Problems

← Back to Fundamentals of Electric Circuits Overview 15.7 Given that F(s) = e⁻2*^s* ∕(s + 1), then f(t) is (a) e⁻2(t−1)u(t − 1) (b) e⁻(t−2)u(t − 2) (c) e^−t u(t − 2) (d) e^−t u(t + 1) (e) e⁻(t−2)u(t)

15.8 The initial value of f(t) with transform

$$F(s) = \frac{s+1}{(s+2)(s+3)}$$

is:

(a) nonexistent	(b) ∞	(c) 0
(d) 1	(e) __1 6

15.9 The inverse Laplace transform of

$$\frac{s+2}{\left(s+2\right)^2+1}$$

Problems

Sections 15.2 and 15.3 Definition and Properties of the Laplace Transform

• 15.1 Find the Laplace transform of 5 sin(at) cos(bt). (Hint: Using the exponential representation for both functions may make this problem easier.)
• 15.2 Determine the Laplace transform of 3.5 cos (5t − 45°).
• 15.3 Obtain the Laplace transform of each of the following functions:

(a) e−2t	(b) e−2t
cos 3tu(t)	sin 4 tu(t)
(c) e−3t	(d) e−4t
cosh 2tu(t)	sinh tu(t)
(e) te−t sin 2tu(t)

• 15.4 Design a problem to help other students better understand how to find the Laplace transform of different time varying functions.
• 15.5 Find the Laplace transform of each of the following functions:
  • (a) t 2 cos(2t + 30°)u(t)

(b)

$$3t^4e^{-2t}u(t)$$

(c)

$$2tu(t) - 4\frac{d}{dt}\delta(t)$$

(d)

$$2e^{-(t-1)}u(t)$$ $$(e) 5u(t/2)$$

(f)

$$6e^{-t/3} u(t)$$

(g) dⁿ ___ dtn δ(t)

is:
\n(a)

$$e^{-t} \cos 2t$$

\n(b) $e^{-t} \sin 2t$
\n(c) $e^{-2t} \cos t$
\n(d) $e^{-2t} \sin 2t$
\n(e) none of the above

15.10 The result of u(t) * u(t) is:

(a) u2 (t)	(b) tu(t)
2 (c) t u(t)	(d) δ(t)

Answers: 15.1b, 15.2a, 15.3d, 15.4d, 15.5a,b,c, 15.6b, 15.7b, 15.8c, 15.9c, 15.10b.

• 15.6 Find G(s) given that g(t) = 2r(t) 2r(t − 2).