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


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This website posts a tutorial on the calculus of one real variable, free! It provides a complete treatment of the 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. You'll hopefully find here any topic that you need help with that's taught in your first or
second course on the calculus of one real variable.


Each chapter is divided into sections. Each section discusses the topics that are the subject of the section and provides a set of
exercises 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. The exercises supply drills on the basic techniques for the topics
discussed in the section, and some are theoretical and/or difficult


If you need clarification or explanation on any part of any section, or have spotted misspellings, or have caught mathematical
errors, or have suggestion, please feel free to email me. Note: In your correspondence, please mention your first name (or
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click here.




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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
                 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  The Derivative
              2.1.1 Rates Of Change
              2.1.2 Tangent Lines And Their Slopes
              2.1.3 The Derivative
       2.2 Differentiability Vs Continuity
       2.3 Rules Of Differentiation
              2.3.1 Differentiation Of Sums, Differences, And Polynomials
              2.3.2 Differentiation Of Products And Quotients
              2.3.3 Differentiation Of Compositions Of Functions - The Chain Rule
              2.3.4 Differentiation Of Inverse Functions
       2.4 Higher-Order Derivatives
       2.5 Implicit Differentiation
       2.6 The Differentials


3. Applications Of The Derivative – Part 1


       3.1 The Mean-Value Theorem
       3.2 Critical Points And Extrema
       3.3 The First-Derivative Test
       3.4 Concavity And Inflection
       3.5 The Second-Derivative Test
       3.6 Sketching Graphs Of Functions
       3.7 Antiderivatives And Indefinite Integrals
       3.8 Motion
       3.9 Differential Equations


4. The Elementary Transcendental Functions


       4.1 The Trigonometric Functions And Their Inverses
              4.1.1 The Trigonometric Functions
               The Trigonometric Functions
               Trigonometric Identities
               Limits Of Trigonometric Functions
               Differentiation Of Trigonometric Functions
               Graphs Of Trigonometric Functions
               The Projectile Motion
               The Simple Harmonic Motion
              4.1.2 The Inverse Trigonometric Functions
               The Inverse Trigonometric Functions
               Differentiation Of The Inverse Trigonometric Functions
       4.2 The Logarithmic And Exponential Functions
              4.2.1 The Natural Logarithm Function
              4.2.2 The Natural Exponential Function
              4.2.3 General Exponential And Logarithmic Functions
              4.2.4 Logarithmic Differentiation
              4.2.5 Growth And Decay
       4.3 The Hyperbolic Functions And Their Inverses
              4.3.1 The Hyperbolic Functions
              4.3.2 The Inverse Hyperbolic Functions


5. Applications Of The Derivative – Part 2


       5.1 Optimization
       5.2 Related Rates
       5.3 Linear Approximations
              5.3.1 Tangent-Line Approximations
              5.3.2 Approximations Of Errors In Measurement
              5.3.3 Approximations Of Roots Of Functions – Newton's Method
              5.4.2 More Indeterminate Forms


6. The Integral


       6.1 Areas And Riemann Sums
              6.1.1 Summation Notation And Formulas
              6.1.2 Areas And Riemann Sums
       6.2 The Definite Integral
       6.3 The Fundamental Theorem Of Calculus
       6.4 Integration By Inspection
       6.5 Techniques Of Integration
              6.5.1 The Method Of Substitution
              The Method Of Substitution
              Integration Of Trigonometric Functions
              Integration Of Powers Of Trigonometric Functions
              The Inverse Trigonometric Substitution
              Other Substitutions
              6.5.2 The Method Of Partial Fractions
              6.5.3 The Method Of Integration By Parts
       6.6 Approximate Numerical Integration
       6.7 Improper Integrals
              6.7.1 Improper Integrals
              6.7.2 Tests For Convergence Of Improper Integrals


7. Applications Of The Integral


       7.1 The Mean-Value Theorem For Integrals
       7.2 Areas Of Plane Regions
       7.3 Volumes
              7.3.1 Finding Volumes By Slicing
              7.3.2 Finding Volumes By Using Cylindrical Shells
       7.4 Distance
       7.5 Arc Length
       7.6 Areas Of Surfaces Of Revolution
       7.7 Applications To Physics
             7.7.1 Work
             7.7.2 Force Exerted By A Fluid
       7.8 Differential Equations – Variables Separable


8. Plane Curves


       8.1 Parametric Curves
             8.1.1 Parametric Curves
             8.1.2 Tangent And Sketching Of Parametric Curves
             8.1.3 Arc Length And Area Of Surface Of Revolution Of Parametric Curves
             8.1.4 Vector Study Of Motion In The Plane
       8.2 The Polar Coordinate System
             8.2.1 The Polar Coordinate System
             8.2.2 Sketching Polar Curves
             8.2.3 Area By Polar Curves
             8.2.4 Arc Length And Area Of Surface Of Revolution Of Polar Curves


9. Infinite Series


       9.1 Infinite Sequences
       9.2 infinite Series
       9.3 The Comparison Tests
       9.4 The Root And Ratio Tests
       9.5 The Integral Test
       9.6 The Alternating-Series And Absolute-Convergence Tests
       9.7. Approximations Of Sums Of Series


10. Representations Of Functions By Power Series


       10.1 Power Series
       10.2 Derivatives And Integrals Of Power Series
       10.3 Taylor Series
       10.4 Applications Of Taylor Series
       10.5 Taylor Polynomials and Taylor Theorem
       10.6 The Binomial Series



Last Updated: 22 Sep 2009


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