Here's a problem whose solution was known before 1630. Newton was born in 1642, so his calculus was not used to solve it, but it yields nicely to derivatives. How would you solve it, with or without calculus?
Given a line, divide it into two parts so that the product of the parts will be a maximum.
To see how Fermat solved it, and to see how calculus problems were solved before calculus. look here: https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1984/0025570x.di021131.02p02223.pdf
?ÿ
That sounds too simple.
You could give a junior high student such a challenge along with a pencil and some graph paper and he/she should be able to plot an approximation of the answer.?ÿ It would be symmetrical about the midpoint of the line.?ÿ Using a line with a length of 100 units at one unit and 99 units the product would be 99.?ÿ At 2 units and 98 units the product is 196. At 49 and 51 (or 51 and 49) the product is 2499.?ÿ At 50 and 50 the product is 2500.?ÿ Thus the maximum will occur at the midpoint.
That's too easy, so apparently I don't understand the problem.
You understand it quite well and that's the way I would begin, except that I would use some technology and graph more than one case.
Problem is, these are solutions of specific examples. What mathematicians want is a general solution so that they know for sure that a correct solution can be found.
In this case, assuming that a specific solution is replicable generally is correct, but that's not always the case.
?ÿ
?ÿ
I think here one example proves it for all lengths, as you can just change to different units and the physical location of the maximum does not change.?ÿ Therefore if you scale to a different actual length it must still have the maximum at the midpoint.
@holy-cow?ÿ The paper he linked delves into the beginnings of calculus and it's development over a span of 200 years.?ÿ It's an intentionally simple example that can be solved by non calculus methods.
?ÿ
Edit:?ÿ I see that others chimed in while I was reading all twelve pages of the paper.?ÿ My head hurts now.
Inference vs deduction. Consider the formula n^2 + n + 41. If you substitute any integer from 0 to 39 in the formula, the result will be a prime number. You could infer that every result is prime, but that's not true. It is true for all of the n's from 0 to 39 and for many others, but it's not true for n = 40.
So, how would you determine unequivocally that the answer to the posted problem is half the length of the line?
