Advanced High-School Mathematics

SECTION 5.1 Quick Survey of Limits 263

An important theorem of

electrical engineering is that

if the input voltage is x(t),

then the output voltage isy=

x∗h(t),whereh(the “impulse

response”) is given explicitly

by

x(t) y(t)

h(t) =







1

τe

−t/τ ift≥ 0

0 ift < 0 ,

and whereτ=RC.

Now assume that R = 1000 Ω and thatC = 2μF (= 2× 10 −^6

Farads). Let

x(t) =







sin 2πft ift≥ 0

0 ift < 0

wheref >0 is the frequency of the signal (in “hertz” (Hz) or units

of (sec)−^1 ). In each case below, compute and graph the output

voltagey(t) as a function of time:

(a) f= 100 Hz

(b) f= 2 kHz, or 2000 Hz

(c) f= 100 kHz

8. (For the courageous student!^6 ) Consider the function

f(t) =







sin(1/t), ift 6 = 0

5 ift= 0,

and set F(x) =

∫x

− 1 f(t)dt. Show that F

′(0) = 0. (Hint: try

carrying out the following steps:

(^6) I am indebted to Robert Burckel for suggesting this problem.