Express the functionf(t)=⎧⎩⎨⎪⎪−7sin(6t),2−9t2,0,0≤t<66≤t<8otherwise𝑓(𝑡)={−7sin(6𝑡),0≤𝑡<62−9𝑡2,6≤𝑡<80,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒using Heaviside functions.Note: You must use the notation u(t−c)𝑢(𝑡−𝑐) rather than uc(t)𝑢𝑐(𝑡) in order for your answer to be accepted by Bolster Academy.
Question
Express the functionf(t)=⎧⎩⎨⎪⎪−7sin(6t),2−9t2,0,0≤t<66≤t<8otherwise𝑓(𝑡)={−7sin(6𝑡),0≤𝑡<62−9𝑡2,6≤𝑡<80,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒using Heaviside functions.Note: You must use the notation u(t−c)𝑢(𝑡−𝑐) rather than uc(t)𝑢𝑐(𝑡) in order for your answer to be accepted by Bolster Academy.
Solution
The function f(t) can be expressed using Heaviside functions as follows:
f(t) = -7sin(6t)u(t) + (2-9(t-6)^2)[u(t-6) - u(t-8)]
Here's the step-by-step breakdown:
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The function is divided into three parts according to the given conditions. Each part is multiplied by a Heaviside function (or a difference of two Heaviside functions) that equals 1 in the corresponding interval and 0 otherwise.
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For 0 ≤ t < 6, the function is -7sin(6t). This is multiplied by u(t), which equals 1 for t ≥ 0 and 0 for t < 0.
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For 6 ≤ t < 8, the function is 2-9(t-6)^2. This is multiplied by [u(t-6) - u(t-8)], which equals 1 for 6 ≤ t < 8 and 0 otherwise.
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For t ≥ 8, the function is 0. This is covered by the term -9(t-6)^2*u(t-8), which equals 0 for t ≥ 8.
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The resulting function is the sum of the three parts, which equals the original function in the given intervals.
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