The bell curve y=e^(-x^2) and its x-axis reflection y=-e^(-x^2) are plotted on a graph. What is the area of the largest circle that can be placed in between the two curves?
I should have done this before the first post, it is rather interesting and not as hard as I expected.
Every point on the upper bell curve can be written as (x,e^(-x^2)).
Whose distance to the origin as a function of x is d(x)=sqrt(x^2+e^(-2x^2))
Though not too complicated, I used Wolfram|Alpha to get the derivative. It's a fraction (and we don't need the denominator, which is never zero.) The numerator can be written as x*(1-2e^(-2x^2))
Setting this equal to zero and solving gives x=0 (the local max) and x^2 = ln(2)/2 (the local mins)
The circle most be small enough to touch the bell curve at these local mins. d(ln(2)/2) = sqrt(ln(2)/2+1/2)
Taking this as the radius of the circle we can find the area sought:
A = pi*(ln(2)+1)/2 or about 2.659589372
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Posted by Jer
on 2018-12-23 16:30:02 |
Solution with more work.
