How do you find the average rate of change of #f(z) = 4 − 3z^2# from [-2,0]?

Question

Evelyn Agard · Accepted Answer

6

The #color(blue)"average rate of change"# of f(z) over an interval between 2 points (a ,f(a)) and (b ,f(b)) is the slope of the #color(blue)"secant line"# connecting the 2 points. in order to determine the average rate of change between the two usage points. #color(red)(|bar(ul(color(white)(a/a)color(black)((f(b)-f(a))/(b-a))color(white)(a/a)|)))# #f(0)=4-3(0)^2=4# #f(-2)=4-3(-2)^2=4-12=-8# The mean rate of variation between (0,4) and (-2,-8) is #(4-(-8))/(0-(-2))=12/2=6# This indicates that the average slope of all the lines tangent to the f(z) graph is 6.

Luke Alvarez · Answer

undefined To find the average rate of change of $ f(z) = 4 - 3z^2 $ from $ z = -2 $ to $ z = 0 $, you use the formula:

$$ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} $$

where $ a $ and $ b $ are the endpoints of the interval. In this case, $ a = -2 $ and $ b = 0 $.

$$ f(-2) = 4 - 3(-2)^2 = 4 - 3(4) = 4 - 12 = -8 $$
$$ f(0) = 4 - 3(0)^2 = 4 $$

Plug these values into the formula:

$$ \text{Average rate of change} = \frac{4 - (-8)}{0 - (-2)} = \frac{12}{2} = 6 $$

So, the average rate of change of $ f(z) = 4 - 3z^2 $ from $ z = -2 $ to $ z = 0 $ is $ 6 $.