Problem 66
Question
Blood Flow It follows from Poiseuille's Law that blood flowing through certain arteries will encounter a resistance of \(R(x)=0.25(1+x)^{4}, \quad\) where \(x\) is the distance (in meters) from the heart. Find the instantaneous rate of change of the resistance at: a. 0 meters. b. 1 meter.
Step-by-Step Solution
Verified Answer
The rate of change is 1 at 0 meters and 8 at 1 meter.
1Step 1: Understand the Problem
We need to find the instantaneous rate of change of the resistance function at specified points. This means we need to find the derivative of the function and evaluate it at the given points.
2Step 2: Differentiate the Function
The resistance function is given as \( R(x) = 0.25(1 + x)^4 \). The derivative, \( R'(x) \), represents the instantaneous rate of change. We use the power rule to differentiate: \( R'(x) = 0.25 \times 4(1 + x)^{3} \times 1 = 1(1 + x)^3 \).
3Step 3: Evaluate the Derivative at 0 Meters
Substitute \( x = 0 \) into the derivative: \( R'(0) = 1(1 + 0)^3 = 1 \). This is the rate of change of resistance at 0 meters.
4Step 4: Evaluate the Derivative at 1 Meter
Substitute \( x = 1 \) into the derivative: \( R'(1) = 1(1 + 1)^3 = 8 \). This is the rate of change of resistance at 1 meter.
Key Concepts
Instantaneous Rate of ChangeResistance FunctionPower Rule
Instantaneous Rate of Change
The concept of the instantaneous rate of change is crucial in understanding the behavior of a function at a specific point. It gives us the rate at which the function is changing exactly at that instance, rather than over an interval. In everyday terms, think of it like the speedometer in your car; while the speedometer shows your speed at a precise moment, the instantaneous rate of change reveals how quickly some event occurs at a single point in time.
To find this rate for a function like the resistance function in Poiseuille's Law, we need to calculate the derivative. The derivative tells us how much the resistance increases or decreases as we move slightly along the artery.
This derivative, noted as \( R'(x) \), allows us to measure the rate at which resistance changes per meter from the heart. By evaluating \( R'(x) \) at specific points, such as 0 or 1 meter, we determine exactly how the resistance behaves at those locations.
To find this rate for a function like the resistance function in Poiseuille's Law, we need to calculate the derivative. The derivative tells us how much the resistance increases or decreases as we move slightly along the artery.
This derivative, noted as \( R'(x) \), allows us to measure the rate at which resistance changes per meter from the heart. By evaluating \( R'(x) \) at specific points, such as 0 or 1 meter, we determine exactly how the resistance behaves at those locations.
Resistance Function
In Poiseuille's Law, the resistance function \( R(x) = 0.25(1+x)^4 \) tells us how resistance varies with distance as blood flows through an artery. Here, \( x \) is the distance from the heart, and the output, \( R(x) \), indicates how strong the force is that resists the flow of blood.
To put it simply, resistance means the obstacles the blood faces as it travels, which can be affected by factors like the size of the artery or the blood's viscosity. In this function, the resistance doesn't remain constant; it changes with distance, and we have a mathematical model showing this relationship.
For example, when \( x = 0 \), meaning right from the heart, the resistance is straightforward to calculate. As \( x \) increases, the expression \( (1 + x)^4 \) shows that resistance grows exponentially. This highlights the complexity encountered further from the heart, where changes can have significant effects on the blood flow.
To put it simply, resistance means the obstacles the blood faces as it travels, which can be affected by factors like the size of the artery or the blood's viscosity. In this function, the resistance doesn't remain constant; it changes with distance, and we have a mathematical model showing this relationship.
For example, when \( x = 0 \), meaning right from the heart, the resistance is straightforward to calculate. As \( x \) increases, the expression \( (1 + x)^4 \) shows that resistance grows exponentially. This highlights the complexity encountered further from the heart, where changes can have significant effects on the blood flow.
Power Rule
The power rule is a fundamental tool in calculus, especially for finding derivatives. This rule simplifies the process of differentiation when dealing with functions of the form \( (x^n) \), making it easier to work with complicated expressions.
According to the power rule, if you have a function \( f(x) = x^n \), then the derivative \( f'(x) \) is \( nx^{n-1} \). This tells us how the slope of the function changes with respect to \( x \).
In the context of our resistance function \( R(x) = 0.25(1+x)^4 \), we apply the power rule to obtain \( R'(x) = 1(1+x)^3 \). By multiplying the original exponent by the coefficient and reducing the power by 1, we streamline finding the derivative.
This technique is crucial for handling the resistance function efficiently, giving us direct insight into how resistance varies at any point along the artery. It's like a mathematical shortcut that gives us deep understanding with less effort, ensuring precision in our calculations.
According to the power rule, if you have a function \( f(x) = x^n \), then the derivative \( f'(x) \) is \( nx^{n-1} \). This tells us how the slope of the function changes with respect to \( x \).
In the context of our resistance function \( R(x) = 0.25(1+x)^4 \), we apply the power rule to obtain \( R'(x) = 1(1+x)^3 \). By multiplying the original exponent by the coefficient and reducing the power by 1, we streamline finding the derivative.
This technique is crucial for handling the resistance function efficiently, giving us direct insight into how resistance varies at any point along the artery. It's like a mathematical shortcut that gives us deep understanding with less effort, ensuring precision in our calculations.
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