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{\bf Syllabus for Math 165 (Fall 1997)}
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{\bf Chapter 1~~ Some Basic Concepts}

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(1)~~Integers, Rational Numbers, Real Numbers

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(2)~~Mathematical Induction

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(3)~~Supremum and Infimum

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(4)~~Absolute Values and The triangle Inequality

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{\bf Chapter 2~~ Integral Calculus}

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(1)~~Functions

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(2)~~Partitions and Step Functions

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(3)~~Integral of Step Functions

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(4)~~Upper and Lower Integrals

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(5)~~The Concept of Integrable Functions

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(6)~~Integrability of Piecewise Monotonic
Functions

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(7)~~Basic Propertities of Integrals

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(8)~~Application of Integrals

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(9)~~Trigonometric Functions

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(10)~Polar Coordinates

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(11)~Area and Volume

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{\bf Chapter 3~~ Continuous Functions}

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(1)~~The Concept of Limit

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(2)~~Basic Poperties of Limit

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(3)~~The Concept of Continuity

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(4)~~Basic Properties of Continuous Functions

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(5)~~Bolzano's Theorem, Intermediate-Value
Theorems and The Extreme-Value Theorem

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(6)~~Integrability of Continuous Functions


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{\bf Chapter 4~~ Differentiable Functions}

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(1)~~The Concept of Differentiability

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(2)~~Derivative and Tangents

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(3)~~Differentiability implies Continuity

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(4)~~Basic Rules of Differentiation

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(5)~~Chain Rule

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(6)~~Applications: Maxima and Minima (First and
Second Derivative Tests)

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(7)~~The Mean-Value Theorem

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{\bf Chapter 4~~The relationship between
Integration and Differentiation}

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(1)~~Primitive Functions

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(2)~~The Fundamental Theorem of Calculus

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(3)~~Techniques of Integrations (Substitution, by
Parts, Partial Fractions

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{\bf Chapter 5~~Inverse Functions The Logarthmic
and Exponential Functions, Inverse Trigonometric
Functions}

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(1)~~Derivative of Inverse functions

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(2)~~The Logarthmic
and Exponential Functions

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(3)~~TInverse Trigonometric
Functions

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(4)~~More on Differentiation and Integrations of
Functions


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