45 CAlculus in Context #4 Math 113 Chapter 4 How does our understanding of derivatives and logistic functions allow us to understand drug accumulation in the human body? Lexapro is an antidepressant that affects the levels of certain chemicals in the brain that may be altered in people suffering from depression or anxiety. A typical starting dose is 10 mg taken once daily. After the medication has been fully metabolized, it is eliminated continuously from the body at a rate of approximately 2.3% per hour. Steady-state concentrations are achieved within 7-10 days of administration. Source: https://pubmed.ncbi.nlm.nih.gov Kこ2.3% 1. A patient with takes their first dose of Lexapro. Use the exponential model P(t) = Poe-ktto show how much Lexapro is left one day (or 24 hours) later. Show your work. PCH) =10e-0.023+ P(24)=10e-0.023 (24 ら(5. 2. This patient takes a second dose of Lexapro 24 hours after the first dose. With the amount remaining from the previous dose (see question #1), how much will they mow have in their bloodstream? Show your work. I DAY こ15,76 3. How much of this amount is left 24 hours after taking that that second dose? Show your work. P(z4) =15,76e-0.023+ 9.01My) 2DAYS 4. Complete this table, displaying how Lexapro accumulates in the body over the course of 10 days. Amount Time Total Amount in Remaining from Previous Doses New Dose (days) Added (mg) Blood Stream P(24) = 19.07e - 0,023 + (t%384) (mg) (Bu) NA (+= 14), 5.76 P(24)320.98e-o,023 + 15.76 1. 10.98 19.07 20.98 3 P(z4)3D 22,08 e (h2=+) S DAYS .0. 12.71 0.0231(24k le DAYS 22.71 23.08 23.29 23,41 23.48 23.52 P(z4) = 22.71e- 9. Plzu) =23.08p -0.0234 = 23.08 8. 10 (4=24). .48 13,52 P(24) =23,29 e-o.023 t 6. P(24) = 23,41 e-o (+=24). Shra't Page 175 fl24) = 23.48e-0.0234 %3D 5. Run (as in for Stat→ edit table States calc→ log → Calaulate Time Total Amount in section 4.7) on your completed data table (see right) to find a model for the total amount in the blood stream as a function of t in hours. Call this model A(t). Store this model in Y1 in your calculator and then write the model Y%=DTitae-bx)) Let's examine what is happening using a different model: (days) Blood Stream (mg) 15.76 2. 3. .07 below. 4. 5. IL'22 9. b=0,86 23.29 23.41 23.48 C=23.30 8. 6. Use your rate of change of the amount in the blood stream as a function of time in hours A'(t).) Show your work in the space below. 6. Use vour derivative rules to find a model for the instantaneous 23,52 7. Use your model for A'(t) and/or the nDeriv function to calculate how quickly the Lexapro quantity is increasing: at 4 days c. at 6 days. at time 2 days b. a. 8. Using your understanding about the logistic model, what will happen to the total amount in the bloodstream if this patient continues this routine long- term? Draw a reasonable sketch of A(t) to support your claim.