BIOMEDICAL: Dose-Response Curves. The relationship between the dosage, x, of a drug and the...

Question:

BIOMEDICAL: Dose-Response Curves. The relationship between the dosage, {eq}x{/eq}, of a drug and the resulting change in body temperature is given by {eq}f(x) = x^2(3-x){/eq} for {eq}0 \leq x \leq 3{/eq}. Make sign diagrams for the first and second derivatives and sketch this dose-response curve, showing all relative extreme points and inflection points.

Sign Diagrams

The sign diagrams are graphs where the relevant information is the sign of the function understudy, for each interval analyzed. If we have a function of one variable {eq}f(x) {/eq}, then the sign diagram is the numeric axis and over it, each interval has a sign associated with it. This diagram can be used to study inequalities.

Answer and Explanation:

We have the response function:

{eq}\begin{equation} f(x)=x^2(3-x) \end{equation} {/eq}

defined in the interval {eq}0\le x\le 3 {/eq}.

The first and second derivatives are:

{eq}\begin{align} f'(x)&=6 x - 3 x^2\\ f''(x)&=6-6x \end{align} {/eq}

The first derivative is positive when:

{eq}\begin{align} x(6-3x)>0\\ x(3x-6)<0\\ \end{align} {/eq}

Solving the equality we have:

{eq}\begin{align} x=0\\ x=2 \end{align} {/eq}

then the sign diagram (Figure 1) will be:

Figure 1: Sign diagrams for the first and second derivative

and the first derivative is bigger than zero in the interval {eq}0<x<2 {/eq}. The points {eq}x=0,x=2 {/eq} are possibles extremes, and using the second derivative test we obtain that {eq}f''(0)>0,f''(2)<0 {/eq}. Therefore {eq}x=0 {/eq} is a minimum, while the point {eq}x=2 {/eq} is a maximum.

For the second derivative we have:

{eq}\begin{align} 6-6x>0\Rightarrow\;\;x<1 \end{align} {/eq}

The sign diagram is presented in Figure 1:

The point {eq}x=1 {/eq} is an inflection point, since the second derivative change sign around this point.

Also, the point {eq}x=3 {/eq} is an extreme of the function (frontier of the domain).

The sketch of the function is presented in Figure 2

Figure 2: Response curve. All important points are presented


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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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