00:01
All right, guys, so here's this problem.
00:02
I think the function s of t is supposed to be this.
00:11
110 minus 80e to the negative 0 .3t, or negative 0 .3t.
00:19
I think that's what the equation is supposed to be.
00:23
And it says, find the rate of change of sales at each time, one year, five years.
00:27
Rate of change goes on and on.
00:31
Does it goes to infinity? does the rate of change? okay.
00:34
So here we go.
00:36
Now, to find the rate of change, we got to take the derivative.
00:41
So s prime of t is going to equal derivative of 100 times to zero.
00:47
That's gone.
00:48
Okay.
00:51
Derivative of an exponential is exactly what it is.
00:55
So it'll be negative 80e.
00:59
To the negative 0 .3t, this, but then we got to times it by the derivative of that exponent.
01:09
The derivative of negative.
01:11
Point 3t is just negative .3.
01:15
Okay.
01:17
So, when we make this all nice and neat, let's put the derivative in a different color, sure, why not? here's our derivative.
01:25
This is the change in sales.
01:27
It's going to be those two negatives multiply together, become a positive.
01:31
0 .3 times 80 is just 24.
01:34
So it's going to be 24e to the negative 0 .3t.
01:41
All right, and that's just an exponential function.
01:45
All right.
01:46
So as we, if we were to graph this thing, or let's do part a and b for.
01:54
So for part a at one year, so when t equals 1, the change.
02:03
We just want to find the rate of change.
02:07
Our sales are, you just put one in for t, so you're going to get negative point three to the one or is e to the negative point three.
02:18
Let's type that, turn the calculator on.
02:21
It's always a good thing.
02:24
So the negative point three and it gives you a number and then you times that by 24.
02:33
So i got 17 .77 let's see so 17 .7 .7.
02:50
Okay, at one year that's how much our sales are increasing.
02:58
We're increasing by 17 .796 ,000.
03:03
Okay, it's going up.
03:05
Now for part b, when t equals five, you plug five in there and so you're going to get e to the negative point three times five and that's something points out who cares and we times that by 24 and we get five point three five five five five one thousand okay so after five years our sales are only increasing by i like 5 ,000 units, okay? and you might be thinking, wait a minute, that doesn't make sense.
03:49
Why? well, think about it's a tech item, right? five years later, the tech goes obsolete like in a day.
03:58
Oh, look, i got the new cell phone next day.
04:00
Oh, there's another new cell phone, you know, and your old, your one -year -old, one -day old phone is like useless.
04:06
Not quite that extreme, but, you know, every year or two, you know, they put out a new cell phone, new tech stuff.
04:10
So that's why five years from now, our sales are only going to be, it's not increasing by much.
04:16
It's increasing by very, very little...