04417nam 2200493 450 99649987090331620230508112459.09783031157653(electronic bk.)9783031157646(MiAaPQ)EBC7134134(Au-PeEL)EBL7134134(CKB)25299477500041(OCoLC)1350690436(PPN)266349307(EXLCZ)992529947750004120230321d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierCalculus off the beaten path a journey through its fundamental ideas /Ignacio ZalduendoCham, Switzerland :Springer,[2022]©20221 online resource (225 pages)Springer undergraduate mathematics seriesPrint version: Zalduendo, Ignacio Calculus off the Beaten Path Cham : Springer International Publishing AG,c2022 9783031157646 Includes bibliographical references and index.Intro -- Preface -- Contents -- Introduction -- 2400 Years of Calculus -- Calculus and Education -- 1 The Real Numbers -- The Rational Line -- Density of Q -- Some Basic Notions -- Irrationality of Q -- From Eudoxus to Dedekind -- The Real Line -- Dyadic Series-A Construction of R -- The Scarcity of Q -- The Completeness of R -- Cardinality -- Exercises -- 2 Sequences and Series -- Sequences -- Limits of Sequences -- Cantor's Nested Intervals Theorem -- Subsequences -- Series -- The Harmonic Series -- Series of Positive Terms -- Series with Positive and Negative Terms -- The Riemann Series Theorem -- Absolute and Unconditional Convergence -- Exercises -- 3 Functions -- The Elementary Functions -- Polynomials -- Circular Functions -- The Exponential Function: Bernoulli's Inequality -- Irrationality of e -- Convergence of k=1∞(1+ak) and of k=1∞ak -- Hyperbolic Functions -- Injectivity and Inverse Functions -- Curves in the Plane: Parametrized Curves -- The Cycloid -- Pythagorean Triples -- Continuity -- Bolzano and Weierstrass -- Limits -- Limits in Ancient Greece: The Area of a Circle -- Three Important Limits -- Exercises -- 4 The Derivative -- Derivative -- Tangents -- Newton-Raphson -- Derivatives of the Elementary Functions -- The Chain Rule -- Derivative of the Inverse Function -- The Derivative of a Parametrized Curve -- First Derivative, Tangent Line, and Growth -- The Mean Value Theorems -- L'Hôpital's Rule -- Snell's Law -- The Brachistochrone -- Exercises -- 5 The Integral -- Measure and Integral -- The Fundamental Theorem of Calculus -- A Pause for Comments -- Buffon's Needle -- Irrationality of π -- Improper Integrals -- Integration and Sums: Linearity of the Integral -- Uniform Convergence-The Weierstrass M-Test -- Gregory's Series -- Integration and Products: Integration by Parts -- Stirling's Formula.Integration and Composition: Integration by Substitution -- A Note on Notation -- Length of Curves. The Catenary -- Area Enclosed by a Simple Closed Curve -- Exercises -- 6 More Derivatives -- Second Derivative, Best-Fitting Parabola, and Curvature -- The Taylor Polynomial of Order Two -- Curvature -- Random Walk and the Gauss Curve -- The Taylor Series -- Exercises -- 7 Convexity and the Isoperimetric Inequality -- The Arithmetic-Geometric Inequality -- Convexity -- Young, Hölder, Jensen, Cauchy-Schwarz… -- The Isoperimetric Inequality -- Exercises -- 8 More Integrals -- Volume -- Double Integrals -- The Basel Problem -- Solids of Revolution -- Integration of e -- Density Functions, Barycenter, and Expectation -- Center of Mass or Barycenter -- Pappus' Theorem -- The Method -- Surface Area -- Normal Distribution. Gauss, Laplace, and Stirling -- Exercises -- 9 The Gamma Function -- The Gamma Function -- Weierstrass' Formula -- Growth of the Harmonic Series, Again -- Exercises -- Bibliography -- Index.Springer undergraduate mathematics series.CalculusCàlculthubLlibres electrònicsthubCalculus.Càlcul515Zalduendo Ignacio1265650MiAaPQMiAaPQMiAaPQ996499870903316Calculus off the Beaten Path2968061UNISA