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A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function $S(t)=A t^{p} e^{-k t}$ 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 a graphing device, use it to graph the drug response curve.

Inflection occurs at $t \approx 28.57,85.71$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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for this program. We can just plug in some specific values into Is the function is off tea and we can compute a second maturity for us with respect to t. So this is the second of directive and the since we want to find the collection points and be careful our domain will be t Pointer, then zero. We only consider this a sour Told me so we set the second of duty because the zero that means this could take a question. You close to zero so we can use calculators, could get to result He he close to 28.57 and a key because those 85 points signaled one. So we have three intervals from zero to to the 8.57 the second it curative um off t Yes, Positive. So the conclave for on the throne on this 28.57 to 85.71 The second the purity of his negative. So it's concave down. And for the last interval on 85.71 Teoh positive infinity their second maturity. If it's called again this ex con que far so we can correlate to have two inflection points. We tried t close to 28.37 and he caused to 85.71 or so If we need to pray this so, in fact, we can just grab face in our coordinate. So we have something like news visit the y axis is s, which is the level of medication, and the x axis will be time so that you were looks like this. The graph of F looks like this. So we have to inflection points, um, reaches around 28 85. That means the level off medication when the test begins. So live off medication grows slowly here, And the gross, um, rapidity here. And, uh, after you after your trip to the local Mexican, it starts decreasing rapidity. And then when the time heats, this inflection points the live off medication, R B krista, slowly and here it gets a syrup

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