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Definite Integral 3

Directions: Using the digits 1 to 9 at most one time each, place a digit in each box to make a solution that is as close to 100 as possible.



How do the upper and lower bounds determine whether the solution is negative or positive?



I only know how to figure this out via brute force and would love to learn if anyone has a conceptual approach to this.

I created this spreadsheet and tried all 9 potential exponent values. The bounds and exponent that result in a value as close to 100 as possible are:

upper bound: 8
lower bound: 6
exponent: 2
value: 98.666…

and a close second place is:

upper bound: 2
lower bound: 1
exponent: 9
value: 102.30

Source: Robert Kaplinsky

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  1. I created a desmos graph to quickly sift through solutions. I integrated leaving the exponent as a variable, and then used “x” and “y” to be the upper and lower bound and set the whole thing equal to 100. I used the slider to change the exponent “c”, and then found solutions for x and y where x is not equal to y and the points are approximately integral. I came up with exponent 2, upper bound 8, lower bound 6; exponent 3, upper bound 5, lower bound 4; and exponent 9, upper bound 2, lower bound 1. The optimal solution is exponent 2, which is 98.6666666…

    Here is the desmos graph I used: https://goo.gl/QEc25d

  2. One of my students did the integral from 0 to 2.9 for x^5. She wondered if she could use it since the problem didn’t restrict them to integers. 😉

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