Avg Value Of A Function: Word Problem

In summary, the conversation discusses finding the average velocity and comparing it to the maximum velocity of blood flow in a blood vessel. The integral form is used to find the average velocity and the maximum velocity is determined by setting the equation for velocity as a function of distance equal to zero. The conversation also suggests drawing a picture to better understand the concept and to make an educated guess as to where the maximum velocity occurs.
  • #1

Homework Statement


http://www.webassign.net/www20/symImages/2/1/9a952b874da3363f9e9d0b7744eaff.gif
The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is v(r), where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood.

(a)Find the average velocity v_ave (with respect to r) over the interval 0 ≤ r ≤ R.

(b)Compare the average velocity v_ave with the maximum velocity v_max.(v_ave/v_max)

So I need to set the problem up, into an integral form, to find the avg velocity for part A. The integration limits for the integral should be from zero to Big R, multiplied by (1 over big R) to get our average value. Now the problem is, every thing in the equation except the (1/4) is a variable. How would I even begin to integrate this?

Thanks.
 
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  • #2
The only variable is r. The rest of those symbols are constants.
 
  • #3
Ah. Well now I've solved part A, how would go about part B?

(a) is PR[tex]^{3}[/tex]/6nlR

I have V_ave, what is V max?
 
  • #4
You have a formula for v as a function of r. At what value of r is it maximum?
 
  • #5
So if you make the equation v(R) instead of v(r), then the (R[tex]^{2}[/tex]-R[tex]^{2}[/tex]) = equal zero, so V_max is zero?
 
  • #6
Why did you do that? Think of it this way - the velocity at which the fluid flows depends on the distance from the center of the pipe. At a particular value of that distance, the velocity is maximum. Try drawing a picture. Plot v as a function of r. Also, first try to guess what it should be. Where do you think the water is flowing fastest? Near the center? Near the edge? Somewhere in between?
 
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What is the definition of "Avg Value Of A Function"?

The average value of a function is the average rate of change of the function over a given interval. It is calculated by dividing the total change in the function over the interval by the length of the interval.

How is the average value of a function calculated?

To calculate the average value of a function, you must first determine the total change in the function over the given interval. This can be done by subtracting the value of the function at the starting point of the interval from the value of the function at the end point of the interval. Next, divide the total change by the length of the interval. The resulting value is the average value of the function over the interval.

What is a word problem involving the average value of a function?

One example of a word problem involving the average value of a function is: "A car travels at a constant speed of 60 miles per hour for 3 hours. What is the average speed of the car over the 3-hour period?" In this problem, the average value of the function would be the average speed of the car, which would be calculated by dividing the total distance traveled (180 miles) by the length of time (3 hours), resulting in an average speed of 60 miles per hour.

What are some real-life applications of finding the average value of a function?

Finding the average value of a function is useful in many real-life situations, such as calculating average speed in a car or finding the average temperature over a given time period. It is also commonly used in economics and finance to calculate average growth rates or average rates of return.

What are some common misconceptions about the average value of a function?

One common misconception about the average value of a function is that it is the same as the median value. However, the median value is the middle value in a set of data, while the average value is the total change divided by the length of the interval. Another misconception is that the average value must be a value that actually appears in the set of data, when in reality it can be a value between two data points or even outside of the data set.

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