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math2400_lecture_1

# math2400_lecture_1 - MATH 2400 LECTURE NOTES PETE L CLARK 1...

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MATH 2400 LECTURE NOTES PETE L. CLARK 1. First Lecture 1.1. The Goal: Calculus Made Rigorous. The goal of this course is to cover the material of single variable calculus in a mathematically rigorous way . The latter phrase is important: in most calculus classes the emphasis is on techniques and applications; while theoretical explana- tions may be given by the instructor – e.g. it is usual to give some discussion of the meaning of a continuous function – the student tests her understanding of the theory mostly or entirely through her ability to apply it to solve problems. This course is very different: not only will theorems and proofs be presented in class by me, but they will also be presented by you, the student, in homework and on exams. This course offers a strong foundation for a student’s future study of mathematics, at the undergraduate level and beyond. As examples, here are three of the fundamental results of calculus; they are called – by me, at least – the three Interval Theorems , because of their common feature: they all concern an arbitrary continuous function defined on a closed, bounded interval. Theorem 1. (Intermediate Value Theorem) Let f : [ a, b ] R be a continuous function defined on a closed, bounded interval. Suppose that f ( a ) < 0 and f ( b ) > 0 . Then there exists c with a < c < b such that f ( c ) = 0 . Theorem 2. (Extreme Value Theorem) Let f : [ a, b ] R be a continuous function defined on a closed, bonuded interval. Then f is bounded and assumes its maximum and minimum values. This means that there exist numbers m M such that a) For all x [ a, b ] , m f ( x ) M . b) There exists at least one x [ a, b ] such that f ( x ) = m . c) There exists at least one x [ a, b ] such that f ( x ) = M . Theorem 3. (Uniform Continuity and Integrability) Let f : [ a, b ] R be a con- tinuous function defined on a closed, bounded interval. Then: a) f is uniformly continuous. 1 b) f is integrable: b a f exists and is finite. Except for the bit about uniform continuity, these three theorems are familiar re- sults from the standard freshman calculus course. Their proofs, however, are not. Most freshman calculus texts like to give at least some proofs, so it is often the case that these three theorems are used to prove even more famous theorems 1 The definition of this is somewhat technical and will be given only later on in the course. Please don’t worry about it for now. 1

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2 PETE L. CLARK in the course, e.g. the Mean Value Theorem and the Fundamental Theorem of Calculus . Why then are the three interval theorems not proved in freshman calculus? Because their proofs depend upon fundamental properties of the real numbers that are not discussed in such courses. Thus one of the necessary tasks of the present course is to give a more penetrating account of the real numbers than you have seen before.
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