12/29/2023 0 Comments Quotient rule calculus proofWhen you find critical number or critical points you are using Fermat’s theorem. Rolle’s theorem makes a major appearance in the MVT and then more or less disappears from the stage. So, Rolle’s theorem could also be called a corollary of Fremat’s theorem. On the other hand, a corollary is a theorem is a result (theorem) that follows easily from the previous theorem. So Fermat’s theorem is a lemma for Rolle’s theorem. So, Fermat’s theorem makes Rolle’s theorem a piece of cake.Ī lemma is a theorem whose result is used in the next theorem and makes it easier to prove. So, by Fermat’s theorem (see this post) the derivative at that point must be zero. The derivative of a constant is zero so any (every, all) value(s) in the open interval qualifies as c.Ĭase II: If the function is not constant then it must have a maximum or minimum in the open interval ( a, b) by the Extreme Value Theorem. (“There exists a number” means that there is at least one such number there may be more than one.)Ĭase I: The function is constant (all of the values of the function are the same as f ( a) and f ( b)). Rolle’s theorem says that if a function is continuous on a closed interval, differentiable on the open interval ( a, b) and if f ( a) = f ( b), then there exists a number c in the open interval ( a, b) such that. But we’ll see an easier way in the next post. It is “legal” to do that, but how do you know to do it? On the other hand, doing things like that is something that has to be done sometimes and students need to know this too. I don’t like this proof because you must know to set up the function h at the beginning. In the final post in this series we will discuss what this all means and how to convince your students of the truth of the MVT without all the symbol pushing that’s required in a proof. On the other hand, we have ended up with a strange equation, which apparently has something to do with mean value, whatever that is. Since I would not like my students not to have any familiarity with proof and definition, I think this is a good place to show them just a little of what it’s all about. The arc from the definition of derivative, through Fermat’s theorem and Rolle’s theorem to the MVT is, I think, a good way to demonstrate how theorems and their proofs work together. So again, we see how one theorem, Rolle’s, leads to another, the MVT. This last equation is very important and will come back in the second act and elsewhere. So we’ll find the derivative and substitute in x = c. Therefore, by Rolle’s theorem there is a number x = c between a and b such that. You can also verify this by substituting first x = a and then x = b into h. In particular, h ( a) = h ( b) = 0 since at the endpoint the two graphs intersect and the distance between them is zero. The function h meets all the conditions of Rolle’s theorem. The equation of the line is in the figure and so we define a new function h( x) = f ( x) – y ( x), this is the vertical distance from f to y. In the figure above we see the graph of f and the graph of the (secant) line, y ( x), between the endpoints of f. The proof, which once you know where to start, is straight forward and rests on Rolle’s theorem. It says a lot more than that which we will consider in the next post. The Mean Value Theorem says that if a function, f, is continuous on a closed interval and differentiable on the open interval ( a, b) then there is a number c in the open interval ( a, b) such that
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