Calculus without Calculus

Published by Newman University on

Sums on board
With exams fast approaching, Newman University’s Senior Lecturer and Course Leader in Mathematics, Andrew Toon discuses a simple idea to help our Mathematics students determine the derivative of any function using the algebra of what are called dual numbers instead of using the many rules of differentiation.

“As some of you may be worried about your calculus skills with exams quickly approaching, I thought I would share with you a simple idea to determine the derivative of any function using the algebra of what are called dual numbers instead of using the many rules of differentiation.

 

“Some of you may be familiar with complex numbers Numbers which can be manipulated using the usual rules of algebra with the understanding that Sum, but don’t worry if you are not!

 

“In a similar way, we can define what are called dual numbers which take the form Sum, where Sum are real numbers but now Sum has the strange property that Sum. Just like the Sum in complex numbers which is understood in terms of its square , we understand and apply dual numbers by requiring  whenever it appears.

“For example

“Also, using Taylor series, we have for example

Sum

“This simple madness allows us to express any function  Sum that has a Taylor series expansion (whose derivatives exist to all orders) in the form:

Sum

“From this we can determine Sum which is simply all terms that Sum once we have fully simplified Sum and used the understanding that  .

 

“Let’s consider a couple of examples:

Sum

However, using dual numbers (and remembering Sum) we have

“Comparing with Sum  as expected.

 

 using the product rule.
“However, using dual numbers (and remembering Sum) we have using the above results

.Sum

 

“Comparing with Sum as expected.

“So, using dual numbers of the form Sum with the understanding that Sum  we can reduce differential calculus to the algebra of such numbers! This gives us a powerful, easy and effective way to evaluate derivatives no matter how complex without needing to know all those many rules of differentiation by using algebra only!

 

“As a final note, you might be wondering how can there be non-zero quantities such that Sum for complex numbers and  (remember Sum) for dual numbers? What you are in fact seeking is a representation of these quantities that makes sense and this is precisely what a mathematics degree will show you, so, do come and join us in this fascinating world!

“I hope you found this useful and interesting, good luck with your exams and happy studies!”
Categories: Mathematics

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