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Extra info for Mathematical Background. Foundations of Infinitesimal Calculus
5. Locally Uniform Derivatives Let f [x] and f [x] be real valued functions defined when x is in a real interval (a, b). Then the following are equivalent: (a) Whenever the hyperreal numbers δx and x satisfy δx ≈ 0, x is finite, and a < x < b with neither x ≈ a nor x ≈ b, the natural extension functions satisfy f [x + δx] − f [x] = f [x] · δx + ε · δx (b) (c) for ε ≈ 0. For every accuracy tolerance θ and every real α and β in (a, b), there is a sufficiently small positive real number γ such that if the real number ∆x satisfies 0 < |∆x| < γ and the real number x satisfies α ≤ x ≤ β, then f [x + ∆x] − f [x] − f [x] < θ ∆x For every real c in (a, b), lim x→c,∆x→0 f [x + ∆x] − f [x] = f [c] ∆x That is, for every real c with a < c < b and every real positive θ, there is a real positive γ such that if the real numbers x and ∆x satisfy [x] − f [c]| < θ.
Condition (a) is almost as easy to establish as the intuitive limit computation. 12. The fraction −1/(4(2+ δx)) is finite because 4(2 + δx) ≈ 8 is not infinitesimal. The infinitesimal δx times a finite number is infinitesimal. 1 1 − ≈0 2(2 + δx) 4 1 1 ≈ 2(2 + δx) 4 This is a complete rigorous proof of the limit. 2 shows that the “epsilon - delta” condition (b) holds. 1 1 1. Prove rigorously that the limit lim∆x→0 3(3+∆x) = 19 . 2. 1 √ 2. Prove rigorously that the limit lim∆x→0 √4+∆x+ = 14 . 2. 3.
Then there is a real θ > 0 such that for every γ > 0 there exist x1 and x2 in [a, b] with |x1 − x2 | < γ and |f [x1 ] − f [x2 ]| ≥ θ. 1 to this implication and select a positive infinitesimal γ ≈ 0. Let x1 = X1 [γ], x2 = X2 [γ] and notice that they are in the interval, x1 ≈ x2 , but f [x1 ] is not infinitely close to f [x2 ]. This contradiction shows that the theorem is true. 2 The Extreme Value Theorem Continuous functions attain their max and min on compact intervals. 4. The Extreme Value Theorem If f [x] is a continuous real function on the real compact interval [a, b], then f attains its maximum and minimum, that is, there are real numbers xm and xM such that a ≤ xm ≤ b, a ≤ xM ≤ b, and for all x with a ≤ x ≤ b f [xm ] ≤ f [x] ≤ f [xM ] Intuitive Proof: The Extreme Value Theorem 45 We will show how to locate the maximum, you can find the minimum.
Mathematical Background. Foundations of Infinitesimal Calculus by K.D.Stroyan