Honors Calculus

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"The document you are currently reading began its life as the lecture notes for a year long undergraduate course in honors calculus." Contents Foreword Spivak and Me What is Honors Calculus? Some Features of the Text Chapter 1: Introduction and Foundations 1. Introduction 1.1. The Goal: Calculus Made Rigorous. 1.2. Numbers of Various Kinds. 1.3. Why do we not do calculus on ℚ? 2. Some Properties of Numbers 2.1. Axioms for a Field. 2.2. Axioms for an ordered field. 2.3. Some further properties of ℚ and ℝ. 2.4. Some Inequalities. Chapter 2: Mathematical Induction 1. Introduction 2. The First Induction Proofs 2.1. The Pedagogically First Induction Proof. 2.2. The (Historically) First(?) Induction Proof. 3. Closed Form Identities 4. Inequalities 5. Extending Binary Properties to n-ary Properties 6. The Principle of Strong/Complete Induction 7. Solving Homogeneous Linear Recurrences 8. The Well-Ordering Principle 9. The Fundamental Theorem of Arithmetic 9.1. Euclid's Lemma and the Fundamental Theorem of Arithmetic. 9.2. Rogers' Inductive Proof of Euclid's Lemma. 9.3. The Lindemann-Zermelo Inductive Proof of FTA. 9.4. A Generalized Euclid's Lemma. Chapter 3: Polynomial and Rational Functions 1. Polynomial Functions 2. Rational Functions 2.1. The Partial Fractions Decomposition. Chapter 4: Continuity and Limits 1. Remarks on the Early History of the Calculus 2. Derivatives Without a Careful Definition of Limits 3. Limits in Terms of Continuity 4. Continuity Done Right 4.1. The formal definition of continuity. 4.2. Basic properties of continuous functions. 5. Limits Done Right 5.1. The Formal Definition of a Limit. 5.2. Basic Properties of Limits. 5.3. The Squeeze Theorem and the Switching Theorem. 5.4. Variations on the Limit Concept. Chapter 5: Differentiation 1. Differentiability Versus Continuity 2. Differentiation Rules 2.1. Linearity of the Derivative. 2.2. Product Rule(s). 3. Optimization 3.1. Intervals and interior points. 3.2. Functions increasing or decreasing at a point. 3.3. Extreme Values. 3.4. Local Extrema and a Procedure for Optimization. 4. The Mean Value Theorem 4.1. Statement of the Mean Value Theorem. 4.2. Proof of the Mean Value Theorem. 4.3. The Cauchy Mean Value Theorem. 5. Monotone Functions 5.1. The Monotone Function Theorems. 5.2. The First Derivative Test. 5.3. The Second Derivative Test. 5.4. Sign analysis and graphing. 5.5. A Theorem of Spivak. 6. Inverse Functions I: Theory 6.1. Review of inverse functions. 6.2. The Interval Image Theorem. 6.3. Monotone Functions and Invertibility. 6.4. Inverses of Continuous Functions. 6.5. Inverses of Differentiable Functions. 7. Inverse Functions II: Examples and Applications 7.1. x⁻ⁿ 7.2. L(x) and E(x). 7.3. Some inverse trigonometric functions. 8. Some Complements Chapter 6: Completeness 1. Dedekind Completeness 1.1. Introducing (LUB) and (GLB). 1.2. Calisthenics With Sup and Inf. 1.3. The Extended Real Numbers. 2. Intervals and the Intermediate Value Theorem 2.1. Convex subsets of ℝ. 2.2. The Strong Intermediate Value Theorem. 2.3. The Intermediate Value Theorem Implies Dedekind Completeness. 3. The Monotone Jump Theorem 4. Real Induction 5. The Extreme Value Theorem 6. The Heine-Borel Theorem 7. Uniform Continuity 7.1. The Definition; Key Examples. 7.2. The Uniform Continuity Theorem. 8. The Bolzano-Weierstrass Theorem For Subsets 9. Tarski's Fixed Point Theorem Chapter 7: Differential Miscellany 1. L'Hôpital's Rule 2. Newton's Method 2.1. Introducing Newton's Method. 2.2. A Babylonian Algorithm. 2.3. Questioning Newton's Method. 2.4. Introducing Infinite Sequences. 2.5. Contractions and Fixed Points. 2.6. Convergence of Newton's Method. 2.7. Quadratic Convergence of Newton's Method. 2.8. An example of nonconvergence of Newton's Method. 3. Convex Functions 3.1. Convex subsets of Euclidean n-space. 3.2. Goals. 3.3. Epigraphs. 3.4. Secant-graph, three-secant and two-secant inequalities. 3.5. Continuity properties of convex functions. 3.6. Differentiable convex functions. 3.7. An extremal property of convex functions. 3.8. Supporting lines and differentiability. 3.9. Jensen's Inequality. 3.10. Some applications of Jensen's Inequality. Chapter 8: Integration 1. The Fundamental Theorem of Calculus 2. Building the Definite Integral 2.1. Upper and Lower Sums. 2.2. Darboux Integrability. 2.3. Verification of the Axioms. 2.4. An Inductive Proof of the Integrability of Continuous Functions. 3. Further Results on Integration 3.1. The oscillation. 3.2. Discontinuities of Darboux Integrable Functions. 3.3. A supplement to the Fundamental Theorem of Calculus. 3.4. New Integrable Functions From Old. 4. Riemann Sums, Dicing, and the Riemann Integral 4.1. Riemann sums. 4.2. Dicing. 4.3. The Riemann Integral. 5. Lesbesgue's Theorem 5.1. Statement of Lebesgue's Theorem. 5.2. Preliminaries on Content Zero. 5.3. Proof of Lebesgue's Theorem. 6. Improper Integrals 6.1. Basic definitions and first examples. 6.2. Non-negative functions. 7. Some Complements Chapter 9: Integral Miscellany 1. The Mean Value Therem for Integrals 2. Some Antidifferentiation Techniques 2.1. Change of Variables. 2.2. Integration By Parts. 2.3. Integration of Rational Functions. 3. Approximate Integration 4. Integral Inequalities 5. The Riemann-Lebesgue Lemma Chapter 10: Infinite Sequences 1. Summation by Parts 2. Easy Facts 3. Characterizing Continuity 4. Monotone Sequences 5. Subsequences 6. The Bolzano-Weierstrass Theorem For Sequences 6.1. The Rising Sun Lemma. 6.2. Bolzano-Weierstrass for Sequences. 6.3. Supplements to Bolzano-Weierstrass. 6.4. Applications of Bolzano-Weierstrass. 6.5. Bolzano-Weierstrass for Subsets Revisited. 7. Partial Limits; Limits Superior and Inferior 7.1. Partial Limits. 7.2. The Limit Supremum and Limit Infimum. 8. Cauchy Sequences 8.1. Motivating Cauchy sequences. 8.2. Cauchy sequences. 9. Geometric Sequences and Series 10. Contraction Mappings Revisited 10.1. Preliminaries. 10.2. Some Easy Results. 10.3. The Contraction Mapping Theorem. 10.4. Further Attraction Theorems. 10.5. Fixed Point Theorems in Metric Spaces. 11. Extending Continuous Functions Chapter 11: Infinite Series 1. Introduction 1.1. Zeno Comes Alive: a historico-philosophical introduction. 1.2. Telescoping Series. 2. Basic Operations on Series 2.1. The Nth Term Test. 2.2. The Cauchy Criterion. 3. Series With Non-Negative Terms I: Comparison 3.1. The sum is the supremum. 3.2. The Comparison Test. 3.3. The Delayed Comparison Test. 3.4. The Limit Comparison Test. 3.5. Cauchy products I: non-negative terms. 4. Series With Non-Negative Terms II: Condensation and Integration 4.1. The Harmonic Series. 4.2. The Condensation Test. 4.3. The Integral Test. 5. Series With Non-Negative Terms III: Ratios and Roots 5.1. The Ratio Test. 5.2. The Root Test. 5.3. Ratios versus Roots. 6. Absolute Convergence 6.1. Introduction to absolute convergence. 6.2. Cauchy products II: Mertens's Theorem. 7. Non-Absolute Convergence 7.1. The Alternating Series Test. 7.2. Dirichlet's Test. 7.3. Cauchy Products III: A Divergent Cauchy Product. 7.4. Decomposition into positive and negative parts. 8. Power Series I: Power Series as Series 8.1. Convergence of Power Series. Chapter 12: Taylor Taylor Taylor Taylor 1. Taylor Polynomials 2. Taylor's Theorem Without Remainder 3. Taylor's Theorem With Remainder 4. Taylor Series 4.1. The Taylor Series. 4.2. Easy Examples. 4.3. The Binomial Series. 5. Hermite Interpolation Chapter 13: Sequences and Series of Functions 1. Pointwise Convergence 1.1. Pointwise convergence: cautionary tales. 2. Uniform Convergence 2.1. Uniform Convergence and Inherited Properties. 2.2. The Weierstrass M-test. 3. Power Series II: Power Series as (Wonderful) Functions Chapter 14: Serial Miscellany 1. ∑ₙ₌₁∞ n⁻² = π²/6 2. Rearrangements and Unordered Summation 2.1. The Prospect of Rearrangement. 2.2. The Rearrangement Theorems of Weierstrass and Riemann. 2.3. Unordered summation. 3. Abel's Theorem 3.1. Statement and Proof. 3.2. An Application to the Cauchy Product. 3.3. Two Identities Justified By Abel's Theorem. 4. The Peano-Borel Theorem 4.1. Statement. 4.2. Proof. 5. The Weierstrass Approximation Theorem 5.1. Statement of Weierstrass Approximation. 5.2. Piecewise Linear Approximation. 5.3. A Very Special Case. 5.4. Proof of the Weierstrass Approximation Theorem. 6. A Continuous, Nowhere Dierentiable Function 7. The Gamma Function 7.1. Definition and Basic Properties. 7.2. The Beta Function. 7.3. Interlude: A Dominated Convergence Theorem. 7.4. Stirling's Formula. Chapter 15: Several Real Variables and Complex Numbers 1. A Crash Course in the Honors Calculus of Several Variables 2. Complex Numbers and Complex Series 3. Elementary Functions Over the Complex Numbers 3.1. The complex exponential function. 3.2. The trigonometric functions. 4. The Fundamental Theorem of Algebra 4.1. The Statement and Some Consequences. 4.2. Proof of the Fundamental Theorem of Algebra. Chapter 16: Foundations Revisited 1. Ordered Fields 1.1. Basic Definitions. 1.2. Some Topology of Ordered Fields. 1.3. A Non-Archimedean Ordered Field. 2. The Sequential Completion 2.1. Sequentially Complete Fields. 2.2. Sequential Completion I: Statement and Applications. 2.3. Sequential Completion II: The Proof. Bibliography