Question

Write a formula for f(t) as a sum of Heaviside functions. Type uc for the Heaviside function that jumps at c

f(t) = t , 0 < t < 2
f(t) = 2t-2, 2 < t < 4
f(t) = 6, 4

compute the Laplace transform F(s)

Answer 1

To write the function f(t) as a sum of Heaviside functions, we need to divide the given range into intervals and express each interval using a Heaviside function. Then, we can compute the Laplace transform F(s) using the properties of the Laplace transform.

Step-by-step explanation:

To write the function f(t) as a sum of Heaviside functions, we need to divide the given range into intervals and express each interval using a Heaviside function.

For 0 < t < 2, f(t) = uc(t-2) * (2t-2)

For 2 < t < 4, f(t) = uc(t-2) * (2t-2) + uc(t-4) * 6

For t > 4, f(t) = uc(t-4) * 6

Now, to compute the Laplace transform F(s), we can use the properties of the Laplace transform. For each Heaviside function, we can apply the corresponding shift property and then take the Laplace transform of the resulting expression.

F(s) = L{f(t)} = L{uc(t-2) * (2t-2)} + L{uc(t-4) * 6}

Using the properties of the Laplace transform and the standard Laplace transforms of functions, we can simplify and compute F(s). Note that uc(t-2) is the unit step function that jumps from 0 to 1 at t = 2, and uc(t-4) is the unit step function that jumps from 0 to 1 at t = 4.