Question

For the function f(x)=2x²−12x+16, what is the average rate of change of f(x) over the interval [−3,−2]?

Answer 1

-22

Step-by-step explanation:

See the attached graph of this function. lets calculate the points on this line at the ends of the given interval [-3,-2]:

f(x)=2x²−12x+16

x f(x)

-3 70

-2 48

The slope at a point on the function is equal to the rate of change of that function. We can find the slope over the specified interval in one of 2 ways: Take the first derivative, or by calculating the slope of a line connecting the two endpoints. We'll do both:

Slope of Line

See the attached graph. We want to average rate of change between the two endpoints. Lets calculate the two endpoints and then find the slope of the line connecting them, which we could say would be the average of the slope for the curve between those two points.

From the information above, the Rise/Run, or slope, is - 22.

First Derivative

The first derivative will give us the slope of the line at point x. Let's take the first derivative and calculate the value for x of -2.5.

f(x)=2x²−12x+16

f'(x) = 4x−12

Assume x = - 2.5 will be a good estimate of the average slope between -3 and - 2.

f'(x) = 4x−12

f'(-2.5) = 4(-2.5)−12

f'(-2.5) = −22

The average rate of change, based on both approaches, is -22.

Answer 2

The average rate of change of the function f(x) over the interval [-3,-2] is 48.

Step-by-step explanation:

The average rate of change of a function over an interval is given by the formula: Average rate of change = (f(b) - f(a))/(b-a), where 'a' and 'b' are the endpoints of the interval. In this case, the interval is [-3,-2], so a = -3 and b = -2. Plugging these values into the formula:

Average rate of change = (f(-2) - f(-3))/(-2-(-3))

Substituting the function f(x) = 2x²−12x+16, we have:

Average rate of change = (2(-2)² - 12(-2) + 16)/(-2-(-3)) = (8 + 24 + 16)/(-2+3) = 48/1 = 48