Question

Find the average rate of change of f(x) = -2x^2 + 5:

a) From 0 to 2
b) From 3 to 5
c) From -2 to 1
d) From -1 to 3

Answer 1

The average rate of change of f(x) = -2x^2 + 5 is -3.0 from 0 to 2, -14.0 from 3 to 5, 3.0 from -2 to 1, and 5.0 from -1 to 3, indicating decreases or increases in function values over those intervals.

Step-by-step explanation:

To find the average rate of change of the function f(x) = -2x2 + 5, we calculate the change in function value over the change in x for the given intervals. The formula for the average rate of change is given by the difference quotient, f(b) - f(a) / (b - a), where [a, b] is the interval.

1. From 0 to 2: Average rate of change = (f(2) - f(0)) / (2 - 0) = (-2(2)2 + 5 - (5)) / 2 = -3.0
2. From 3 to 5: Average rate of change = (f(5) - f(3)) / (5 - 3) = (-2(5)2 + 5 - (-2(3)2 + 5)) / 2 = -14.0
3. From -2 to 1: Average rate of change = (f(1) - f(-2)) / (1 - (-2)) = (-2(1)2 + 5 - (-2(-2)2 + 5)) / 3 = 3.0
4. From -1 to 3: Average rate of change = (f(3) - f(-1)) / (3 - (-1)) = (-2(3)2 + 5 - (-2(-1)2 + 5)) / 4 = 5.0

Remember, a positive result indicates an overall increase in the function values over the interval, while a negative result indicates a decrease.