Averaging Polynomial & Non-Linear Functions: Examples & Explanations

In summary: The charge density of a sphere varies as \beta_t, but when I tried average, it doesn't work although it seems to be a linear variation. It doesn't work because when you average a function, you're taking the integral over a range and dividing that range by the number of elements in that range. When you're averaging a function, thedependent variable is usually changing (like the charge density in our case) so when you take the integral, the integrand is changing constantly.
  • #1
WiFO215
420
1
Now we've all been taught how to use the average. Let me give 2 examples to those who don't know.

Example 1: Say an object moves with velocity 3t in the time t=0 till t=2. Find distance covered.
Initial velocity = 0.
Final velocity = 6 disp. unit/ time unit.
Avg. Velocity = 3 disp. unit/ time unit.
Distance covered = Avg. velocity x time = 6 disp. units.
Using s = ut +1/2 a[tex]t^{2}[/tex] we get 6 again. Amazing!

Example 2: Force acting on a box of mass 1 unit is 3t in the time t=0 till t=2. Find work done by the Force. Box is initially at rest to your frame.
No other forces act on it.

Initial force = 0
Final force = 6 units.
Avg. force = 3 units.
Now avg. accn. = 3 units [mass = 1]
As in previous sum, displacement = 6 units.
Work done = 3 x 6 = 18 units. This comes out fine if you work it out the normal way also.

Now onto my questions.

If you noticed both were linear variations. How do I find the average of any polynomial function? I would find that VERY useful. For instance I found out for a cos/sin function average is 1/[tex]\sqrt{2}[/tex] of the co-efficient of the cos function. Isn't that fantastic?

Also one more. I was given a problem that the charge density of a sphere varies as [tex]\beta[/tex]t. But when I tried average, it doesn't work although it seems to be a linear variation.

Why doesn't it work?
 
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  • #2
Can you expand a bit on the problem at hand?
 
  • #3
You use integral calculus in the general case. Just like a simple average is the sum of the elements divided by the number of elements, a generalized average is the quotient of two integrals.
 
  • #4
To calculate the average (also called mean), integrate the function over the range of interest and divide by that range.
 
  • #5
berkeman said:
You use integral calculus in the general case. Just like a simple average is the sum of the elements divided by the number of elements, a generalized average is the quotient of two integrals.

For example, see the end of this:

http://math.cofc.edu/lauzong/Math105/Section%205.4%20Applying%20Definite%20Integral.pdf


.
 
Last edited by a moderator:
  • #6
I see. Thats awesome! Average is such a nice way of going about the problem. What about the sphere of charge? Why can't I average that?
 

1. What is the difference between a polynomial and a non-linear function?

A polynomial function is an expression that contains constants, variables, and exponents, and the terms are combined using addition, subtraction, and multiplication. Non-linear functions, on the other hand, include terms with variables raised to non-integer powers or involve trigonometric or exponential functions.

2. How do you determine the degree of a polynomial function?

The degree of a polynomial function is determined by the highest exponent in the expression. For example, a function with the term x^3 has a degree of 3.

3. Can you give an example of an averaging polynomial function?

One example of an averaging polynomial function is a moving average, where a set of data points is averaged over a specified number of points. For instance, a 3-point moving average of the data set [2, 4, 6, 8, 10] would result in the values [4, 6, 8], where each value is the average of 3 consecutive data points.

4. How do you find the average rate of change of a non-linear function?

The average rate of change of a non-linear function can be found by calculating the slope of the secant line between two points on the function. This is done by finding the difference in the function values between the two points and dividing it by the difference in the corresponding x-values.

5. What are some real-world applications of averaging polynomial and non-linear functions?

Averaging polynomial and non-linear functions have various applications in fields such as economics, engineering, and physics. Some examples include calculating average temperature changes, determining average stock market trends, and predicting the average growth rate of a population.

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