Part ii) 5. Let e be a positive number and consider the function f.(x) defined byif 0 €.The graph of this function is shown in Figure 65. It is clear that forevery e > 0 we have Jo f.(x)dx = 1. Show that1-e-peL[f.(x)] =-peandlim L[f.(x)]=1.Strictly speaking, lim0 f.(x) does not exist as a function, soL[lim.0 f.(x)]is not defined; but if we throw caution to the winds, then8(x) = lim f.(x)1/ɛFIGURE 65 ii. Prove the second translation theorem (in time): If F(s) = L(f (t)), thenL(ua(t)f(t – a)) = e_a$ F(s) (a > 0).-as

Name: Part ii) 5. Let e be a positive number and consider the function f.(x)
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Description: Part ii) 5. Let e be a positive number and consider the function f.(x) defined byif 0 €.The graph of this function is shown in Figure 65. It is clear that forevery e > 0 we have Jo f.(x)dx = 1. Show that1-e-peL[f.(x)] =-peandlim L[f.(x)]=1.Strictly

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Answered: ii. Prove the second translation…

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ii. Prove the second translation theorem (in time): If F(s) = L(f(t)), then L(ua(t)f(t – a)) = e-a® F(s) (a > 0).

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.5: Transformation Of Functions

Problem 4SE: When examining the formula of a function that is the result of multiple transformations, how can you...

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

Question

Part ii)

5. Let e be a positive number and consider the function f.(x) defined by
if 0 <x< €
f.(x)=-
%3D
if x > €.
The graph of this function is shown in Figure 65. It is clear that for
every e > 0 we have Jo f.(x)dx = 1. Show that
1-e-pe
L[f.(x)] =-
pe
and
lim L[f.(x)]=1.
Strictly speaking, lim0 f.(x) does not exist as a function, so
L[lim.0 f.(x)]is not defined; but if we throw caution to the winds, then
8(x) = lim f.(x)
1/ɛ
FIGURE 65

ii. Prove the second translation theorem (in time): If F(s) = L(f (t)), then
L(ua(t)f(t – a)) = e_a$ F(s) (a > 0).
-as

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