# calculus

Posted by Paul on Friday, February 19, 2010 at 3:57am.

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?

* Calculus - Damon, Friday, February 19, 2010 at 10:07am

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

* Calculus - Paul, Wednesday, February 24, 2010 at 11:12am

But how did you arrive at "as N gets big this looks like e^x(-NcosNx)/N^2?????

1. 0
2. 1
1. Professor Damon is describing the integral, not the function.

Have you graphed this function? For various N?

posted by bobpursley
2. Yes I have, but I don't understand why the integral would go to zero. And the description of the interval.

posted by Paul
3. is it because every interval of one the integrals approach 1?

posted by Paul
4. I meant to say zero

posted by Paul
5. But then would this confirm the limiting behavior of the graph because at every interval up to infinity it would approaching zero?

posted by Paul

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