In exercise 15, Express the given function f in terms of unit step functions and use Theorem 8.4.1 tofind L.f /. Where indicated by C/G , graph f . sin , 0<1 <15.C/G f)= 2 sint.cos t.12 x. Theorem 8.4.i Let g be defined on (0, 00). Suppose r 2 0 and L (g(t + ) cists for s > Sp. ThenLufr-r)g(0) exists Jor s > *q, UndL(u(1 -t)g(t)) = L (gr + ).Proof By definition,L (u(t – T)g(t)) =eu(t - t)g() dt.From this and the definition of u(f - r),L (u(t-t)g()) =e"(0) dt +"g(() dt.The first integral on the right equals zero. Introducing the new variable of integration x=t-rin thesecond integral yieldsL(u(1-T)g()) =g+t)d =,g(r + t)dx.Changing the name of the variable of integration in the last integral from x to r yieldsL(u(-t)g(1))e"g(t +t)dr = eL(g(t+ r)). I

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Answered: sin t. 15. C/G f()= 2 sint. cost, 12 x.

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sin t. 15. C/G f()= 2 sint. cost, 12 x.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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Question

In exercise 15, Express the given function f in terms of unit step functions and use Theorem 8.4.1 to find L.f /. Where indicated by C/G , graph f .

sin , 0<1 <
15.
C/G f)= 2 sint.
cos t.
12 x.

Theorem 8.4.i Let g be defined on (0, 00). Suppose r 2 0 and L (g(t + ) cists for s > Sp. Then
Lufr-r)g(0) exists Jor s > *q, Und
L(u(1 -t)g(t)) = L (gr + ).
Proof By definition,
L (u(t – T)g(t)) =
eu(t - t)g() dt.
From this and the definition of u(f - r),
L (u(t-t)g()) =
e"(0) dt +
"g(() dt.
The first integral on the right equals zero. Introducing the new variable of integration x=t-rin the
second integral yields
L(u(1-T)g()) =
g+t)d =,
g(r + t)dx.
Changing the name of the variable of integration in the last integral from x to r yields
L(u(-t)g(1))
e"g(t +t)dr = eL(g(t+ r)). I

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