HP 50g Graphing Calculator - Manual (Page 490)

Page 16-13
Θ Differentiation theorem for the n-th derivative.
Let f (k)
o
= d k f/dx k |
t = 0
, and f
o
= f(0), then
L{d
n
f/dt
n
} = s
n ⋅F(s) – s n-1 ⋅f
o
−…– s⋅f (n-2)
o
– f
(n-1)
o
.
Θ Linearity theorem. L{af(t)+bg(t)} = a⋅L{f(t)} + b⋅L{g(t)}.
Θ Differentiation theorem for the image function. Let F(s) = L{f(t)}, then d n
F/
ds n = L{(-t) n ⋅f(t)}.
Θ Integration theorem. Let F(s) = L{f(t)}, then
Θ Convolution theorem. Let F(s) = L{f(t)} and G(s) = L{g(t)}, then
Example 2 – As a follow up to Example 1, the acceleration a(t) is defined as
a(t) = d
2
r/dt
2
. If the initial velocity is v
o
= v(0) = dr/dt|
t=0
, then the Laplace
transform of the acceleration can be written as:
A(s) = L{a(t)} = L{d 2 r/dt 2 }= s 2 ⋅R(s) - s⋅r
o
– v
o
.
Example 3 – Let f(t) = e
–at
, using the calculator with ‘EXP(-a*X)’ ` LAP, you
get ‘1/(X+a)’, or F(s) = 1/(s+a). The third derivative of this expression can be
calculated by using:
‘X’ ` ‚¿ ‘X’ `‚¿ ‘X’ ` ‚¿ μ
The result is
‘-6/(X^4+4*a*X^3+6*a^2*X^2+4*a^3*X+a^4)’, or
d
3
F/ds
3
= -6/(s
4
+4 ⋅a⋅s 3
+6 ⋅a 2 ⋅s 2
+4 ⋅a 3 ⋅s+a 4
).
Now, use ‘(-X)^3*EXP(-a*X)’ ` LAP μ. The result is exactly the same.
{ }
). (
1
) (
0
s F
s
du u f
t
⋅ = ∫ L