## Working AnalysisThe text is for a two semester course in advanced calculus. It develops the basic ideas of calculus rigorously but with an eye to showing how mathematics connects with other areas of science and engineering. In particular, effective numerical computation is developed as an important aspect of mathematical analysis. * Maintains a rigorous presentation of the main ideas of advanced calculus, interspersed with applications that show how to analyze real problems * Includes a wide range of examples and exercises drawn from mechanics, biology, chemical engineering and economics * Describes links to numerical analysis and provides opportunities for computation; some MATLAB codes are available on the author's webpage * Enhanced by an informal and lively writing style |

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### Contents

Foundations | 3 |

Sequences of Real Numbers | 29 |

Continuity | 57 |

The Derivative | 85 |

Higher Derivatives and Polynomial | 121 |

Solving Equations in One Dimension | 151 |

Integration | 171 |

i | 261 |

The Derivative in Rn | 295 |

Solving Systems of Equations | 323 |

Quadratic Approximation and Optimization | 385 |

Constrained Optimization | 439 |

Integration in | 489 |

Applications of Integration to Differential | 555 |

Appendix II | 599 |

631 | |

### Common terms and phrases

assume calculate Cauchy sequence chain rule change of variable Chapter compact compute constant constraint continuous function converges uniformly convex coordinates critical point curve deduce defined denote Df(x differential equation error estimate example exercise exists fixed point formula function f(x functional iterates given graph Hence implicit function theorem implies improper integral intermediate value theorem inverse function theorem Jordan domain Lagrange equations least upper bound Lemma Let f(x Let G limit linear approximation Lipschitz mapping maximizer mean value theorem minimizer minimum monotone Newton's method norm one-to-one open interval open set parameters partition positive definite power series problem quadratic rational numbers real numbers rectangle result satisfies Section sequence xn Show Simpson's rule solution steepest descent subset Suppose tangent Taylor polynomial tends to zero trapezoid rule uniformly continuous unique vector