find the rms value of the function
i=15(1-e^-1/2t) from t=0 to t=4
If we let r = rms of f(t), then
4r^2 = Int(15 - 15e^-1/2t)[0,4]
= 15t + 30e^(-t/2)[0,4]
= [60 + 30/e^2] - [0 + 30]
= 60 + 30/e^2 - 30
= 30 + 30/e^2
=34.06
so, r = 2.92
see wikipedia on root mean square
To find the RMS (Root Mean Square) value of a function, you'll need to follow these steps:
Step 1: Square the function.
Step 2: Integrate the squared function over the specified interval.
Step 3: Divide the result by the length of the interval.
Step 4: Take the square root of the quotient obtained in step 3.
Let's apply these steps to find the RMS value of the given function:
Step 1: Square the function.
i² = 15²(1 - e^(-1/2t))^2
Step 2: Integrate the squared function over the specified interval.
We'll integrate i² from t = 0 to t = 4:
∫[0,4] 15²(1 - e^(-1/2t))^2 dt
Since solving this integral analytically might be quite complex, we can use numerical methods to approximate it. One common method is the trapezoidal rule. Here's an example of how you could use it:
Divide the interval [0,4] into smaller segments. Let's say we divide it into n segments. The width of each segment, Δt, would be 4/n.
Now, our integral becomes:
∫[0,4] 15²(1 - e^(-1/2t))^2 dt ≈ Δt/2 * [15²(1 - e^(-1/2(0)))² + 2 * ∑[i=1 to n-1] 15²(1 - e^(-1/2(ti)))² + 15²(1 - e^(-1/2(4)))²]
Step 3: Divide the result by the length of the interval.
Since our interval is from 0 to 4, the length of the interval is 4.
Thus, the RMS value would be:
RMS = sqrt( (∫[0,4] 15²(1 - e^(-1/2t))^2 dt) / 4 )
You can use numerical methods or software tools like Python libraries (e.g., scipy) to evaluate the integral and compute the RMS value.