Question

A circle has center that the origin and radius r. What is r when the circle is tangent e^(-x^2) (but no intersection with e^(-x^2))? explain.... I can graph it and estimate it, but I don't know how to do this by hand.

Answer 1

Not sure what tools you have to use, but you might try to minimize the distance from (0,0) to e^(-x^2). That is

z^2 = x^2 + e^(-2x^2)
2z z' = 2x - 4x e^(-2x^2)
Since z will never be zero, you just need

x = 2xe^(-2x^2)
e^(-2x^2) = 1/2
x^2 = ln2/2
So, y^2 = e^(-ln2/2) = 1/2

x^2+y^2 = ln2/2 + 1/2

See the graphs at

http://www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%3Dln2%2F2+%2B1%2F2,+y+%3D+e%5E(-x%5E2)

Answer 2

Thank you