Calculus How To

Auxiliary Function

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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

Auxiliary functions “u” and “v” come in handy when the expression you want to differentiate or integrate is complicated.

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:
auxiliary function

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”.

For example, the following proof of the mean value theorem (Chung, 2007). reaches a conclusion for an auxiliary function g so that Rolle’s theorem can be applied:

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

References

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

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Stephanie Glen. "Auxiliary Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/auxiliary-function/
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