Lista de Exercícios - Função Degrau - Boyce e DiPrima

Lista 8 Função degrau extraídos do livro de Boyce e DiPrima Problems In each of Problems 1 through 4 sketch the graph of the given function on the interval t 0 1 gt ut 2ut 6u1 In each of Problems 5 through 8 a Sketch the graph of the given function b Express ft in terms of the unit step function ut 5 ft 0 0 t 3 2 3 t 5 2 5 t 7 1 t 7 6 ft 0 0 t 1 1 1 t 2 1 2 t 3 3 3 t 4 0 t 4 7 f t 0 t 2 e 7u2 t 2 8 f t 0 t 2 2 2 t 5 7 t 5 t 7 0 t 7 In each of Problems 9 through 12 find the Laplace transform of the given function 9 ft 0 I 2 t 22 t 2 10 ft 0 t t 2π 0 t 2π 11 ft ut 2ut 6u1t 12 ft r 3u7t 1 2u7t In each of Problems 13 through 16 find the inverse Laplace transform of the given function 13 14 15 16 Fs 31 s 24 Fs e2s s2 s 2 Fs 1 s2 1 e2s Fs s2 2s 2 e3s c es 17 Suppose that Fs Lf t exists for s a 0 a Show that if c is a positive constant then Lf tc 1 c F s c s ca b Show that if k is a positive constant then c Show that if a and b are constants with a 0 then 2L 1 Fs b ebac L 1Fks 1 k F L Fta b ebsc a 2 gt f 1 tu1t where t t2 3 gt f 1 3tu1t where f t sin t 4 gt 1 1u1t 21 2u2t 1 3u3t In each of Problems 18 through 20 use the results of Problem 17 find the inverse Laplace transform of the given function 18 19 20 21 22 23 Fs e2 1 s 2 s 1 Fs 2s 1 4s2 4s 5 Fs 923 124 3 In each of Problems 21 through 23 find the Laplace transform of given function In Problem23 assume that termbyterm integrat of the infinite series is permissible ft 0 0 I 1 0 1 1 0 0s I 1 0 1 s 2 0 2 s 3 0 s 3 f r 1 1 ur See Figure 638 t1 FIGURE 638 The function ft in Problem 23 a square wave 24 Let f satisfy ft T ft for all 0 and for so fixed positive number T f is said to be periodic with period T 0 t ω Show that 25 26 In each of Problems 25 through 28 use the result of Problem 24 find the Laplace transform of the given function 25 ft 0 0 t 1 ft 2 ft 0 1 t 2 Compare with Problem 23 26 ft 1 0 t 1 ft 2 ft 1 1 t 2 See Figure 639 27 ft 1 0 t 1 ft 1 ft See Figure 6310 28 ft sin t 0 t π ft π ft See Figure 6311 29 a If fl 1 ut find Lf l Sketch the graph of y f t Compare with Problem 21 b Let gt F t ut where the function f is defined in part a Sketch the graph of y gt and find Lgt Use your expression for Lgt to find an explicit formula for gt Hint See Problem 28 in Section 62 c Let ht g1 gl 2 where g is defined in part b Sketch the graph of y ht and find Lht Use your expression for Lht to find an explicit formula for ht 30 Consider the function p defined by pt 1 0 t 1 pt 2 pt 2 t 1 t 2 a Sketch the graph of y pt b Find Lpt by noting that p is the periodic extension of the function h in Problem 29 then use the result of Problem 24 c Find L pt by noting that pt ft dt where f is the function in Problem 26 then use Theorem 621 FIGURE 639 The function ft in Problem 26 a square wave FIGURE 6310 The function ft in Problem 27 a sawtooth wave FIGURE 6311 The function ft in Problem 28 a rectified sine wave edisciplinasuspbr