Question

Answer all the problems please.

(1 point) The graph of f(t) is given below (Click on graph to enlarge) a. Represent f(t) using a combination of Heaviside step functions. Use h(t - a) for the Heaviside function shifted a units horizontally f(t) = help (formulas) b. Find the Laplace transform F(s) = L {f(t)) for s 0. F(s) = L {f(t)) = help (formulas)

Answer 1

(4, 0) (7, a) y - 0/9_3 = x - 4/3 y = 3 (x - 4) f (t) = {0 t lessthanorequalto 3 (t - 4) u lessthanorequalto t lessthanorequalto 7 a t lessthanorequalto t lessthanorequalto 8 a) f (t) = 3 (t - 4) (U_u (t) - u > (t)) + a (u > (t) - u_8 < (t_)) L [f (t)] = 3/s^2 e^-4s - 3/s^2 e^-7s - a e^-7s/s + a e^-7s/s b) = 3/s^2 [e^-4s - e^-7s] - 9 e^-8s/s (2) f (s) = e^-7s/s^2 - 4s + 3s - 12 = e^-7s/(s - 4) (s + 3) = -e^-7s [t/s + 3 - 1/s - 4 s (t) = 1/7 e^-3 (t - 7) u > (t) Therefore f (t) = 1/7 H (t - 7) [e^4 (t - 7) - e^-3 (t - 7)] (2) f (s) = -6 e^4s/s - 4 e^-7s/s - 5 e^-8s/s = -(6 e^-4s + 4 e^-7s + 5 e^-8s)/s