The Second Shifting Theorem and the Unit Step Function
This video is on how to use Laplace transforms for piecewise continuous functions without integrating. The first part of the video is on the unit step function, which is what we use to transform piecewise functions into individual functions. Then once we have used the unit step function, we can use the second shifting theorem to find the Laplace transform of piecewise functions.
00:00 Introduction
01:45 Defining the Unit Step Function
02:57 Unit Step Functions for Closed Intervals
06:52 Piecewise functions using the unit step
12:23 Second Shifting Theorem Intro
13:03 Second Shifting theorem proof
15:05 Second Shifting Corollary
15:30 Example 1: Laplace Transform
19:17 Example 2: Laplace Transform
23:58 Example 3: Inverse Laplace Transform
25:04 Example 4: Inverse Laplace Transform
26:20 Conclusions