Question

If f(s) = „’f(t) and n = 1, 2, 3, . . . , then „’tnf(t) = (-1)ndndsnf(s). Evaluate the given Laplace transform. Write your answer as a function of s.

Answer 1

To evaluate the given Laplace transform „’tnf(t) = (-1)ndndsnf(s), we can use the property of the Laplace transform. Substitute n = 1 into the expression to find the Laplace transform of „’t f(t).

Step-by-step explanation:

To evaluate the given Laplace transform, we'll use the property of the Laplace transform that states if f(s) = f(t), then the Laplace transform of f(t) is F(s). In this case, we have f(s) = f(t), so the Laplace transform of f(t) is F(s). Now, we'll substitute n = 1 into the given expression „’tnf(t) = (-1)ndndsnf(s). This gives us „’t f(t) = (-1)d1d1sf(s).

Now, let's evaluate the Laplace transform of „’t f(t). Take the derivative of F(s) with respect to s to get F'(s), and then multiply it by (-1) to get the Laplace transform of „’t f(t) as (-1)F'(s). Therefore, the given Laplace transform is (-1)F'(s).