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Calculus:
An intuitive approach
Wai Yan Pong
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Front Matter
I
Calculus I
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1
Differentiation
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1.1
Differentiability
1.2
Derivatives
1.2.1
Rules of Differentiation
1.2.2
Implicit Differentiation
1.3
Derivatives of Elementary Functions
1.3.1
Derivatives of Algebraic Functions
1.3.2
Derivatives of Exponential and Logarithmic Functions
1.3.3
Derivatives of trigonometric functions and their inverses
1.4
Exercises
2
Integration
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2.1
Anti-derivatives
2.2
Integration by Substitution
2.3
Fundamental Theorem of Calculus
2.4
Exercises
II
Calculus II
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3
Techniques of Integration
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3.1
Integration by Parts
3.2
Trigonometric Integrals
3.2.1
Products of Trigonometric Functions
3.2.2
Powers of Trigonometric Functions
3.2.3
Half-Angle Substitution
3.3
Trigonometric Substitutions
3.4
Integrals of Rational Functions
3.4.1
Partial Fraction Decomposition
3.4.2
The reduced cases
3.5
Exercises
4
Improper Integrals
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4.1
Definitions and Examples
4.2
Comparison Test for Integrals
4.3
Exercises
5
Sequence and Series
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5.1
Sequences
5.1.1
The Basics
5.1.2
Algebraic Operations on Sequences
5.1.3
Convergence of Sequence
5.2
Convergence Tests for Sequences
5.3
Series
5.4
Convergence Tests for Series
5.5
Power Series
5.5.1
Applications to summing series
5.6
Taylor Series
5.7
Exercises
Back Matter
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A
Trigonometric Identities
B
Some Important Limits
C
Power Series of Some Common Analytic Functions
D
Binomial Series
E
Hints and Answers to Selected Exercises
References
Index
Colophon
Section
2.3
Fundamental Theorem of Calculus
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