Properties of Laplace Transformations - One Line Questions
1.
If L{f(t)} = F(s), then the Laplace transform of t*f(t) is: —
-dF(s)/ds
2.
The Laplace transform of t*f(t) is given by: —
-dF(s)/ds
3.
If L{f(t)} = F(s), then L{t*f(t)} is given by: —
-dF(s)/ds
4.
If L{f(t)} = F(s), then L{t^n f(t)} is given by: —
(-1)^n * d^n/ds^n [F(s)]
5.
The Laplace transform of a periodic function f(t) with period T is given by: —
(1 / (1 - e^(-sT))) * ∫₀ᵀ e^(-st) f(t) dt
6.
If L{f(t)} = F(s), what is L{f(at)}? —
(1/a)F(s/a)
7.
If L{f(t)} = F(s), then L{f(at)} is: —
(1/a)F(s/a)
8.
What is the Laplace transform of e^(at)cos(ωt)? —
(s-a)/((s-a)^2 + ω^2)
9.
If L{f(t)} = F(s), then L{f(t)/t} is: —
∫<0xE2><0x82><0x9B> F(σ) dσ
10.
If L{f(t)} = F(s), then L{f(t)/t} is given by: —
∫<0xE2><0x82><0x9B> F(σ) dσ
11.
What is the Laplace transform of the Dirac delta function δ(t)? —
1
12.
The Laplace transform of e^(at) is: —
1/(s-a)
13.
What is the Laplace transform of t*e^(at)? —
1/(s-a)^2
14.
What is the Laplace transform of e^(-at)? —
1/(s+a)
15.
If f(t) is a unit step function, L{f(t)} is: —
1/s
16.
What is the Laplace transform of t? —
1/s^2
17.
The Laplace transform of the unit ramp function t is: —
1/s^2
18.
The Laplace transform of t^2 is: —
2/s^3
19.
What is the Laplace transform of t^3? —
6/s^4
20.
L{sinh(at)} is equal to: —
a/(s^2 - a^2)
21.
What is the Laplace transform of a constant 'a', i.e., L{a}? —
a/s
22.
The property L{f(t)/t} = ∫<0xE2><0x82><0x9B> F(σ) dσ, where F(s) = L{f(t)}, is known as: —
Division by t Property
23.
If L{f(t)} = F(s), what is the Laplace transform of f(t-a)u(t-a)? —
e^(-as)F(s)
24.
If L{f(t)} = F(s), then L{f(t-a)u(t-a)} is: —
e^(-as)F(s)
25.
The second shifting theorem states that if L{f(t)} = F(s), then L{f(t-a)u(t-a)} equals: —
e^(-as)F(s)
26.
If L{f(t)} = F(s), what is the Laplace transform of e^(at)f(t)? —
F(s-a)
27.
The Laplace transform of the integral of a function, L{∫₀ᵗ f(τ) dτ}, is related to L{f(t)} = F(s) by: —
F(s)/s
28.
If L{f(t)} = F(s), then L{∫₀ᵗ f(τ) dτ} is: —
F(s)/s
29.
If L{f(t)} = F(s), then L{∫₀ᵗ f(τ) dτ} is: —
F(s)/s
30.
The Convolution Theorem states that if L{f(t)} = F(s) and L{g(t)} = G(s), then L{f(t) * g(t)} (where * denotes convolution) is: —
F(s)G(s)
31.
The Laplace transform of the convolution of two functions f(t) and g(t) is: —
F(s)G(s)
32.
The property that states L{f(t-a)u(t-a)} = e^(-as)F(s) is known as: —
Second Shifting Theorem
33.
The property L{e^(at)f(t)} = F(s-a) is known as the: —
First Shifting Theorem
34.
The property relating L{f(t)} = F(s) to L{e^(at)f(t)} is the: —
First Shifting Theorem
35.
The Laplace transform of the integral ∫₀ᵗ g(τ) dτ is: —
G(s)/s
36.
The property L{f(t)/t} = ∫<0xE2><0x82><0x9B> F(σ) dσ is valid when: —
lim<0xE2><0x82><0x9B>→∞ F(σ) exists and is 0
37.
The Laplace transform of t*sin(ωt) can be found using the property: —
Multiplication by t
38.
The Laplace transform of t*e^(at) is found using: —
Multiplication by t property and First Shifting Theorem
39.
The Laplace transform of t^n, where n is a non-negative integer, is: —
n!/s^(n+1)
40.
What is L{t^n}? (n is a non-negative integer) —
n!/s^(n+1)
41.
The property that L{∫₀ᵗ f(τ) dτ} = F(s)/s implies that the Laplace transform of an integral corresponds to division by: —
s
42.
The property L{∫₀ᵗ f(τ) dτ} = F(s)/s means that integration in the time domain corresponds to division by ____ in the s-domain. —
s
43.
What is the Laplace transform of f''(t) given L{f(t)} = F(s)? —
s^2F(s) - sf(0) - f'(0)
44.
If L{f(t)} = F(s), what is L{f''(t)}? —
s^2F(s) - sf(0) - f'(0)
45.
The Laplace transform of the hyperbolic cosine, cosh(at), is: —
s/(s^2 - a^2)
46.
What is the Laplace transform of cosh(at)? —
s/(s^2 - a^2)
47.
L{cos(ωt)} is equal to: —
s/(s^2 + ω^2)
48.
If L{f(t)} = F(s), what is the Laplace transform of f'(t)? —
sF(s) - f(0)
49.
What is the Laplace transform of the derivative of f(t), f'(t), given the initial condition f(0)? —
sF(s) - f(0)
50.
If L{f(t)} = F(s), what is L{f'(t)}? —
sF(s) - f(0)