8.3 The Delta Iiiipiilse 163 d(t)f(t) clisappcarr for t I LlxN only OEie negative dwivative of f ( t ) at t = remains We m n iiitrrprcxt the derivative h(t) as c? distrib tlir va~iieof tiir drrivative - J ( o ) from D timction f i t ) ,using the s r Tht~tlrrivation of h(t) is revcaled its dfwt on tlie contiiiimus faiiglc functions with a limit o illri as tliey are iiat cliff crciitiable, b r e ( w l d , however, lime in(r functioii with a, limit LVe only ch impulse as tuiotlrw inipu fiinction I W C R I ~ ~itF ~ i i ph icularly simple to irilegrate (8.11) The corrtymrrdiiig limit for R bell-shaprd impnlse fiinction cZ( L ) with c h r acteristic width T is illiistiatrd iii P’igurc 8.10 (top) In contrast to thc funct>iorr,llir dcrivative d(t) of the bell-slritpd functiori can be kxxmL for T -+ theu !teach to t,lie dcrivativc h(1) of 6(t) show~iin F i g u ~ e8.10 (bclow) thc double impulse upvartis and dovlmvartls it i s d s o c d c d a douIt shoidd also be emphasised here that the graphical rcprrscntniori on thc riglib-hand \itit> of Fig exact description of distrihutioiis is only p fiinctioiis (see (8.13) and (8.23)) - lim d(t) T-+ I -b-t (8.24)