mean value theorems
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Short note by Phil dated 10.14.04, prompted by reading about Gaussian quadrature in a Schaum book. It summarizes the first pages of Gen-Bin Huang's paper "Topics in Mean Value Theorems": the MVT for integrals, the generalized weighted version, the classical MVT, and Cauchy's extended MVT, with their equivalences. Sheets A-D give proofs of the Cauchy MVT, a graphic of the classical MVT, the classical MVT from Rolle, and Rolle's Theorem.
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Mean Value Theorems PhL 10.14.04
I encountered these while reading up in my Schaum book about Gaussian Quadrature. I was able to find simple proofs of all items in just a few web pages.
Mr Gen-Bin Huang wrote a 28 page paper called "Topics in Mean Value Theorems" where he studies issues relating to the actual values of the type values, an advanced topic for me. But in his first 6 pages he makes a nice statement of four forms of the Mean Value Theorem, and then shows they are pairwise equivalent. I have kept only these pages of his paper. Here are his four forms of the MVT and his names and numbers:
1.1 the MVT for integrals. This states that the mean value of an integral lies in the range of the function that you are integrating.
1.2 the generalized MVT for integrals. This is similar to 1.1, but there is a positive definite weighting function inside the integral, and the (b-a) of 1.1 is replaced by an integral of the weight function. In other words, 1.1 is a special case of 1.2 where the weight function is unity.
1.3 the classical MVT. This is the situation shown in the picture of sheet B below. There is some point on a curve segment where the slope matches the overall rise over run. This theorem is proven on sheet C using Rolle's Theorem.
1.4 Cauchy's MVT. This one involves two arbitrary functions f and g with the restriction that g' 0 over the domain. I don't have a graphic interpretation of this Cauchy MVT which is also called the Extended MVT. Again, if you set g(x) = x, you find that 1.3 is a special case of 1.4. This theorem is proven on Sheet A.
Now, Mr Huang shows quickly that (1.1) (1.3) and also that (1.2) (1.4). Since we have proofs of 1.3 and 1.4 on sheets C and A, we then have proofs of all four MVT's.
However, the proof of 1.3 on sheet C requires Rolle's Theorem, so we provide on sheet D a proof of Rolle. All these proofs are quite simple.
A Proof of the Cauchy (extended) MVT
B Graphic picture showing the interpretation of the Classical MVT
C proof of the Classical MVT from Rolle
D proof of Rolle's Theorem