MAA Textbooks : Calculus Deconstructed: a Second Course in First-Year Calculus.
Nitecki, Zbigniew H.
MAA Textbooks : Calculus Deconstructed: a Second Course in First-Year Calculus. - 1st ed. - 1 online resource (509 pages) - AMS/MAA Textbooks Series ; v.16 . - AMS/MAA Textbooks Series .
cover -- copyright page -- title page -- Preface -- Contents -- Precalculus -- Numbers and Notation in History -- Numbers, Points, and the Algebra of Inequalities -- Intervals -- Sequences and their Limits -- Real Sequences -- Limits of Real Sequences -- Convergence to Unknown Limits -- Finding Limits -- Bounded Sets -- The Bisection Algorithm (Optional) -- Continuity -- Continuous Functions -- The Intermediate Value Theorem and Inverse Functions -- Extreme Values and Bounds for Functions -- Limits of Functions -- Discontinuities -- Exponentials and Logarithms -- Epsilons and Deltas (Optional) -- Differentiation -- Slope, Speed and Tangents -- Formal Differentiation I: The Algebra of Derivatives -- Formal Differentiation II: Exponentials and Logarithms -- Formal Differentiation III: Inverse Functions -- Formal Differentiation IV: The Chain Rule and Implicit Differentiation -- Related Rates -- Extrema Revisited -- Geometric Application of Limits and Derivatives -- Mean Value Theorems -- L'Hôpital's Rule and Indeterminate Forms -- Continuity and Derivatives (Optional) -- Integration -- Area and the Definition of the Integral -- General Theory of the Riemann Integral -- The Fundamental Theorem of Calculus -- Formal Integration I: Manipulation Formulas -- Formal Integration II: Trigonometric Functions -- Formal Integration III: Rational Functions and Partial Fractions -- Improper Integrals -- Geometric Applications of Riemann Sums -- Riemann's Characterization of Integrable Functions (Optional) -- Power Series -- Local Approximation of Functions by Polynomials -- Convergence of Series -- Unconditional Convergence -- Convergence of Power Series -- Handling Power Series -- Complex Numbers and Analytic Functions (Optional) -- The Rhetoric of Mathematics (Methods of Proof) -- Answers to Selected Problems -- Bibliography -- Index -- About the Author. Back cover.
Calculus Deconstructed is a thorough and mathematically rigorous exposition of single-variable calculus for readers with some previous exposure to calculus techniques but not to methods of proof. This book is appropriate for a beginning Honors Calculus course assuming high school calculus or a "bridge course" using basic analysis to motivate and illustrate mathematical rigor. It can serve as a combination textbook and reference book for individual self-study. Standard topics and techniques in single-variable calculus are presented in context of a coherent logical structure, building on familiar properties of real numbers and teaching methods of proof by example along the way. Numerous examples reinforce both practical and theoretical understanding, and extensive historical notes explore the arguments of the originators of the subject.No previous experience with mathematical proof is assumed: rhetorical strategies and techniques of proof (reductio ad absurdum, induction, contrapositives, etc.) are introduced by example along the way. Between the text and exercises, proofs are available for all the basic results of calculus for functions of one real variable.
9781614446026
Calculus -- Textbooks.
Calculus -- Problems, exercises, etc.
Electronic books.
QA303.2.N58 2009eb
515
MAA Textbooks : Calculus Deconstructed: a Second Course in First-Year Calculus. - 1st ed. - 1 online resource (509 pages) - AMS/MAA Textbooks Series ; v.16 . - AMS/MAA Textbooks Series .
cover -- copyright page -- title page -- Preface -- Contents -- Precalculus -- Numbers and Notation in History -- Numbers, Points, and the Algebra of Inequalities -- Intervals -- Sequences and their Limits -- Real Sequences -- Limits of Real Sequences -- Convergence to Unknown Limits -- Finding Limits -- Bounded Sets -- The Bisection Algorithm (Optional) -- Continuity -- Continuous Functions -- The Intermediate Value Theorem and Inverse Functions -- Extreme Values and Bounds for Functions -- Limits of Functions -- Discontinuities -- Exponentials and Logarithms -- Epsilons and Deltas (Optional) -- Differentiation -- Slope, Speed and Tangents -- Formal Differentiation I: The Algebra of Derivatives -- Formal Differentiation II: Exponentials and Logarithms -- Formal Differentiation III: Inverse Functions -- Formal Differentiation IV: The Chain Rule and Implicit Differentiation -- Related Rates -- Extrema Revisited -- Geometric Application of Limits and Derivatives -- Mean Value Theorems -- L'Hôpital's Rule and Indeterminate Forms -- Continuity and Derivatives (Optional) -- Integration -- Area and the Definition of the Integral -- General Theory of the Riemann Integral -- The Fundamental Theorem of Calculus -- Formal Integration I: Manipulation Formulas -- Formal Integration II: Trigonometric Functions -- Formal Integration III: Rational Functions and Partial Fractions -- Improper Integrals -- Geometric Applications of Riemann Sums -- Riemann's Characterization of Integrable Functions (Optional) -- Power Series -- Local Approximation of Functions by Polynomials -- Convergence of Series -- Unconditional Convergence -- Convergence of Power Series -- Handling Power Series -- Complex Numbers and Analytic Functions (Optional) -- The Rhetoric of Mathematics (Methods of Proof) -- Answers to Selected Problems -- Bibliography -- Index -- About the Author. Back cover.
Calculus Deconstructed is a thorough and mathematically rigorous exposition of single-variable calculus for readers with some previous exposure to calculus techniques but not to methods of proof. This book is appropriate for a beginning Honors Calculus course assuming high school calculus or a "bridge course" using basic analysis to motivate and illustrate mathematical rigor. It can serve as a combination textbook and reference book for individual self-study. Standard topics and techniques in single-variable calculus are presented in context of a coherent logical structure, building on familiar properties of real numbers and teaching methods of proof by example along the way. Numerous examples reinforce both practical and theoretical understanding, and extensive historical notes explore the arguments of the originators of the subject.No previous experience with mathematical proof is assumed: rhetorical strategies and techniques of proof (reductio ad absurdum, induction, contrapositives, etc.) are introduced by example along the way. Between the text and exercises, proofs are available for all the basic results of calculus for functions of one real variable.
9781614446026
Calculus -- Textbooks.
Calculus -- Problems, exercises, etc.
Electronic books.
QA303.2.N58 2009eb
515