00:01
C of t equals 20t times e to the negative 0 .04t gives the concentration in milligrams per milliliter of a drug in the bloodstream t minutes after an injection.
00:09
First we want to find the instantaneous rate of change of the concentration after 10 minutes.
00:15
So instantaneous rate of change means we're going to need to find the derivative of this function with respect to t.
00:23
So we need to find the derivative of c or c prime of t and we're going to use that by using the product rule since we have two different factors 20t times e to the negative 0 .04t.
00:35
So that would be the derivative of that first factor with respect to t so the derivative of 20t times the second factor e to the negative 0 .04t plus that first factor times the derivative with respect to t of that second factor and that's just using the product rule.
00:59
So the derivative of 20t is just 20 and that's times e to the negative 0 .04t plus 20t.
01:11
The derivative of an exponential function e to the negative 0 .04t would just be itself so times e to the negative 0 .04t but then we need to multiply by the derivative of that what's in the exponent using the chain rule which would just be negative 0 .04.
01:30
So that gives us 20 times e to the negative 0 .04t and then 20 times negative 0 .04 that would be minus or negative 0 .8 and we have the t times e to the negative 0 .04t.
01:51
There's our derivative.
01:54
Now we want to find that rate of change at 10 minutes so that means we would plug 10 in for t.
02:01
That would give us 20 times e to the negative 0 .04 times 10 minus 0 .8 times 10 times e to the negative 0 .04 times 10 and if we plug that in a calculator that's going to give us 8 if we rounded the nearest whole number and that would be 8 milligrams per milliliter per minute.
02:46
That gives the concentration per minute at that instant after 10 minutes.
02:53
And next we want to find the concentration or the maximum concentration and when it occurs.
02:57
So when is this function maximized? so that's when the derivative c prime of t would equal 0.
03:06
So we take that function 20 e to the negative 0 .04t minus 0 .8t times e to the negative 0 .04t and set that equal to 0.
03:27
We can do that first by factoring out this e to the negative 0 .04t.
03:35
So e to the negative 0 .04t and then that leaves us with 20 minus 0 .8t.
03:44
I want to know when does that equal 0.
03:47
So we have two different factors and we have this in factored form so we could use the zero factor property.
03:53
We'll set each factor equal to 0.
03:56
So an exponential function of this form will never equal 0 so we don't have to worry about that one.
04:04
Then we need to set the factor of 20 minus 0 .8t equal to 0.
04:09
So that means 0 .8t would equal 20 and if we divide both sides by 0 .8, 20 divided by 0 .8 is 25.
04:22
So that is when the maximum occurs and we could confirm that that's a maximum if we can show that the second derivative is negative there showing that the function would be concave down...