Proposition 2. (Comparison Test for improper integrals) Let f(t) and g(t) be functions on the domain (0,0). Suppose first that f(t) < g(t) for all t>a, and that g(t)dt converges. Then fa f(t)dt converges. Similarly, suppose that f(t) < g(t) for all t>a, and that f f(t)dt diverges. Then g(t)dt diverges. (You should think for a minute about why these should intuitively be true.) Problem 7: A) A function is of exponential type if there exists constants C, M and to with M> 0 not depending on t such that for all t > to, |ƒ(t)| < Met. Prove that if a function f(t) is piece-wise continuous on [0, ∞] and is of exponential type with constants M and C as above, then (L{f})(s) exists for all s > C. (Hint: first, get rid of a "finite" part of the integral, then use the above proposition.) B) Show that f(t) = et² is not of exponential type. (Hint: for any C>0, t² - Ct > 0 for all t> C. p(t) C) Show that for any C> 0 and any polynomial p(t), lim = 0. Note that this implies that p(x) is of exponential type. t-x Problem 8: Show that if f(t) is a function of exponential type (i.e. the Laplace Transform of f(t) actually exists), then (L{f'(t)})(s) = s(L{f})(s) — ƒ(0). (Hint: integration by parts.)