Lecture 21

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Laplace Transforms Improper Integrals, Piecewise Functions, and Integral Transforms
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MATLAB Quiz Thursday, March 13 Same Time As Your Section (APM B432) Alternate Time Sign Ups Are Online DON’T FORGET!!!! (NO MAKEUPS)
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Laplace Transforms “Change of Variables” For Ordinary Differential Equations Need to review 2 ideas from Calculus
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Review Topic 1: Improper Integrals An Integral with Infinity In Limit is Called “Improper” Z 1 0 f ( t ) dt Evaluated with a limit lim C ! 1 Z C 0 = f ( t ) dt Two Behaviors: Converge Diverge Integral Goes To A Finite Constant Integral Goes To 1
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Review Topic 2: Piecewise Continuous Functions A Function is Called “Piecewise Continuous” On An Interval [a,b] if: The interval can be divided into sub intervals , , t n < t < b , t 0 < t < t 1 a < t < t 0 t i < t < t i +1 And the function is continuous on each interval And lim t ! t i + f ( t ) lim t ! t i - f ( t ) Both exist (BUT DON’T HAVE TO EQUAL) for any t i
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Integral Transform Introducing a “Kernel Function” K ( s , t ) f ( t ) K ( s , t ) Z β dt = F ( s ) Now a function of s Is Given By An “Integral Transform” of a Function f ( t )
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Integral Transform
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