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How do you find the average net change?

Published in Calculus Concepts 3 mins read

The provided material only explains how to calculate net change, not the average net change. Therefore, I will explain both.

Net Change Explained

The net change between two points, let's call them a and b, is the difference in the function's output (or the y values) at those points. This can be calculated using the following formula:

Net Change = f(b) - f(a)

As the reference video explains, you take the function value at point b (f(b)) and subtract the function value at point a (f(a)) to find the net change between those two points.

Example from Video

The reference video ([Part of a video titled Net Change & Average Rate of Change - YouTube]()) specifically uses the example of finding the net change between x=2 and x=3. In this case, a=2 and b=3, so the formula becomes:

Net Change = f(3) - f(2)

Average Net Change Explained

While net change is the overall change from one point to another, the average net change considers the rate at which that change happens over that interval. In other words, the average rate of change (or average net change) is how much the function's output changes, on average, for every change in the input variable.

Formula for Average Net Change

To calculate this, we need to divide the net change by the change in the input variable:

Average Net Change = (f(b) - f(a)) / (b - a)

This formula finds the slope of the line between two points on the graph of the function.

Calculating the Average Net Change

Here's how to apply this formula in steps:

  1. Identify the points: Determine the a and b values (the start and end of your interval).
  2. Evaluate the function: Calculate f(a) and f(b), the function values at points a and b.
  3. Find the Net Change: Calculate f(b) - f(a).
  4. Calculate the change in input: Calculate b - a.
  5. Divide: Divide the net change by the change in the input to get the average rate of change: (f(b) - f(a)) / (b - a).

Example

Let's say we have a function f(x) = x2, and want to find the average net change between x=1 and x=3:

  1. Points: a = 1, b = 3
  2. Evaluate:
    • f(1) = 12 = 1
    • f(3) = 32 = 9
  3. Net Change: 9 - 1 = 8
  4. Change in Input: 3 - 1 = 2
  5. Average Net Change: 8 / 2 = 4

Therefore, the average net change from x=1 to x=3 for the function f(x) = x2 is 4.

Summary

Concept Formula Explanation
Net Change f(b) - f(a) The difference in function values between two points.
Average Net Change (f(b) - f(a)) / (b-a) The average rate at which the function's output changes for every unit of input change.

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