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?????
When you have the function f(x) = e^x * sin(Nx), where N is a positive integer, graphing the function for different values of N allows you to observe and describe how the graph behaves.
To graph the function, you can use a graphing calculator or a software program, or plot the points manually. However, since the interval is from 0 to 1, it's relatively easy to approximate the graph visually.
Here's how you can describe the graph for different values of N:
1. For N = 1:
- The function f(x) = e^x * sin(x) is a sinusoidal curve that oscillates between -e and e.
- The graph starts at (0, 0) and ends at (1, e * sin(1)).
- The curve passes through the x-axis multiple times.
2. For N = 2:
- The function f(x) = e^x * sin(2x) shows two cycles of oscillations within the interval [0, 1].
- The graph still starts at (0, 0) but ends at (1, e * sin(2)).
- The curve oscillates more quickly compared to N = 1.
3. For N = 3:
- The function f(x) = e^x * sin(3x) exhibits three cycles of oscillations within the interval [0, 1].
- The graph still starts at (0, 0) and ends at (1, e * sin(3)).
- The curve oscillates even more quickly compared to N = 2.
As N approaches infinity, the frequency of the oscillations increases infinitely as well. This means that the graph becomes increasingly "wiggly" and the amplitude of the oscillations may vary. The graph seemingly becomes more dense with oscillations.
Regarding the integral's value, as N approaches infinity, the integral converges to a limit. This is because the oscillations between -e and e will balance each other out. The integral over the interval [0, 1] will tend towards 0.
In summary, the graph exhibits more oscillations as N increases, appearing more dense and "wiggly." As N approaches infinity, the integral's value tends to 0, which is confirmed by the increasingly dense oscillations on the graph.