Averaging Polynomial & Non-Linear Functions: Examples & Explanations

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Discussion Overview

The discussion revolves around the concept of averaging polynomial and non-linear functions, with specific examples provided for linear variations in velocity and force. Participants explore methods for calculating averages, particularly in the context of integral calculus, and raise questions about the challenges of averaging certain functions, such as charge density in a sphere.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant provides examples of calculating average velocity and work done using linear functions, noting the consistency of results.
  • The same participant questions how to find the average of polynomial functions and mentions a specific case for sine and cosine functions.
  • Another participant suggests using integral calculus to find generalized averages, comparing it to the simple average calculation.
  • Further clarification is offered on the method of integrating a function over a specified range to calculate the average.
  • A participant expresses curiosity about the difficulties encountered when trying to average the charge density of a sphere, seeking further explanation.

Areas of Agreement / Disagreement

Participants generally agree on the use of integral calculus for calculating averages, but the discussion remains unresolved regarding the specific challenges of averaging the charge density of a sphere.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the nature of the functions being averaged and the specific conditions under which the averaging methods apply. The exact reasons for the difficulties with the charge density averaging are not fully explored.

Who May Find This Useful

This discussion may be useful for students and professionals in physics and mathematics who are interested in understanding the methods of averaging functions, particularly in the context of applied problems and theoretical concepts.

WiFO215
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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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Can you expand a bit on the problem at hand?
 
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.
 
To calculate the average (also called mean), integrate the function over the range of interest and divide by that range.
 
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:
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?
 

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