   Chapter 10.4, Problem 10E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Medication For Problems 9 and 10, consider that when medicine is administered, reaction (measured in change of blood pressure or temperature) can be modeled by R = m 2 ( c 2 − m 3 ) where c is a positive constant and m is the amount of medicine absorbed into the blood (Source: R. M. Thrall et al., Some Mathematical Models in Biology, U.S. Department of Commerce, 1967).The rate of change of reaction R with respect to the amount of medicine m is defined to be the sensitivity.(a) Find the sensitivity, S.(b) Find the amount of medicine that is being absorbed into the blood when the sensitivity is maximum.

(a)

To determine

To calculate: The sensitivity which is the rate of change of reaction R w.r.t amount of medicine.

Explanation

Given Information:

The equation for reaction is given as

R=m2(c2m3)

Where c is a constant and m is the amount of medicine absorbed in the blood and sensitivity S is the rate of change of reaction R with respect to the amount of medicine m.

Formula used:

To find the maximum value, calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to 0. the relevant value is the stationary value that satisfies the provided conditions.

If the second derivative of the provided function is less than zero, then substituting the value of the independent variable will give the maximum value of the equation.

The power rule is used for a function in which the expression can be written as every term raised to a power (be it fractional, positive or negative). For the function f(x)=xn, the derivative is

ddx[xn]=nxn1

The product rule is used when the function is a product of two differentiable terms. If p(x)=f(x)g(x), then the derivative is

p'(x)=f'(x)g(

(b)

To determine

To calculate: The amount of medicine being absorbed at maximum sensitivity which is the rate of change of reaction R w.r.t amount of medicine.

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