POSSIBLE SOLUTION(S)
POSSIBLE SOLUTION(S)
1) Argue, however, that w(t) = t does NOT count the number of oscillations per second and is thus not the frequency of oscillation.
Input :=
Plot[Sin[2 Pi t^2], {t, 2, 3}, AspectRatio -> Automatic,
PlotPoints -> 50]
Output = -Graphics-
Since there are f = 5 cycles for t in [2, 3] and f=5 does not lie between w(2)=2 and w(3)=3 and w(t) is monotone on [2, 3], w(t) must not be counting the number of oscillations per second.
2) Indeed, if W(t) = 2t, then [W(2)=4] <= [f=5] <= [6=W(3)]. Furthermore, the average of W(t) on [2, 3] is f = 5 which makes sense since the frequency W is increasing linearly.
3)
Input :=
w = t^2;
Plot[Sin[2 Pi w t], {t, 2, 3}, PlotPoints -> 50]
Output = -Graphics-
Counting very carefully, we find f=19 cycles in the one second for t on [2, 3]. If W(t) = 3t^2, then [W(2)=12] <= [f=19] <= [W(3)=27]. Indeed, the average of W(t) on [2, 3] is
Input :=
Integrate[3 t^2, {t, 2, 3}]
Output = 19
4)
Input := w = 200 + (100/(2 Pi)) Sin[2 Pi t];
Input := W[t_] = 200 + 100 t Cos[2 Pi t] + (100/(2 Pi)) Sin[2 Pi t];
Input :=
Plot[Sin[2 Pi w t], {t, 1, 1.04}, PlotPoints -> 50]
Output = -Graphics-
It looks like about 12 and 1/6 cycles occur for t in [1, 1.04] which means that we are averaging about f=228.125 cycles per second.
Input := f = (12 + 1/6)/0.04
Output = 304.167
which means that we are averaging about f=304.167 cycles per second.
Input := leftside = W[1] //N
Output = 300.
Input := rightside = W[1.04] //N
Output = 304.691
Input :=
NIntegrate[W[t], {t, 1, 1.04}]/0.04
Output = 302.909
Recognizing that we were only able to make a crude approximation (the 1/6 of a cycle part), we did a pretty good job! As we have seen before, W(t) counts the number of cycles per second at time t. Also, over the interval [1, 1.04], the average number of cycles per second, about 303, is pretty darn close to our estimate of 304!
5) Given w(t), W(t) = D(t w(t)).
6) Given W(t), w(t) = Integrate[W(t)]/t.
7)
Input := ClearAll[W, t];
Input := W[t_] = 263 + (526 - 263)/3 t
Output =
263 t
263 + -----
3
Input :=
Plot[W[t], {t, 0, 3},
AxesLabel -> {"seconds", "frequency"}]
Output = -Graphics-
Input := w = Simplify[Integrate[W[t], t]/t]
Output =
263 (6 + t)
-----------
6
Input := g = Sin[2 Pi w t]
Output =
263 Pi t (6 + t)
Sin[----------------]
3
8)
Input :=
Play[g, {t, 0, 3}]
Output = -Sound-
Input := W = 263 2^(t/3);
Input :=
Plot[W, {t, 0, 3}]
Output = -Graphics-
Input := w = Simplify[Integrate[W, t]/t]
Output =
t/3
789 2
--------
t Log[2]
Input := h = Simplify[Sin[2 Pi w t]] //N
Output =
0.333333 t
Sin[7152.06 2. ]
Input :=
Play[h, {t, 0, 3}]
Output = -Sound-