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CALCULUS A TUTORIAL |
By Pheng Kim Ving, BA&Sc,
MSc
Email: pheng@phengkimving.com
Toronto - Canada
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Welcome To CALCULUS OF ONE REAL VARIABLE!!
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.
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CONTENTS |
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
Last Updated: 14 May 2015.