Average Rate of Change of f(x) on [a,b]
The average rate of change of a function f(x) on the interval [a, b] is determined by finding the slope of the secant line that passes through the points (a, f(a)) and (b, f(b))
The average rate of change of a function f(x) on the interval [a, b] is determined by finding the slope of the secant line that passes through the points (a, f(a)) and (b, f(b)).
To calculate the average rate of change, follow these steps:
1. Determine the values of f(a) and f(b) by plugging in a and b into the function f(x). Note that f(a) and f(b) represent the y-coordinates of the points on the graph of f(x).
2. Calculate the change in y-coordinates by subtracting f(a) from f(b). This will give you the vertical change or the “rise” between the two points.
Change in y = f(b) – f(a)
3. Calculate the change in x-coordinates by subtracting b from a. This will give you the horizontal change or the “run” between the two points.
Change in x = b – a
4. Divide the change in y by the change in x to find the average rate of change.
Average Rate of Change = (f(b) – f(a)) / (b – a)
For example, let’s say we have the function f(x) = 2x + 3, and we want to find the average rate of change on the interval [1, 4].
Step 1:
Calculate f(a) and f(b):
f(1) = 2(1) + 3 = 5
f(4) = 2(4) + 3 = 11
Step 2:
Calculate the change in y:
Change in y = f(4) – f(1) = 11 – 5 = 6
Step 3:
Calculate the change in x:
Change in x = 4 – 1 = 3
Step 4:
Divide the change in y by the change in x:
Average Rate of Change = (6) / (3) = 2
Therefore, the average rate of change of f(x) = 2x + 3 on the interval [1, 4] is 2.
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