The exponential x (t) = e^ at for t is less than or equal to 0 and zero otherwis

Question

The exponential x (t) = e^ at for t is less than or equal to 0 and zero otherwise is a very common analog signal. Likewise, y[n] = alpha^ n for integers n is less than or equal to 0 and zero otherwise is a very common discrete-time signal. Let us see how they are related. Do the following using MATLAB:
Suppose that a current x (t) = e^-0.5t for t less than or equal to 0 and zero otherwise is applied to a discharge capacitor of capacitance C = 1 F at t = 0. What would be the voltage in the capacitor at t = 1 sec?
x (t) = e^(-0.5t)
t = 0
0 applied to discharge capacitor
C = 1 @ t = 0
C =? @ T = 1
e.) How would you obtain an approximate result to the above problem using a computer? Explain

Explanation / Answer

A comment on time, typically you look for t>0, and 0 otherwise. Okay, the current going through the capacitor is x(t). NOTE: the current through a capacitor is the following: Since C = 1F (or 1), we get: Which implies the following following function: This can be computed in MATLAB by using Simpson's Rule. I don't know if your version of MATLAB has this; if it doesn't, you can Wiki the rule to give you the process, and you can write the M-file. It will discretely calculate the integral from any two points for a contiunous function along the interval (a,b). The time intverval will be from 0 to 1. You'll need at least 4 time increments to calculate (I reccommend 21 points for the more accurate results).

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