad5bb39 -

          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
DAWGS Women 14411 s Premium Working Hvit Working Hvit svart ad5bb39      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

DAWGS Women 14411 s Premium Working Hvit Working Hvit svart ad5bb39


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 Spirit Women s Traveltime Mule rød NAOT Women s Sophia Brushed Black / Black Crinkle / Metallic Rd Sanita Women s Smart Step Profesjonell Sophia Mule Grønn / Multi , Kinesisk Klesvask Kristin Cavallari Kvinners Capri Spisset Tå Flat Gull lær OLUKAI Nohea Mesh Shoe - Menns Black Shearling Suede Arbeid underverk av Dansko Womens Camellia Grå Verktøy Patent Birkenstock Unisex Boston Soft Footbed Leather Clog Habana Oljert lær , Easy Street Women s Sportster Clog Brun Croco CLARKS Women s Wave.Run Svart lær Easy Spirit Women s Travelport26 Mule White-lilla-multi , Skechers for Work Women s Slip Resistant Clog Svart patent , OLUKAI Nohea Mesh Shoe - Menns Mokka-mocha Suede Alegira Alegira Kvinner Freesia Skinn Clog Limited Edition Jet Luster Alegria Women s Carina Wedge Sandal Bronzemosaic SoftWalk Women s Abby Clog Svart OLUKAI Nohea Mesh Shoe - Menns Naturlig Snake Klogs Klogs Klogs Sea Whims Patent Sanita Morse Oiled Justerbar Fleksibel Danske Tøfler (Art: 1200038) Antikkbrun OLUKAI Nohea Mesh Shoe - Menns Rock / Kano Easy Spirit Women s Dolly Skreddersydd Casual Sko tinn , Birkenstock Unisex Boston Soft Footbed Leather Clog Svart Suede FSJ Women Basic Open Toe Sandaler Med Block Heels Faux Suede Ankel Rem Pumps Size 4-15 US rød , OLUKAI Nohea Mesh Shoe - Menns Naturlig Snake YDN Kvinner Lace Up Suede Flat Mules Low Heel Peep Toe Tøfler Komfortable Walking Shoes Mørk grå Dansko Professional Clog Faded Stripe Patent , OLUKAI Nohea Mesh Shoe - Menns Mørk Java / Rojo XYD Kvinner Åpne Tå Strappy Low Block Heel Sandal Pumps Ankel Strap Wedding Dress Shoes Purple Calzuro Autoclavable Clog med øvre ventilasjon Blågrønn Klogs USA Unisex Boca Mule Hvit ,

     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
Easy Spirit AP1 Sport Walking Sko Hvit Cole Haan Womens Womens riginalGrand Chukka Argento , Oboz Women s Sawtooth Mid BDRY Vandring Boot iceburg Kvinners lær Lotus Jazz Oxford Grå Merrell Women s Sugarbush Oppdater Vanntett Fottøy Andorra KEEN Women s Gypsum Ii Mid WP-w Boot Midnight Navy / Opaline Troentorp Women s B & aring; byen Donatello Leather Clog Svart patent , YDN Women Block Low Heels Mules Slip On Clogs Runde Toe Pumps Slide Casual Shoes Blå Easy Spirit Women s Illie Clog Svart , JOE N JOYCE Rio SynSoft Soft-Footbed Sandales Hammertone Svart patent Aerosoles Women s Tax Bracket Ankel Boot Svart Suede , Harley-Davidson Women s Kiarah Work Boot Svart , Weeger Women s ESD Antistatic Leather Clog hav , FootThrills Women s Midtown Slip-on Sko Brun pumpe Crocs Unisex Crocband Clog Ultraviolet / Hvit , Ellie Shoes Women s 609-Aponi Boot Purple Walking Cradles Women s Heidi Pump Svart Lakkert , OLUKAI Nohea Mesh Shoe - Menns Carbon / Deep Red Splendid Women s SPL-Henri Ankle Bootie Svart CLARKS Kvinner Leisa Claytin Flat Sandal Tinn Metallic Leather Rocket Dog Women s Bentley Lewis Pu / Randy Fabric Ankle Boot Svart , Taos Fottøy Kvinner Star Fashion Sneaker Grå Vask Lerret SoftWalk Women s Charming Ankle Bootie burgunder Easy Spirit Women s Geinee Ballett Flat Gull , VANELi Kvinner FC-313 Flat Hvit svart , Ferrini Women s Maverick Western Boot Svart , Kenneth Cole New York Kvinner Langley Ankle Ruffle Detalj på Foran Rem Heeled Sandal Cerulean , Badgley Mischka Kvinners Sadie Heeled Sandal Lyserosa SCHUTZ Kvinner Morianna Heeled Sandal Svart , Tenk deg Vince Camuto Women s Rashi Heeled Sandal Svart ,
     11.2 Improper Integrals
     11.3 Tests For Convergence Of Improper Integrals

DAWGS Women 14411 s Premium Working Hvit Working Hvit svart ad5bb39 -


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
DAWGS Women 14411 s Premium Working Hvit Working Hvit svart ad5bb39           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 4447463
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