Please see the link below: M320_Weekly_Blog
In class we discussed Darboux’s Theorem and the oddity that it implies. Differentiability of a function does not imply the continuity of its derivative, but it does imply the Intermediate Value Theorem holds for its derivative. For this blog, I would like to explore a function for which this oddity occurs. Consider the function discussed […]
We began class with challenge problem presentations. We introduced Theorem 5.2.5 (Chain Rule) and went over some of its implications. One thing that is important to note about the usage of Chain Rule is that we can control the inputs of a function, specifically so that they are not equal, but we cannot do this […]
Without Loss of Generality In our proof of the Interior Extremum Theorem, we advanced our argument by supposing that in order to find a sign for so that the order limit theorem could be used. In doing this we put a restriction on the element in our proof that is not necessarily true for all […]
Problem: Let be a differentiable function on an interval . Show that if on , then is one-to-one. Provide an example to show that the converse of the statement need not be true. Proof: Suppose is a differentiable function on some interval . Let for some arbitrary . By Theorem 5.2.3, is also continuous on and so […]
Corrections
Proof by Contradiction Suppose is differentiable on an interval with , and has more than 1 fixed point. Let such that are fixed points where . By the definition of fixed point from Weekly HW 6, and . By Theorem 5.2.3, since is differentiable on , then is continuous on . Thus, we can apply the Mean Value Theorem so […]
The Intermediate Value Theorem: In the book, we have the formal definition stating: Let f : [a, b] → R be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c ∈ (a, b) where f(c) = L. To prove […]
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