MIT18_03S10_c23 - 18.03 Class 23, April 2, 2010 Step and...

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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(t-a) turns on at t = a . If a < b , u(t-a) - u(t-b) turns on at t = a and off again at t = b : it's a "window." We can use u(t-a) to turn on another function: u(t-a) 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(t-b) - u(t-a)) f(t) (2) (u(t-a) - u(t-b)) f(t-a) (3) (u(t-a) - u(t-b)) f(t) (4) u(t-a) f(t-a) - u(t-b) f(t-b) (5) none of these Ans: (3). [2] From bank accounts to delta functions.
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Bank 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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This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c23 - 18.03 Class 23, April 2, 2010 Step and...

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