Why is this?

04072022, 02:12 AM
Post: #1




Why is this?
Likely simple math, but I will ask anyway…I was messing around with graphing various functions. When I graphed x*e^(x^2), I discovered that the max was equal to sin (45 degrees). Why?


04072022, 03:37 AM
Post: #2




RE: Why is this?
(04072022 02:12 AM)lrdheat Wrote: Likely simple math, but I will ask anyway…I was messing around with graphing various functions. When I graphed x*e^(x^2), I discovered that the max was equal to sin (45 degrees). Why? Sin(45^{o}) simply equals Sqrt(2)/2, which holds no mystery whatsoever. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

04072022, 05:05 PM
Post: #3




RE: Why is this?  
04082022, 01:40 AM
Post: #4




RE: Why is this?
Yes, and this is more intuitive to me. Is it purely coincidental that the max
of x*e^(x^2) is (sqrt 2)/2, or is there a direct relationship to the sine function? I’m thinking that I am missing some basic math… 

04082022, 02:12 AM
(This post was last modified: 04082022 02:44 AM by Albert Chan.)
Post: #5




RE: Why is this?
f = x * exp(x^2) is product of linear growth and exponential decay. A peak is expected.
Peak of f also at peak of ln(f) = ln(x)  x^2 (ln(f))' = 1/x  2x = 0, we have x^2 = 1/2, or x = √(1/2)  g = x^(1/x) = exp(ln(x)/x) ln(x) does not grow as fast as x, so again g is decaying after a peak. Peak of g also at peak of ln(g) = ln(x)/x (ln(g))' = (x*(1/x)  1*ln(x)) / x^2 = (1  ln(x)) / x^2 = 0, we have ln(x) = 1, or x = e 

04082022, 02:28 AM
Post: #6




RE: Why is this?
(04082022 01:40 AM)lrdheat Wrote: Yes, and this is more intuitive to me. Is it purely coincidental that the max  The derivative of y = x * e^(x^2) is y' = e^(x^2) * (1  2 * x^2)  To find the extrema of y you must solve y' = 0.  The exponential term is never 0 and the polynomial term 1  2 * x^2 is 0 for x = Sqrt(2) / 2, which is where y has a maximum.  There's no relationship whatsoever to the sine function, just numerical coincidence with the numerical value of sin(45 deg), that's all. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

04082022, 02:32 AM
Post: #7




RE: Why is this?
Yes, it makes sense as far as the idea of an increase and then a decay in these functions. That the max of x*e^(x^2) occurred at x=(sqrt 2)/2 suggested a trigonometric relationship to me, but how? It is beginning to seem coincidental now, and there is not a direct trigonometric relationship…


« Next Oldest  Next Newest »

User(s) browsing this thread: 1 Guest(s)