A drug response curve describes the level of medication in...

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function $ S(t) = At^p e^{-kt} $ is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, $ A = 0.01 $, $ p = 4 $, $ k = 0.07 $, and $ t $ is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have graphing device, use it to graph the drug response curve.

Transcript

00:01 Called the surge function and it's equal to some number a times t to the p power times e to the negative k times t power.

00:12 S is the level of medication in the bloodstream.

00:15 T is the number of minutes that have elapsed.

00:19 For this particular search function, a is .01 times t to the p power where p is given to be four.

00:33 And then that multiplies e to the negative kt.

00:37 K is given to be .07, so that's e to the negative 0 .07, or actually 0 .07, e to the negative 0 .07, e to the negative 0 .07, e to the negative 0 .07, t.

00:53 So we are going to graph this function using desmos, and then we are going to look for the inflection points and kind of describe what's going on with this search function at those inflection points.

01:14 Okay, so using desmos, i graphed the search function.

01:20 Desmos prefers to work with the variable x.

01:23 So instead of the variable t, which was representing minutes, we're just using the variable x.

01:29 And since the number of minutes that have elapsed since the injection, should be greater than or equal to zero.

01:41 Graphing this function for time less than zero really didn't mean anything, so to make the graph look a little less confusing, i restricted this function to being graphed off when the number of minutes, x is greater than equal to zero.

02:00 The goal here is to estimate the time, which would be the horizontal x, estimate the time at which inflection points occur.

02:12 Now, an inflection point occurs when the second derivative of the function is zero.

02:22 Basically, it's signals, an inflection point signals when the first derivative will change from increasing to decreasing.

02:36 If the second derivative is zero, that means the first derivative reaches a maximum point, meaning the first derivative or the steepness, the slope, if you will, was increasing, reached a maximum point, now we'll start decreasing.

02:52 Or vice versa, an inflection point being where the second derivative is zero, an inflection point could also be where the first derivative reaches a minimum.

03:03 The first derivative is decreasing, reaches a minimum, and then starts increasing.

03:10 Now, if we're looking for the first derivative to be at a maximum point or minimum point, that means, for example, if we say the first derivative is reaching a maximum, that means the first derivative was increasing, reaches a maximum point, will start decreasing.

03:30 Well, the first derivative lets you know just how steep the function itself is increasing or decreasing.

03:39 So if we're looking for where the first derivative is going to change from increasing or decreasing, we want to look for a change in the steepness of the curve.

03:52 Let me go back to our whiteboard here and give you an idea of what an inflection point would look like.

04:00 Just so you have an idea what we're looking for it.

04:02 If i have a function, it's going up steeply, okay, but then at this point, it's still going to be increasing, but at a slower rate, okay, this would be an inflection point.

04:18 The function is still increasing, but the rate at which it's increasing, as indicated by the first derivative, is changing.

04:26 Here the function is increasing at a quicker and quicker rate, so the slope or the first derivative is increasing.

04:34 Then at this inflection point, the second derivative is zero.

04:39 The first derivative reaches a maximum.

04:42 The function itself still increases, but now at a slower rate, because f prime of x now is starting to decrease.

04:50 So this would be the graph of the function f, and f prime, of x would change from the slope would at this inflection point changes from increasing and increasing to where the slope or the derivative is now decreasing okay so f prime of x changes from positive to negative the second derivative at this inflection point would be zero.

05:42 So this is what happens at an inflection point.

06:08 At an inflection point, the function will still be increasing.

06:13 And of course, this can also be shown for a function that is decreasing through an inflection point.

06:19 Let me give a little idea what that looks like.

06:21 If i have a function that's decreasing, reaches an inflection point, still decreases, but now at a slower rate.

06:36 This would also be an inflection point.

06:39 At an inflection point, the second derivative is zero, and the first derivative changes signs...