00:01
So on this problem, we're given this function for measles, pathogenous, and curve, and we're asked to find using seven subintervals in their midpoints, to estimate the total amount of infection needed to develop symptoms of measles.
00:21
Okay, so first of all, we need to find where the critical point is on this curve.
00:30
And so we multiply this out and we get minus t cubed plus 20 t squared plus 21 t.
00:52
We'll multiply all that out.
00:54
Now differentiate to get the derivative.
00:58
So f prime at t gives us, we bring the exponent down, that's minus three and then subtract one from the exponent.
01:06
That's t squared then.
01:08
Bring the two down that's plus 40 t plus 21 all right so we set this equal to zero to find the critical points point right because remember the critical point is where the derivative will be zero use the quadratic equation since this is a squared polynomial so then t is minus b it's minus 40 plus or minus the square root of b squared that's 40 squared minus 4 times a minus 3 times c all over 2 times a okay so that gives us that we'll have we'll have let's see minus and minus cancel and when i cancel them with the plus minus it still becomes plus minus so that's gone and so i could write this as one -third times well two goes into 40 20 times plus the square root when you work through all of that of 463 so that is approximately 13 .8 um they give us t in days okay so this is 14 days now take the second derivative so we can find out if this is a minimum or a maximum.
03:19
Second derivative of that first, in other words, the derivative of the first derivative, we'll bring that two down, so that's minus 6t plus 40.
03:29
So the second derivative evaluated at 13 .8 is going to be less than zero, right? it's going to be negative.
03:43
So this curve is, concave down at this point, which makes this a maximum...