Please help..
Determine the value of "g" so that the average rate of change of the function h(x)=x^2+3x+2 on the interval -3≤x≤g is -1. Thank you.
To determine the value of "g" such that the average rate of change of the function h(x)=(x^2+3x+2) over the interval -3≤x≤g is -1, we need to use the formula for average rate of change.
The average rate of change of a function over an interval is calculated by finding the difference in the function values at the endpoints of the interval, divided by the difference in the input values (x-values) of those points.
In this case, the specific average rate of change we are given is -1, and the interval is -3≤x≤g.
Let's calculate the average rate of change using the given formula:
Average rate of change = (Change in function values) / (Change in input values)
For the function h(x) = x^2 + 3x + 2, we need to find the value of h at the endpoints of the interval.
At x = -3:
h(-3) = (-3)^2 + 3(-3) + 2 = 9 - 9 + 2 = 2
At x = g:
h(g) = g^2 + 3g + 2
The change in function values is h(g) - h(-3), and the change in input values is g - (-3) = g + 3.
So, we have the following equation for the average rate of change:
(-1) = (h(g) - h(-3)) / (g + 3)
Substituting the function values calculated earlier:
-1 = (g^2 + 3g + 2 - 2) / (g + 3)
Simplifying the equation:
-1 = (g^2 + 3g) / (g + 3)
To solve this equation, we can rearrange it:
g^2 + 3g = -g - 3
Combining like terms:
g^2 + 4g + 3 = 0
Now, we can factor the quadratic equation:
(g + 1)(g + 3) = 0
Setting each factor equal to zero and solving for g:
g + 1 = 0 or g + 3 = 0
Solving each equation:
g + 1 = 0 implies g = -1
g + 3 = 0 implies g = -3
Therefore, the possible values for "g" are -1 or -3.