Timeless problems

After a late night Christmas party and the usual early morning presents by the tree, this Christmas was slightly different.

My wife’s cousin, who just came to US a couple of days ago, is with us for the weekend before heading to Florida on Sunday for his Masters in Aeronautics. I happened to notice his book on “Flight Stability and Automatic Control” and was drawn to the calculus - derivatives and integration used in the text book. And, it soon lead to reminiscing about some math problems.

I am usually intrigued by Math problems that younger students ask as it allows me an opportunity to indulge in a common language despite the age difference! He posed the following.

1. Is it possible for velocity of an object to be 0 and acceleration to be non-zero ?
2. What is larger - x or 1/x ?
3. A vehicle moves 1Km at 30 Kmph from A to B. What velocity would it have to move from B to C (another 1km distance) so the average velocity from A to C is 60 Kmph ?
4. What is the derivative of Sqrt(2+sqrt(2+sqrt(2+sqrt(2+..…  infinite series
5. And then, once he assessed me with that, he got on to the real one.. derivative of Sqrt(x+sqrt(x+sqrt(x+sqrt(x+..… infinite series

I don’t consider myself a math whiz, but take pride in my interest in basic math and like solving problems like these from time to time. After answering all of them, I must have gained the right to ask and it was my turn.

I am not particularly good at recalling problems from my memory, with the exception of this one below. That was because I created this problem myself, as I was learning Integration around 12^th grade – almost a quarter century ago!

I asked him - which of the following, if any,  is/are integrable ? And, if they are, solve them.

1. 1/(1+x^4)
2. x/(1+x^4)
3. x^2/(1+x^4)
4. x^3/(1+x^4)
5. x^4/(1+x^4)
6. x^5/(1+x^4)

As soon as I said that, my 7 year old daughter who is still struggling in getting her 2 digit additions right – accidentally blurted out the correct answer – “All of them”. She probably figured the pattern in how I quiz her already J. I told her she was the first one who ever answered that question correctly. (and definitely the one to answer without even solving them !)

And, after a good discussion in solving all of them with the 21 year old student, he told me – “You created this problem even before I was born”. That was a profound observation. And, one of the reasons why I love Math problems – the timeless nature of the problems and more importantly – their solutions. Strangely, the solutions provide a sense of stability in this day and age where everything changes at a frantic pace. I cannot say how many other solutions remain the same over such a long period of time!

I look forward to asking the same question in about a decade when my daughter learns calculus. I am sure I will remind her she got it right her first time :)

After writing this article, I prod her – “how did she answer it correctly when I asked her about the “integrable” question?. She says - All of those are “Incredible” – the word she heard !  Math problems and solutions are incredible indeed– with their unvarying answers in this ever-changing world!