How do you find the average rate of change of #y = 9x^3 - 2x^2 + 6# between x = -4 and x = 2?

Question

William Abdalla · Accepted Answer

#-80#

The #color(blue)"average rate of change"# of y = f(x) 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 typical rate of change.

#color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)((f(b)-f(a))/(b-a))color(white)(a/a)|)))#

Here, b = 2 and a = -4.

#f(-4)=9(-4)^3-2(-4)^2+6=550#

#f(2)=9(2)^3-2(2)^2+6=70#

The typical rate of change from (2,70) and (-4,550) is

#(70-550)/(2-(-4))=(-480)/6=-80#

This indicates that between (-4,550) and (2,70), the average slope of all the lines tangent to the graph of f(x) is - 80.

Ezra Adams · Answer

undefined To find the average rate of change of $ y = 9x^3 - 2x^2 + 6 $ between $ x = -4 $ and $ x = 2 $, first calculate the values of the function at $ x = -4 $ and $ x = 2 $, then use the formula for average rate of change:

$$ \text{Average rate of change} = \frac{\text{Change in } y}{\text{Change in } x} $$

1. Substitute $ x = -4 $ into the function:
$$ y(-4) = 9(-4)^3 - 2(-4)^2 + 6 $$

2. Substitute $ x = 2 $ into the function:
$$ y(2) = 9(2)^3 - 2(2)^2 + 6 $$

3. Calculate $ y(-4) $ and $ y(2) $.

4. Calculate the change in $ y $:
$$ \text{Change in } y = y(2) - y(-4) $$

5. Calculate the change in $ x $:
$$ \text{Change in } x = 2 - (-4) $$

6. Divide the change in $ y $ by the change in $ x $ to find the average rate of change.