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?
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?
If you graph the function for those N, you will be able to see what the graph does.
Does changing N change the limit?
I am wondering why you are asking others this question: Did you graph it?
But how can you tell the limit?
On the graph? YOu look at the graph as x > limits. Look at the same graph for various N.
To describe the graph of the function f(x) = e^x * sin(Nx) on the interval [0,1], where N is a positive integer, we need to understand the behavior of the function and how it changes as N varies.
When N is a positive integer, the graph of the function will exhibit periodic behavior due to the presence of the sine function. The function will oscillate between positive and negative values, and the frequency of the oscillations will be determined by the value of N. In other words, as N increases, the graph will appear more "compressed" or "squished" horizontally.
Now let's consider the behavior of the graph and the value of the integral as N approaches infinity. As N gets larger and larger, the oscillations of the graph will become more rapid and closely spaced. This implies that the graph will approach a horizontal line, especially near the limits of the interval [0,1]. So, as N approaches infinity, the graph of the function will become closer and closer to the function f(x) = e^x, which is an exponential function.
Regarding the value of the integral, as N increases, the oscillations of the function become more closely spaced, resulting in more positive and negative areas canceling each other out. This leads to the integral approaching a value of zero as N tends to infinity. This behavior confirms the limiting behavior of the integral's value, indicating that as N increases, the net area under the curve becomes smaller and smaller.
Now let's consider the specific cases when N equals 5, 10, and 100:
1. When N = 5: The graph will have five complete oscillations within the interval [0,1]. The function will have a zigzag pattern, with positive and negative regions.
2. When N = 10: The graph will have ten complete oscillations within the interval [0,1]. The function will have a more compressed zigzag pattern compared to the N = 5 case.
3. When N = 100: The graph will have one hundred complete oscillations within the interval [0,1]. The function will appear as a highly compressed zigzag pattern with very rapid oscillations.
Overall, as N increases, the graph becomes more closely packed with oscillations, resulting in a more compressed appearance.
In summary, the graph of the function f(x) = e^x * sin(Nx), when N is a positive integer, exhibits periodic oscillations. As N approaches infinity, the graph approaches a horizontal line and the value of the integral approaches zero. The specific values of N, such as 5, 10, and 100, impact the number of oscillations and the compression of the graph.