Question

Find the average rate of change for f(x)=−2x²+4x+6 on the two given intervals:[−3,−1] is [ [2, 5] ) is

a) For [−3,−1]: ?, For [2,5]: ?
b) For [−3,−1]: ?, For [2,5]: ?
c) For [−3,−1]: ?, For [2,5]: ?
d) For [−3,−1]: ?, For [2,5]: ?

Answer 1

The average rate of change for the function f(x)=-2x²+4x+6 is 12 for the interval [−3,−1] and -10 for the interval [2, 5].

Step-by-step explanation:

The average rate of change of a function over an interval is computed by finding the difference in function values at the endpoints of the interval and dividing by the length of the interval. Let's calculate the average rate of change for the function f(x) = -2x² + 4x + 6 over the intervals [−3,−1] and [2,5].

Interval [−3,−1]

First, compute the function values at the endpoints:
f(−3) = -2(−3)² + 4(−3) + 6 = -18 - 12 + 6 = -24
f(−1) = -2(−1)² + 4(−1) + 6 = -2 - 4 + 6 = 0

The average rate of change is the change in f divided by the change in x, so:
Average rate of change on [−3,−1] = (f(−1) - f(−3))/(−1 − (−3)) = (0 - (-24))/2 = 24/2 = 12.

Interval [2,5]

Next, compute the function values at the endpoints:
f(2) = -2(2)² + 4(2) + 6 = -8 + 8 + 6 = 6
f(5) = -2(5)² + 4(5) + 6 = -50 + 20 + 6 = -24

So, the average rate of change is:
Average rate of change on [2,5] = (f(5) - f(2))/(5 − 2) = (-24 - 6)/3 = -30/3 = -10.