00:01
Okay, so we're told that since january 1st, 1960, the population of this area is described by this equation.
00:08
P equals 21 ,000 times 0 .95 to the power of t, where p is the population of the city t years after the start of 1960.
00:18
We want to find the rate at which the population was changing on january 1st, 1982.
00:24
So this is going to be corresponding to t equals 22 because january 1st, 1982, because january 1st, 1982, is 22 years after 1960.
00:35
So we want to find d .p.
00:38
By dt at t equals 22.
00:43
This is our goal because the rate of change at which the population of changing is d .p.
00:48
By dt.
00:49
So let's firstly find d .p.
00:51
By dt.
00:51
So if i just copy this equation to bring it down below.
00:56
And not zoom, sorry, we want to paste.
01:00
Paste.
01:00
Okay, so immediately it's probably not obvious how to differentiate this.
01:07
So the easiest thing when you have something to a power of t like this, so you might be familiar with t to the power of something like t squared or t -q, but here we have something to the power of t, which is slightly different.
01:20
And the easiest way to go about doing this is to take the log of both sides.
01:24
So if you take the log of p, this is equal to the log of 21 ,000 times, 0 .0.
01:33
Let me put this in brackets.
01:34
9.
01:35
Not that in brackets...