The table shows values of the viral load vt in hiv patient...

The table shows values of the viral load $ V(t) $ in HIV patient 303, measured in RNA copies/mL, $ t $ days after ABT-538 treatment was begun. (a) Find the average rate of change of $ V $ with respect to $ t $ over each time interval: (i) $ [4, 11] $ (ii) $ [8, 11] $ (iii) $ [11, 15] $ (iv) $ [11, 22] $ What are the units? (b) Estimate and interpret the value of the derivative $ V'(11) $.

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00:01 So in this problem, we're given this table of data, which is the viral load v of t in this patient, a t days after a treatment has begun.

00:15 So the t is in days, and this viral load is rna copies per milliliter.

00:31 Okay.

00:32 And first of all, we're asked to find the average rate of change of v with respect to the over each of these different time intervals.

00:39 So the first one's from 4 to 11.

00:43 So the average rate of change, i'll abbreviate aroc, will be v at 11 minus v at 4 over 11 minus 4.

00:59 Well, from the table, this is 9 .4 minus 53 over 7, which is a negative 43 .6 over 11.

01:23 So then we're asked to do the interval 8 to 11.

01:30 So the average rate of change here.

01:34 Again, v11 minus v8 over 11 minus 8.

01:42 And so this is 9 .4 minus 18.

01:50 Over 3 and that is a negative 8 .6 over 3.

02:02 Then we're asked to do from 11 through 15 and that average rate of change will be v of 15 minus v of 11 over 15 minus 11 which is 5 .2 minus 9 .4 over 4.

02:27 So that is a negative 4 .2 over 4.

02:42 Okay.

02:43 And then we're asked to do 11 to 22.

02:56 This average rate of change will be v of 22 minus v of 11 over 22 minus 11...

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