Average Growth Rates
The average growth rate of a function f(x) is the average rate of increase (or decrease) over an interval of the x-axis. Here is how to calculate the average growth rate:
The x-axis interval can be described as [a, b]. The average growth rate is:
( f(b) – f(a) ) / (b – a)
Example: if f(x) = x2, the average growth rate of f(x) on the interval [2, 4] is:
( 42 – 22 ) / (4 – 2) = (16 – 4) / (4 – 2) = 12/2 = 6
You have probably heard of average growth rates before. If f(x) = mx + b, the average growth rate is the slope. If you have studied distance-time problems, the average growth rate is the average speed.
For each of the functions below, calculate the average growth rates on two intervals: [0, 1] and [1, 2]. First think: what do you expect the answers to be. Do the answers fit your expectations?
1. f(x) = 2x + 5
2. f(x) = |x – 1|
3. f(x) = x – 1
4. f(x) = sqrt(x) (the square root of x). For this problem, take the square root of 2 to be 1.414.
5. f(x) = -(x – 1)2