18.03 Class 23
, April 2, 2010
Step and delta
[1] Step function
u(t)
[2] Rates and
delta(t)
[3] Regular, singular, and generalized functions
[4] Generalized derivative
[5] Heaviside and Dirac
Two additions to your mathematical modeling toolkit.
 Step functions [Heaviside]
 Delta functions [Dirac]
[1] Model of on/off process: a light turns on; first it is dark, then it
is light. The basic model is the Heaviside unit step function
u(t) = 0
for
t < 0
1
for
t > 0
Of course a light doesn't reach its steady state instantaneously; it takes
a small amount of time. If we use a finer time scale, you can see what
happens. It might move up smoothly; it might overshoot; it might move up
in fits and starts as different elements come on line. At the longer time
scale, we don't care about these details. Modeling the process by
u(t)
lets us just ignore those details. We simplify the model by supposing
that
u(t) = 0
for all
t < 0
no matter how near to zero,
u(t) = 1
for all
t > 9
no matter how near to zero,
and
u(0)
is left undefined.
u(ta)
turns on at
t = a .
If
a < b , u(ta)  u(tb)
turns on at
t = a
and off again at
t = b :
it's a "window."
We can use
u(ta)
to turn on another function:
u(ta) f(t)
is zero when
t < a
and agrees with
f(t)
when
t > a .
Q1:
What is the equation for the function which agrees with
f(t) between
a
and
b
( a < b )
and is zero outside this window?
(1)
(u(tb)  u(ta)) f(t)
(2)
(u(ta)  u(tb)) f(ta)
(3)
(u(ta)  u(tb)) f(t)
(4)
u(ta) f(ta)  u(tb) f(tb)
(5)
none of these
Ans: (3).
[2]
From bank accounts to delta functions.
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View Full DocumentBank account equation:
x' + Ix = q(t)
x = x(t) = balance
(K$)
I = interest rate
((yr)^{1})
q(t) = rate of savings (K$/yr)
I am happily saving at the rate
K$1/yr.
The concept of rate can be
clarified] by thinking about the cumulative total, Q(t) (from some starting
time,
perhaps
t = 0);
Q'(t)
=
q(t)
or
Q(t)
=
integral_0^t q(u) du
At
t = 1
I won $2K at the race track! I deposit this into the account.
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 Spring '09
 vogan
 Addition, Derivative, Continuous function, Heaviside step function, Delta Functions

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