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Use, it still takes time for physicians to fully accept and start prescrlblng the medication: The eccepiance by physicians approaches J 8 new cancer medication is a...

Question

Use, it still takes time for physicians to fully accept and start prescrlblng the medication: The eccepiance by physicians approaches J 8 new cancer medication is approximaled by Iha equation below: Complele parts (a) through (c):Jo par monthper monlh:AP

use, it still takes time for physicians to fully accept and start prescrlblng the medication: The eccepiance by physicians approaches J 8 new cancer medication is approximaled by Iha equation below: Complele parts (a) through (c): Jo par month per monlh: AP



Answers

Pharmaceutical firms invest significantly in testing new medications. After a drug is approved by the Federal Drug Administration, it still takes time for physicians to fully accept and start prescribing the medication. The acceptance by physicians approaches a limiting value of $100 \%,$ or $1,$ after time $t,$ in months. Suppose that the percentage $P$ of physicians prescribing a new cancer medication after $t$ months is approximated by $P(t)=100\left(1-e^{-0.4 t}\right)$ a) What percentage of doctors are prescribing the medication after 0 months? 1 month? 2 months? 3 months? 5 months? 12 months? 16 months? b) Find $P^{\prime}(7),$ and interpret its meaning. c) Sketch a graph of the function.

Okay so we're given this function pft which tells us the percentage of doctors that are prescribing a new medicine after t. Months. And so what we're gonna do is we're going to find P. F. one which is equal to 100 Yet 100 -100 times E. To the negative point to. And now we can just plug this into a calculator. So 100 times E. To the negative point to. It's equal to 81 .87. Sophie Take 100 and subtract that. We get 18 point 1 2. So 18.12% of doctors are prescribing the medication after one month and now we're going to figure out What p. of sixes so do the same thing. We can say 100 -100 times E. To the negative 0.2 times six would be the negative 1.2. So we have 100 times e. to the negative 1.2 which is equal to about 30. And then if we save 100- that We get 69.88 so 69.88% of doctors are prescribing this medicine after six months. So it's increasing as we go farther and farther in time. And now what we're gonna figure out is the derivative of pft P. Prime of T. Which is equal to the derivative of 100 times one minus E. To the negative point to T. So the derivative of this. Um It's gonna be easier done at least in my opinion if we actually distribute this 100. So we have d. 100 -100 E. To the negative point to T. And now we can actually find the derivative. So the derivative of any constant is zero. So there's 100 goes to zero. And then the derivative of E. To the negative 00.2 T. Would be negative point to Times E. To the negative point to T. And then we also are multiplying by 100. So these two negative signs are going to cancel out. We're gonna get positive 20.2 times 100 which is 20. So we have 20 E. To the negative point to T. As our derivative of our function. PFT. And now what we want to figure out is how long it will take for 90% of doctors to actually use this medicine. So we're gonna let pFT equal 90 and we're just going to solve her teeth. So 90 is equal to 100 -100 E. To negative point to T. And so the first thing that we can do is I'm going to add this 100 over. So 100 is the negative point to T. Plus 90 is equal to 100. And then I'm gonna minus this 90 over. So we get 100 ft to the negative 1000.2 T. Is equal to 10. And now we can divide by 100. So you get E. To the negative point to T. Is equal to 1/10. And now what we want to do is we want to take each side um E. To each side. So we're gonna take E. And raise it to the power of each of these numbers. So we're sorry we don't want to do that. We want to take the natural log of both sides. Since we already have E raised to some power. If we had a natural log here we would want to take E. Um raise to both sides. But we want to take the natural log on both sides since we have E raised to some power. So we're gonna say the natural log of E. Negative 0.2 T. Equal to the natural log 1/10. So we can bring out this exponents. So we have a negative point to T. Times the natural log of E. Which is just one. So we just have a negative 10.2. T. Is equal to the natural log of 1/10. So then T. Is equal to negative five times the natural log of 1/10. And negative five is the same thing as dividing by a negative point to. So now we can plug this into a calculator. The natural log of one divided by 10 Is equal to negative 2.3. And then if we multiply by negative five we get 11.5. So T. is equal to 11.51. So it takes about 11.51 months for Doctors for 90% of doctors to be prescribing the medicine. And then the last thing that we're gonna do is we're gonna find the limit as T. Goes to infinity of our function. Pft Which was 100 -100 E. To the negative point to T. And so what we can do here is we can split up this limit and say the limit as T. Goes to infinity of 100 minus 100 E. To the negative point duty is equal to the limit as T. Goes to infinity of 100- the limit as T. Goes to infinity Of 100 times e. To the negative point to T. So the limit is he goes to um sorry the limited he goes to infinity of 100 is going to be just 100. So we have 100 And now we can take a 100 out of this. So you can save 100 -100 times the limit. As T. Goes to infinity of E. I'm actually gonna say one over E to the point to T. And now we can see that we have a E. raised to some power in the denominator. And so as he goes to infinity our denominator is going to increase. Its going to be increasing in going towards infinity and our numerator is going to be staying at one. So this is going to be equal to zero. So this is equal to 100 -100 Time zero. Just just equal to 100. So what this means is that as time goes on This certain medicine is going to be adopted by all doctors. So after a certain amount of time 100% of doctors are going to be prescribing this medicine

We know that those is it called the rate Multiply every time doings is equal to rate. Multiply by time. I'm. Therefore, time is equal to doors divided by rate. By putting the values we have, those is equal to attorney. Six Gray divided by Oh no, Let's write down rate the value of great and it's 16 multiplied by 10 to the dollar, minus three green ha a minute into one day, divided by 1440 minutes 1440 minutes and sold ing this We have Time T is equal to 1.5 six dear and time is approximately cool, too. 1.6 Dan.

Uh, So we have a model here for the percent of generic drugs that are, um, being given as generics by CBS, and we have a peaceful eyes function, so we have to linear equations, and you just write it down. And we know that this top linear model is used when our time frame is from 8 to 11 and are bottom one is used when our time frame is from 12 to 14 and so on. And this one is corresponding with 2008 this 1, 2014 and everything in between. And we want to write those down. We want to find out what they are. So we have for 2008, 9, 10, 11, 12, 13, 14. And so I'm going into my calculator. And I'm putting Weiss of one as my first model 2.77 x plus the 45.2. And then why so, too? I'm plugging in 1.95 x plus 55.9, and then I'm going to make a table to table set and I'm going to start my table. The start at eight, and I'm gonna have it go up by ones, and I got that table in front of me now. And so now I know this is the percent that they would have during these different years. And so for the first model I need to use for 89 10, 11, I need to use my top linear model. And so for me, I'm going to write down the results of my wife's of one equation. So I get 67 0.36%. I get 70 0.13% for 10 2010 72.9% for 11. I get 75.67% and now I need to switch to my other model because my other model is for when I'm at 12 so I can look at my wife's up to for 12. And that gives me 79.3% for 13. That up just a tad bit. I have 81.25%. And then for my last one, I have 83.2% are generics. So again, you just have ah piece wise function in your domain to use Here is the eight through 11 and your domain to use for the second function is the 12 through 14, and using a table is very, very helpful rather than just plugging things in.

Alright for this problem we're told that a body assimilates a 12 hour cold tablet at a rate model by D. C by DT equals eight minus lawn of T square T squared minus two T plus four. Where T. Is between zero and 12 and D. C. By D. T. Is measured in milligrams per hour. T. Is the time in hours. We want to use Simpson's rule with N equals eight to estimate the total amount of the drug absorbed into the body during the 12 hours. So one second here I'm just going to get things set up properly here so you can see that I have set up. First of all I state this is Simpson's rule then we set up. Okay A equals zero vehicles 12. The time step is b minus a over N. N. Is equal to eight. First of all, I'll print out what the evaluating points are going to look like. So we'll have b minus a over three and up front so we'll have one half in terms of function at zero plus four times a function at 3/2 and so on. I'm going to let you read that so I don't run out of breath here now. We just need to plug in. Okay, X equals or f of X equals eight minus log X squared minus two X plus four. Then execute that code again. And this is what you should have symbolically from plugging everything into the function. Then the numerical value you get should be 58.9125.


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