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Express the functionf(t)={4cos(9t),0,7โ‰คt<8otherwise๐‘“(๐‘ก)={4cosโก(9๐‘ก),7โ‰ค๐‘ก<80,๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’using Heaviside functions.Note: You must use the notation u(tโˆ’c)๐‘ข(๐‘กโˆ’๐‘) rather than uc(t)๐‘ข๐‘(๐‘ก) in order for your answer to be accepted by Bolster Academy.

Question

Express the functionf(t)={4cos(9t),0,7โ‰คt<8otherwise๐‘“(๐‘ก)={4cosโก(9๐‘ก),7โ‰ค๐‘ก<80,๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’using Heaviside functions.Note: You must use the notation u(tโˆ’c)๐‘ข(๐‘กโˆ’๐‘) rather than uc(t)๐‘ข๐‘(๐‘ก) in order for your answer to be accepted by Bolster Academy.

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Solution

The Heaviside function, also known as the step function, is defined as:

u(t) = {1 for t >= 0, 0 for t < 0}

We can use this function to express the given function f(t) in terms of Heaviside functions. The given function f(t) is defined as 4cos(9t) for 7 โ‰ค t < 8 and 0 otherwise.

We can express this using Heaviside functions as follows:

f(t) = 4cos(9t) * [u(t - 7) - u(t - 8)]

Here, u(t - 7) is 1 for t >= 7 and 0 for t < 7, and u(t - 8) is 1 for t >= 8 and 0 for t < 8.

So, the term [u(t - 7) - u(t - 8)] is 1 for 7 โ‰ค t < 8 and 0 otherwise, which matches the condition for f(t) to be 4cos(9t).

Therefore, f(t) = 4cos(9t) * [u(t - 7) - u(t - 8)] is the expression of the given function using Heaviside functions.

This problem has been solved

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