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An auxiliary function isn’t a “real” function per se, it’s a placeholder. You usually create an auxiliary function to make calculations easier, or as a “stand in” function for a proof.
Auxiliary Functions in Integration
For example, let’s say you wanted to integrate the function
y = 2x(x2 + 3)70
The function is much easier to integrate if you use an auxiliary function “u” in place of x2 + 3:
This particular use of “u” as an auxiliary function is called u substitution. A similar procedure is integration by parts (the UV rule), which uses a second auxiliary function “v” in addition to u.
Auxiliary Function in Proofs
When constructing a proof, sometimes you want to use an auxiliary function as a means to an end. In other words, inserting one into your proof may help you make a logical leap from “a” to “b”.
Let g: [a, b,] be the real-valued function
Where g is a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b) with
We have g(b) = g(a) = 0, so by Rolle’s Theorem, there exists a number (ξ) ∈ (a, b) such that g′(ξ) = 0, or equivalently:
Note that I’ve introduced a proof here to demonstrate the idea of an auxiliary function. It isn’t intended to be a rigorous proof; If you do want a more comprehensive idea of the workings behind the proof, refer to S.K. Chung’s Basic Calculus, p.275
Chung, S.K.(2007). Basic Calculus. Retrieved September 25, 2020 from: http://www.math.nagoya-u.ac.jp/~richard/teaching/f2016/BasicCalculus.pdf
Fischer, I. Basic Calculus Refresher. Retrieved September 25, 2020 from: http://pages.stat.wisc.edu/~ifischer/calculus.pdf
Stephanie Glen. "Auxiliary Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/auxiliary-function/
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