Let f(t) be a continuous function for t > 0. The Laplace transform is another important method in applications of calculus. It is defined as F(s) = | f(t)e¬stdt for any the real number s for which the integral converges. You may have to determine of s for which the integral converge below. range (1) Find the Laplace transform of f(t) = et. For which values of s does the integral converge? = t. For which values of s does the (2) Find the Laplace transform of f(t) integral converge? (3) Suppose that 0 < f(t) < Meat and 0 < f'(t) < Keat for t > 0, where f'(t) is also continuous and a a real number. If we write G(s) for the Laplace transform of f' (t), and F(s) for the Laplace transform of f(t), show that for this specific choice of f(t), we have G(s) = sF(s) – f (0) for any s > a. Hint: Make sure to evaluate all of these as improper integrals! You will also need to use l'Hopital's rule to evaluate some of the limits.

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