Calculus
posted by Paul on .
I have the function f(x)=e^x*sinNx on the interval [0,1] where N is a positive integer. What does it mean describe the graph of the function when N={whatever integer}? And what happens to the graph and to the value of the integral as N approaches infinity? Does the graph confirm the limiting behavior of the integral's value?

well, e^0 is 1
and e^.5 = 1.64
and e^1 is 2.72
so it is a sine wave with increasing amplitude as you approach 1 and frequency increasing with N
The integral of e^ax sin bx dx is
[e^ax/(a^2+b^2)] [a sin bx b cos bx}here a = 1 and b = N
so
[e^x/(1+N^2)] [sin Nx  N cos Nx]
as N gets big
this looks like
e^x (N cos Nx)/N^2
or
(e^x/N)(cos Nx)
e^x is that small constant and cos Nx ranges between 1 and + 1 so as N gets big this goes to zero like 1/N 
Now how would you describe the graph of this function when say N=5, N=10, and N=100?
And what does it mean does the graph confirm the limiting behavior of the integral's value? 
But how did you arrive at "as N gets big this looks like e^x(NcosNx)/N^2?????