## Mathematical Analysis and Applications: An IntroductionStarting with a discussion of Real Numbers and Functions, this text introduces standard topics of Differential and Integral Calculus together with their Applications such as Differential Equations, Numerical Analysis, Approximation Methods. |

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### Contents

RATIONAL NUMBERS | 5 |

INDUCTION PRINCIPLE | 19 |

REAL NUMBER SYSTEM | 33 |

SETS AND FUNCTIONS | 49 |

SEQUENCES | 69 |

CONTINUOUS FUNCTIONS I | 89 |

CONTINUOUS FUNCTIONS II | 107 |

INFINITE SERIES | 131 |

TAYLORS THEOREM APPROXIMATION | 189 |

POWER SERIES MAPS | 205 |

DIFFERENTIAL EQUATIONS I | 231 |

DIFFERENTIAL EQUATIONS II | 251 |

RIEMANN INTEGRATION I | 277 |

RIEMANN INTEGRATION II | 301 |

RIEMANN INTEGRATION III | 313 |

INDEX | 338 |

### Common terms and phrases

approximation Archimedean property axiom called Cauchy Chapter closed set compact interval computation Consider constant continuous function Corollary cosx curve decimal defined definition denoted derivative Differential Equations domain evaluate Examples Continued Exercises exists exponential function expression finite fixed formula function f(x geometric given equation graph Hence homogeneous equation induction inequality infinite series irrational keep in mind limit point linearly independent Maclaurin Mathematics Mean Value Theorem method monotone increasing natural numbers non-empty non-zero notation Note obtained one-one open set particular solution polynomial positive integer positive number power series power series map problems PROOF Proposition proved radius of convergence real numbers remainder term result Riemann integral Rolle's Theorem sequence xn Similarly sinx solving sub-interval subset Suppose Taylor's polynomial Taylor's Theorem trigonometric functions variables words write Wronskian zero