18.03 Muddy
Card
responses,
March
10,
2006
1.
The
commonest
question
concerned
the
idea
and
utility
of
operators.
I’ll
say
something now.
You can look ahead at the “exponential shift law” if you want, to
see one use later.
An
operator
modiFes a
function
in
some
way.
D
diﬀerentiates,
so
Dx
=
x
˙.
[The
independent
variable
isn’t
indicated
in
the
notation,,
and
has
to
be
gleaned
from
the context.
In fact in one of the lectures I muddied the waters further by writing
Dx
2
= 2
x
, so in that instance the independent variable must have been
x
.
If it had
been
t
, the correct formula would have been
Dx
2
= 2
xx
˙, which could aslo be written
Dx
2
= 2
xDx
.]
You
can
multiply
operators
and
add
them.
Multiplication
means
“compose”:
so
D
2
means “do
D
twice,” or take the second derivative.
If we want to
express, say, ¨
x
+
bx
˙ +
kx
as the eﬀect of an operator on the function
x
, we’ll need a
symbol for the operator which leaves
x
alone, the identity operator.
Some books use
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Staff
 Derivative, Exponential Functions, Laplace, exponential shift law, exponential input signal

Click to edit the document details