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leanprover-community.github.io | ||
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thehousecarpenter.wordpress.com
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| | | | | As usual, this post can be viewed as a PDF. Theorem (Boundedness Theorem). A continuous real-valued function on a closed interval is bounded. Proof. Suppose f is such a function and [a,?b] is its domain. First, observe that for every c???[a,?b], since f is continuous at c, there is a positive $latex \delta \in {\mathbb... | |
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mathscholar.org
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| | | | | [AI summary] The text presents a detailed, self-contained proof of the Fundamental Theorem of Calculus (FTC) using basic principles of calculus and real analysis. It breaks the proof into two parts: Part 1 establishes that the integral of a continuous function defines a differentiable function whose derivative is the original function, and Part 2 shows that the definite integral of a continuous function can be computed as the difference of an antiderivative evaluated at the endpoints. The proof relies on lemmas about continuity, differentiability, and the properties of integrals, avoiding advanced techniques. The text is structured to provide a clear, step-by-step derivation of the FTC for readers familiar with calculus fundamentals. | |
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www.jeremykun.com
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| | | | | Last time we defined and gave some examples of rings. Recapping, a ring is a special kind of group with an additional multiplication operation that "plays nicely" with addition. The important thing to remember is that a ring is intended to remind us arithmetic with integers (though not too much: multiplication in a ring need not be commutative). We proved some basic properties, like zero being unique and negation being well-behaved. | |
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www.pl-enthusiast.net
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| | | This post describes the structure, goals, and content of CMSC 330, UMD's sophomore-level programming languages course. This is part 1. | ||
