082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017
112 Chapter 2 Engineering functionsExample2.26 Atrainof impulses isgiven byf(t) = δ(t)+3δ(t −1)+2δ(t −2)Depict the traingraphically.Solution Figure 2.58 shows the representation. In Section 22.8 we shall call such a function aseries of weighted impulseswhere the weights are1,3and 2,respectively.f (t)3210 1 2 tFigure2.58A trainofimpulsesgiven byf(t) = δ(t)+3δ(t −1)+2δ(t −2).Engineeringapplication2.19ImpulseresponseofasystemAn impulse signal is sometimes used to test an electronic system. It can be thoughtofasgiving the systemavery harsh joltforavery shortperiodoftime.It is not possible to produce an impulse function electronically as no practicalsignalcanhaveaninfiniteheight.However,anapproximationtoanimpulsefunctionisoftenused,consistingofapulsewithalargevoltage,V,andshortduration,T.Thestrength of such an impulse isVT. When this pulse signal is injected into a systemthe output obtained isknown astheimpulse response ofthe system.The approximation is valid provided the width of the pulse is an order of magnitudelessthanthefastesttimeconstantinthesystem.IfthevalueofTrequiredissmallinordertosatisfythisconstraint,thenthevalueofV mayneedtobelargetoachievethecorrectimpulsestrength,VT.Oftenthiscanruleoutitsuseformanysystemsasthe value ofV would thenbelargeenoughtodistort the systemcharacteristics.EXERCISES2.4.91 Sketch the impulsetrain given by(a) f(t) = δ(t −1)+2δ(t −2)(b) f(t) =3δ(t)+4δ(t −2)+δ(t −3)
Review exercises 2 113REVIEWEXERCISES21 Statethe ruleand sketch the graphofeach ofthefollowingfunctions:(a) f(x)=7x−2(b) f(t)=t 2 −2 0t5(c) g(x)=3e x +4 0x2(d) y(t) = (e 2t −1)/2 t 0(e) f(x)=x 3 +2x+5 −2x22 Statethe domainand range ofthe functions inQuestion1.3 Determine the inverse ofeach ofthe followingfunctions:(a) y(x) =2x (b) f(t) =8t −3(c) h(x) = 2x +1 (d)m(r)=1−3r3(e) H(s) = 3 s +2(g) f(t) =e 2t(i) g(v) = ln(v +1)4 Iff(t)=e t find(a) f(2t)(b) f(x)(c) f(λ) (d) f(t − λ)(f) f(v)=lnv(h) g(v) =lnv+1(j) y(t) = 3e t−25 Ifg(t) =ln(t 2 +1)find(a)g(λ),(b)g(t − λ).6 Sketchthe following functions:{ 0 t<0(a) f(t) =0.5t t0⎧⎨4 t<0(b) f(t) = t 0t<3⎩2t 3t4(c) f(t)=|e t |−3t 37 Thefunction f (x) isperiodicwith a periodof2, andf(x)=|x|,−1x1.Sketchffor−3x3.8 Givena(t) =3t,b(t) =t +3andc(t) =t 2 −3writeexpressions(a) b(c(t))(c) a(b(t))(e) a(b(c(t)))(b) c(b(t))(d) a(c(t))(f) c(b(a(t)))9 Sketchthe following functions,stating anyasymptotes:(a) y(x) = 3 +xx(b) y(x) =2xx 2 −1(c) y(x) =3−e −x (d) y(x) = ex +1e x10 Simplify each expression asfar aspossible.(a) e 2x e 3x(c) e 4 e 3 e(e)( )e x 2e −x(b) e x e 2x e −3xe x(d)e −x( )2(f) ln3x +lnx(g) 3lnt+2lnt 2 (h) e lnx(i) ln(e x ) (j) e lnx2(k) e 0.5lnx2 (l) ln(e 2x )(m) e 2lnx (n) ln(e 3 ) +ln(e 2x )11 Solve the following:(a) e 4x = 200 (b) e 3x−6 = 150(c) 9e −x = 54 (d) e (x2) = 601(e) = 0.16+e−x 12 Solve the following:(a) 0.5lnt = 1.2(b) ln(3t +2) = 1.4(c) 3ln(t−1)=6(d) log 10 (t 2 −1) = 1.5(e) log 10 (lnt) = 0.5(f) ln(log 10 t) = 0.513 Express eachofthe following in terms ofsinhx andcoshx:(a) 7e x +3e −x (b) 6e x −5e −x(c) 3ex −2e −x 1(d)2 e x +e −xe x(e)1+e x14 Express eachofthe following in terms ofe x ande −x :(a) 2sinhx +5coshx(b) tanhx +sechx(c) 2coshx − 3 4 sinhx1(d)sinhx −2coshx(e) (sinhx) 2
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112 Chapter 2 Engineering functions
Example2.26 Atrainof impulses isgiven by
f(t) = δ(t)+3δ(t −1)+2δ(t −2)
Depict the traingraphically.
Solution Figure 2.58 shows the representation. In Section 22.8 we shall call such a function a
series of weighted impulseswhere the weights are1,3and 2,respectively.
f (t)
3
2
1
0 1 2 t
Figure2.58
A trainofimpulsesgiven by
f(t) = δ(t)+3δ(t −1)+2δ(t −2).
Engineeringapplication2.19
Impulseresponseofasystem
An impulse signal is sometimes used to test an electronic system. It can be thought
ofasgiving the systemavery harsh joltforavery shortperiodoftime.
It is not possible to produce an impulse function electronically as no practical
signalcanhaveaninfiniteheight.However,anapproximationtoanimpulsefunction
isoftenused,consistingofapulsewithalargevoltage,V,andshortduration,T.The
strength of such an impulse isVT. When this pulse signal is injected into a system
the output obtained isknown astheimpulse response ofthe system.
The approximation is valid provided the width of the pulse is an order of magnitudelessthanthefastesttimeconstantinthesystem.IfthevalueofT
requiredissmall
inordertosatisfythisconstraint,thenthevalueofV mayneedtobelargetoachieve
thecorrectimpulsestrength,VT.Oftenthiscanruleoutitsuseformanysystemsas
the value ofV would thenbelargeenoughtodistort the systemcharacteristics.
EXERCISES2.4.9
1 Sketch the impulsetrain given by
(a) f(t) = δ(t −1)+2δ(t −2)
(b) f(t) =3δ(t)+4δ(t −2)+δ(t −3)
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