Achieving Infinite Resolution_A Gentle Look at the Role of Infinity in Calculus

Description

Contents Preface 1 The Ubiquitous Infinity 1.1 Infinitely many numbers and infinity in numbers 1.2 What might we find at the end of an infinite process? 1.3 Infinity concealed: from areas inside circles to logarithms 1.4 ∞, 0, and nothingness in analysis 2 Sets, Functions and Logic: the building blocks 2.1 Sets and relations 2.2 Functions 2.3 Basic logical concepts and operations 3 Infinities: infinitely many of them! 3.1 Bijections, cardinality and Hilbert's hotel 3.2 An infinity of infinities: Cantor's theorem 3.3 Countable or uncountable? 4 Sequences: taking infinitely many baby steps 4.1 Infinite lists of numbers 4.2 Sequence types and plots 4.3 Convergent sequences: infinite resolution lenses 4.4 Divergent sequences: wandering about, near or far 5 The Real Numbers: mostly irrational, but not lawless 5.1 Rational numbers: a tiny but pervasive minority 5.2 From the rationals to the reals, in baby steps 5.3 The real numbers, at last! 5.4 Characteristics of the set of real numbers: completeness and more 5.5 The uncountability of real numbers: the irrational majority 6 Infinite Series: adding how many numbers? 6.1 A tale of two series and other oddities 6.2 Infinite series as sequence limits 6.3 The geometric series: beauty in simplicity 6.4 Testing for convergence (without calculating the sum) 6.5 The effects of sign changes and Riemann's rearrangement theorem 6.6 The real numbers revisited ... rational, irrational and transcendental 7 Derivatives: changing by infinitely little 7.1 Measuring and calculating the velocity - without speedometers 7.2 Derivative and the tangent line: here comes infinity 7.3 Derivative formulas and higher derivatives 7.4 When derivatives fail to exist: more often than may seem 7.5 Continuity and singularities 7.6 Newton's method: fast convergence with a risk of singularities 7.7 The Mean Value Theorem and the shapes of functions 7.8 What about the ε and δ? 8 Integrals: not just areas 8.1 From acceleration to velocity to position to ... area? 8.2 The integral: partition, add, take limit! 8.3 The Fundamental Theorem of Calculus: opposites annihilate! 8.4 Logarithmic and exponential functions: from areas to exponents 8.5 Numerical approximations of integrals: all you need is a computer 8.6 The improper Riemann integral: infinity made explicit 9 Infinite Series of Functions: the wonders never cease! 9.1 The geometric series as a function series: amazing power at a low cost 9.2 Unexpected encounters with infinity 9.3 Exploring the hidden infinity: a thought experiment 9.4 Infinite sequences of functions: explore infinity's realm 9.5 Infinite series of functions: the magical infinity show! 9.6 Power series and Taylor expansions: transcendental functions explained 9.7 Fine tuning the infinite: l'Hôpital's rule 9.8 Trigonometric series and Fourier expansions 9.9 Continuous yet nowhere differentiable: Koch's snowflake and Weierstrass's function 10 Infinity as the Link between Human Intuition and Reality 10.1 Missing links in the human understanding of nature 10.2 The second missing link and numerical algorithms 10.3 The first missing link: human intuition 10.4 Connecting the components 11 Appendices 11.1 Appendix: Archimedes's area argument and a modern derivation 11.2 Appendix: A Trigonometry refresher 11.3 Appendix: The proof of Cantor's power-set theorem 11.4 Appendix: Cantor's construction of the real numbers 11.5 Appendix: Discontinuity in a space of functions 11.6 Appendix: The derivative formula for sin x 11.7 Appendix: Proofs of some limit and derivative theorems in the text References and Further Reading A-S S-W Index abc def ghi jklmn opqr st uvwz