MTH 241 Section 3.4 Average Rate of Change
How fast are y-values changing with respect to the x-values?
Average Rate of Change =
Δy
Δx
For linear functions: y = mx + b,
Δy = m, the slope.
Δx
In general: The average rate of change over the interval [a, b] is:
Δf =
Δx
f(b) - f(a)
b-a
Note that the average rate of change is really the slope of_______________________
EXAMPLE: A linear case:
quantity, q
10
20
30
Cost C(q)
15
30
45
Find Δx, Δy, the rate of change and interpret these values.
EXAMPLE: A non­linear case:
A revenue function is R(q) = ­20q2 + 160q
Graph on your calculator or grid paper.
a) Use values from the calculator to find the average rate of change between q = 0 and q = 5.
b) Find the average rate of change over the interval [1, 5].
In general: The average rate of change over the interval [a, b] is:
Δf =
Δx
f(b) - f(a)
b-a
Another notation for the same formula:
The average rate of change over the interval [a, a + h] =
f(a + h) - f(a)
h
Refer to previous example where Revenue = ­20q2 + 160q.
For the interval [1, 5] a = 1 and 5 = a + h. Therefore h = ____
EXAMPLE: Calculate the average rate of change of the given function over the intervals [a, a + h] where h = 1, .1, .01, .001 and .0001 f(x) = x2
2 (where a = 1)
EXAMPLE: Calculate the average rate of change on the given interval.
PRACTICE:
Problems #12 and #33

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