Explanation Given: The given function is f ( t ) = − 1 + 2 ∑ k = 1 ∞ ( − 1 ) k + 1 U ( t − k ) . Concept used: If F ( s ) = L { f ( t ) } and a > 0 , then L { U ( t − a ) } = e − a s s , where, U is a unit step function. The Laplace transformation of 1 is L { 1 } = 1 s . Calculation: Expand the given function f ( t ) by putting the value of k = 1 , 2 , 3 , 4 , 5 , … and so on as shown below. f ( t ) = − 1 + 2 ∑ k = 1 ∞ ( − 1 ) k + 1 U ( t − k ) = − 1 + 2 { ( − 1 ) 1 + 1 U ( t − 1 ) + ( − 1 ) 2 + 1 U ( t − 2 ) + ( − 1 ) 3 + 1 U ( t − 3 ) + ( − 1 ) 4 + 1 U ( t − 4 ) … } = − 1 + 2 { U ( t − 1 ) − U ( t − 2 ) + U ( t − 3 ) − U ( t − 4 ) … } = − 1 + 2 U ( t − 1 ) − 2 U ( t − 2 ) + 2 U ( t − 3 ) − 2 U ( t − 4 ) … That is f ( t ) = − 1 + 2 U ( t − 1 ) − 2 U ( t − 2 ) + 2 U ( t − 3 ) − 2 U ( t − 4 ) … (1) Simplify the Equation (1) as follows. f ( t ) = − 1 + 2 U ( t − 1 ) − 2 U ( t − 2 ) + 2 U ( t − 3 ) − 2 U ( t − 4 ) … = ( − 1 ) − ( − 1 ) U ( t − 1 ) + U ( t − 1 ) − U ( t − 2 ) + ( − 1 ) U ( t − 2 ) − ( − 1 ) U ( t − 3 ) … = { ( − 1 ) − ( − 1 ) U ( t − 1 ) } + { U ( t − 1 ) − U ( t − 2 ) } + { ( − 1 ) U ( t − 2 ) − ( − 1 ) U ( t − 3 ) } … On further simplification, f ( t ) = { ( − 1 ) − ( − 1 ) U ( t − 1 ) } + { U ( t − 1 ) − U ( t − 2 ) } + { ( − 1 ) U ( t − 2 ) − ( − 1 ) U ( t − 3 ) } … (2) The Graph for the equation (2) can be drawn as follows

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 4, Problem 33CR

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To sketch: The graph for the given function and also find the Laplace transform L{f(t)}.

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Chapter 4, Problem 32CR

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The graph for the given function and also find the Laplace transform L { f ( t ) } .

Solution Summary: The author illustrates the Laplace transform of the given function f(t), where U is a unit step function.

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To sketch: The graph for the given function and also find the Laplace transform L{f(t)}.

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Chapter 4 Solutions