Medicine in the blood: rates from data…

Name: ______________________ Date: ____________ Medicine in the blood: rates from data and from a model (9 points)

Table

The concentration of a medicine in a patient's blood is modeled by a differentiable function

C

, where

C(t)

is measured in milligrams per liter (mg/L) and

t

is measured in hours after the medicine is taken. Selected values of

C(t)

are shown in the table.

Concentration C(t), in milligrams per liter, at selected times t, in hours
$t$ (hours)	$0$	$1$	$3$	$4$	$6$	$8$
$C(t)$ (mg/L)	$0$	$7.4$	$12.2$	$12.0$	$9.9$	$7.3$

(a) Use the data in the table to estimate $C'(5)$ . Show the computations that lead to your answer, and indicate units of measure. [2 points]
(b) Interpret the meaning of $C'(5) \approx -1.05$ in the context of the problem. [1 point]
(c) Must there be a time $t$ , for $1 < t < 3$ , at which $C(t) = 10$ ? Justify your answer. [2 points]
(d) For $0 \le t \le 8$ , the concentration is also modeled by $M(t) = 10te^{-0.3t}$ . Find $M'(5)$ . Show the setup, and indicate units of measure. [2 points]
(e) Using the model $M$ , is the rate of change of the concentration increasing or decreasing at $t = 6$ ? Give a reason for your answer. [2 points]

The question