LEADER 04417nam 2200493 450 001 996499870903316 005 20230508112459.0 010 $a9783031157653$b(electronic bk.) 010 $z9783031157646 035 $a(MiAaPQ)EBC7134134 035 $a(Au-PeEL)EBL7134134 035 $a(CKB)25299477500041 035 $a(OCoLC)1350690436 035 $a(PPN)266349307 035 $a(EXLCZ)9925299477500041 100 $a20230321d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCalculus off the beaten path $ea journey through its fundamental ideas /$fIgnacio Zalduendo 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (225 pages) 225 1 $aSpringer undergraduate mathematics series 311 08$aPrint version: Zalduendo, Ignacio Calculus off the Beaten Path Cham : Springer International Publishing AG,c2022 9783031157646 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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 ? -- 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. 327 $aIntegration and Composition: Integration by Substitution -- A Note on Notation -- Length of Curves. The Catenary -- Area Enclosed by a Simple Closed Curve -- Exercises -- 6 More Derivatives -- Second Derivative, Best-Fitting Parabola, and Curvature -- The Taylor Polynomial of Order Two -- Curvature -- Random Walk and the Gauss Curve -- The Taylor Series -- Exercises -- 7 Convexity and the Isoperimetric Inequality -- The Arithmetic-Geometric Inequality -- Convexity -- Young, Hölder, Jensen, Cauchy-Schwarz? -- The Isoperimetric Inequality -- Exercises -- 8 More Integrals -- Volume -- Double Integrals -- The Basel Problem -- Solids of Revolution -- Integration of e -- Density Functions, Barycenter, and Expectation -- Center of Mass or Barycenter -- Pappus' Theorem -- The Method -- Surface Area -- Normal Distribution. Gauss, Laplace, and Stirling -- Exercises -- 9 The Gamma Function -- The Gamma Function -- Weierstrass' Formula -- Growth of the Harmonic Series, Again -- Exercises -- Bibliography -- Index. 410 0$aSpringer undergraduate mathematics series. 606 $aCalculus 606 $aCàlcul$2thub 608 $aLlibres electrònics$2thub 615 0$aCalculus. 615 7$aCàlcul 676 $a515 700 $aZalduendo$b Ignacio$01265650 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a996499870903316 996 $aCalculus off the Beaten Path$92968061 997 $aUNISA