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



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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.


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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
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
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


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


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
11.2 Improper Integrals
11.3 Tests For Convergence Of Improper Integrals


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
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 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


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Last Updated: 14 May 2015.


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