The aortic pressure-diameter relation in a particular...

The aortic pressure-diameter relation in a particular patient who underwent cardiac catheterization can be modeled by the polynomial $$\begin{aligned} D(p)=0.000002 p^{3}-& 0.0008 p^{2}+0.1141 p+16.683 \\ & 55 \leq p \leq 130 \end{aligned}$$ where $D(p)$ is the aortic diameter (in millimeters) and $p$ is the aortic pressure (in mmHg). Determine where this function is increasing and where it is decreasing within the interval given.

The aortic pressure-diameter relation in a particular patient who underwent cardiac catheterization can be modeled by the polynomial
$$\begin{aligned} D(p)=0.000002 p^{3}-& 0.0008 p^{2}+0.1141 p+16.683 \\ & 55 \leq p \leq 130 \end{aligned}$$
where $D(p)$ is the aortic diameter (in millimeters) and $p$ is the aortic pressure (in mmHg). Determine where this function is increasing and where it is decreasing within the interval given.

Key Concepts

Domain Analysis

Domain analysis in the context of this problem involves working within a specific interval over which the function is defined or of interest. It ensures that the conclusions drawn about the function's behavior, such as intervals of monotonicity, are relevant to the condition or constraints provided in the model.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points often indicate potential local maxima, minima, or points of inflection and play a key role in analyzing and partitioning the domain of the function into intervals of increasing or decreasing behavior.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. It is a fundamental tool in calculus used to analyze the behavior of functions, including determining intervals of increase or decrease.

Polynomial Function

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. These functions are widely used in modeling because of their smooth and predictable behavior, making them useful for approximating relationships in physical and engineering contexts.

Monotonicity

Monotonicity refers to the property of a function to be either entirely non-increasing or non-decreasing over an interval. By examining the sign of the derivative, one can determine where a function is increasing (positive derivative) or decreasing (negative derivative), which is crucial for understanding the behavior of the function.

Transcript

00:01 For this problem, we have been given a function showing the relationship between the aortic diameter in millimeters and the aortic pressure.

00:10 Now, we want to find out where this function is increasing and where it is decreasing.

00:17 To do that, we start by finding the derivative.

00:19 So the derivative of this function is going to equal 0 .1, 2, 3, 4, 5 zeros with a 6.

00:33 P squared minus 0 .0016p plus 0 .114.

00:43 I'm going to put a little green box around this.

00:45 We are going to reference this a couple times in this problem.

00:48 So just so we don't lose it in amongst everything else.

00:51 This is the equation for the derivative.

00:54 Now, to know where our function is increasing and decreasing, it can be it'll change whether it's increasing or decreasing at points where this derivative equals 0.

01:04 So i'm going to set this equal to zero and solve.

01:07 Now, since these are not very nice, neat numbers here, lots of decimals, we're going to go to the quadratic formula to find my zeros for this function.

01:17 So as a reminder, this is my a, this is my b, including that negative sign, and this my c.

01:27 So quadratic formula says p equals negative b, so the opposite of b, plus or minus the square root of b squared minus four times a times c all over to a.

01:56 And if you put that into your calculator, you'll get two values for p.

02:01 P either equals 259 .3 or p equals 7 .3.

02:11 Make that a little bit easier to read...

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