Calculus for Cranks, 2021

Calculus for Cranks, 2021.
 
   This course, Calculus for Cranks, will be somewhat unusual. It is a proof-based treatment of calculus, intended for those who have already demonstrated a strong grounding in the calculus, as treated in a high school course. You might ask, “What’s the point?” “Will I learn anything new?” “Will it be at all useful for applied work?” “Are formal proofs just voodoo that has no impact on ’the right answers?” Mathematicians usually defend a course like this in a philosophical vein. We are teaching you how to think and the ability to think precisely and rigorously is valuable in whatever held you pursue. There is some truth in this idea, but we must be humble in its application and admit that the value of being able to think does depend somewhat on what is being thought about. You will be learning to think about analysis, the theoretical underpinning of calculus. Is that worth thinking about?




Induction for derivatives of polynomials.
from Theorem 1.2.3 by using the definition of the composition of polynomials to break any composition into a sum of compositions with individual powers. We‘ll leave this as an exercise, but we won’t actually need more than Theorem 1.2.3 for our purposes.

A remark: This has been a really horrible way of proving the product rule and chain ride. It only works for polynomials. And it makes the whole subject appear as if it is a kind of list of random identities. The moral is that the definition of the derivative and the proof of the rules of differentiation from the definition are useful even if the only functions you will ever differentiate are polynomials, because they make the subject more conceptual. We will cover that in detail later.

Contents.
Preface.
1: Induction and the Real Numbers.
1.1 Induction.
1.2 Induction for derivatives of polynomials.
1.3 The real numbers.
1.4 Limits.
2: Sequences and Series.
2.1 Cauchy sequences and the Bolzano-Weierstrass and Squeeze theorems.
2.2 In nite series.
2.3 Power series.
3: Functions and Derivatives.
3.1 Continuity and limits.
3.2 Derivatives.
3.3 Mean Value Theorem.
3.4 Applications of the Mean Value Theorem.
3.5 Exponentiation.
3.6 Smoothness and series.
3.7 Inverse function theorem.
4: Integration.
4.1 De nition of the Riemann integral.
4.2 Integration and uniform continuity.
4.3 The fundamental theorem.
4.4 Taylor's theorem with remainder.
4.5 Numerical integration.
5: Convexity.
5.1 Convexity and optimization.
5.2 Inequalities.
5.3 Economics.
6: Trigonometry, Complex Numbers, and Power Series.
6.1 Trigonometric functions by arclength.
6.2 Complex numbers.
6.3 Power series as functions.
7: Complex Analysis.
7.1 Roots of polynomials.
7.2 Analytic functions.
7.3 Cauchy's theorem.
7.4 Taylor series of analytic functions.
A: Appendix.
A.1 How to Use This Book.
A.2 Logic, Axioms, and Problem Solving.
A.3 Further Reading.
Index.



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