1. equation (6), we write Then, from Examples 5 and 7, we obtain L{f(t)} 2. f(t) = 6-t, (1², Problems In each of Problems 1 through 3, sketch the graph of the given function. In each case determine whether f is continuous, piecewise continuous, or neither on the interval 0 ≤ t ≤ 3. 1², f(t)=2+t, (t-1)-¹, 1, t², 1, = L{ f(t)} = 5£{e-²¹} - 3£{sin(41)}. 0 ≤ t ≤ 1 1< t <2 2 0. a. f(t) = t b. f(t) = 12 c. f(t) = t", where n is a positive integer 5. Find the Laplace transform of f(t) = cos(at), where a is a real constant. Recall that In each of Problems 16 through 18, find the Laplace transform of the given function. In each of Problems 6 through 7, use the linearity of the Laplace lov transform to find the Laplace transform of the given function; a and b are real constants. 6. f(t) = cosh(bt) 7. f(t) = sinh(bt) Recall that 1 cos(bt) gibt + e-ibt) and sin(bt) = (eibt - e-ibt). žleibe 1 2i In each of Problems 8 through 11, use the linearity of the Laplace transform to find the Laplace transform of the given function; a and b are real constants. Assume that the necessary elementary integration formulas extend to this case. 16. f(t)= Isitial to go 1≤t<∞0 0 ≤ t < 1 11 < 2 2≤t<∞0 to noituldo, In each of Problems 19 through 21, determine whether the given integral converges or diverges. 3. f(t) = 3-t, 7011 weroo 4. Find the Laplace transform of each of the following functions: no: 19. S (t² + 1)-¹dt 17. f(t) = 8. f(t) = sin(bt) 9. f(t) = cos(bt) 10. f(t) = eat sin(bt) 11. f(t) = eat cos(bt) In each of Problems 12 through 15, use integration by parts to find the Laplace transform of the given function; n is a positive integer and a is a real constant. 18. f(t) = worl = 20. 21. 22. Suppose that f and f' are continuous for t≥ 0 and of exponential order as t→∞. Use integration by parts to show that 1 (eb¹ + e−br) and sinh(bt) = -(e¹¹ - e-bt). be an if F(s) = £[ƒ(1)), then lim F(s) = 0. The result is actually true L{f(t)}, Sintienos no 12: 818 under less restrictive conditions, such as those of Theorem 6.1.2. 1, 0≤t -1. a. Show that, for p > 0, 88 = √²² T(p+1) = b. Show that IT (1) = 1. c. T(p+1) = p (p). e-xxPdx. If p is a positive integer n, show that T(n + 1) = n!. possible to show that I' Since I'(p) is also defined when p is not an integer, this function provides an extension of the factorial function to nonintegral values of the independent variable. Note that it is also consistent to define 0! = 1. d. Show that, for p > 0, p(p+1)(p+2) (p+n-1) = Thus I (p) can be determined for all positive values of p if T (p) is known in a single interval of unit length-say, 0 < p ≤ 1. It is T(p+n) T(p) 3 (1) = √T. Find I (²) 2 and I 11 (1) 2

Laplace transform of f(t) = cos(at), where a is a real +e-b) and sinh(br) = =(e" e-b) oblems 6 through 7, use the linearity of the Laplace nd the Laplace transform of the given function; a and b nts. (0)1-(0)32= (0)13 = (b + cosh(bt) sinh(hr) - 21. 22. expe if F Janon und TOYO 23. T(F TER