HP 48gII User Manual page 491

Example 1 – The velocity of a moving particle v(t) is defined as v(t) = dr/dt,
where r = r(t) is the position of the particle. Let r
the transform of the velocity can be written as V(s) = L{v(t)}=L{dr/dt}= s⋅R(s)-r
•
Differentiation theorem for the second derivative. Let f
(df/dt) = df/dt| , then L{d
o t=0
Example 2 – As a follow up to Example 1, the acceleration a(t) is defined as
2 2
a(t) = d r/dt . If the initial velocity is v
transform of the acceleration can be written as:
A(s) = L{a(t)} = L{d
•
Differentiation theorem for the n-th derivative.
(k) k k
Let f = d f/dx |
o t = 0
n
L{d f/dt
•
Linearity theorem. L{af(t)+bg(t)} = a⋅L{f(t)} + b⋅L{g(t)}.
•
Differentiation theorem for the image function. Let F(s) = L{f(t)}, then
n n n
⋅f(t)}.
d F/ds = L{(-t)
–at
Example 3 – Let f(t) = e
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
3
d F/ds
L{df/dt} = s⋅F(s) - f
2 2 2 ⋅F(s) - s⋅f
f/dt } = s
= v(0) = dr/dt|
o
2 2 2
⋅R(s) - s⋅r
r/dt }= s
, and f = f(0), then
o
n n n-1
⋅F(s) – s ⋅f −...– s⋅f
} = s
o
, using the calculator with 'EXP(-a*X)' ` LAP, you
3 4 3 2 ⋅s
= -6/(s +4⋅a⋅s +6⋅a
.
o
= r(0), and R(s) =L{r(t)}, then,
o
= f(0), and
o
– (df/dt) .
o o
, then the Laplace
t=0
– v .
o o
(n-2) (n-1)
– f .
o o
2 3 ⋅s+a 4
+4⋅a ).
Page 16-13
.
o