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.
h(-3) = 9 - 9 + 2 = 2
h(g) = g^2 + 3g + 2
average rate of change = (g^2 + 3g + 2 = 2)/(g+3)
= (g^2 + 3g)/(g+3)
= -1
g^2 + 3g = -g - 3
g^2 + 4g + 3 = 0
(g+1)(g+4) = 0
g= -1 or g = -4
but -3≤x≤-4 is not a valid interval
so g = -1
checking:
if g = -1
f(-1) = 0 and avg rate = (0-2)/(-1+3) = -1
if g=-4
g(-4) = 6, and avag rate = (6-2)/(-4+3) = -4 ≠ -1
To determine the value of "g," we need to find the average rate of change of the function and set it equal to -1.
The average rate of change of a function over an interval is given by the difference in the function values divided by the difference in the input values.
Step 1: Find the value of h(x) at the endpoints of the interval.
Let's substitute -3 and g into the function h(x) to get the function values at the endpoints:
h(-3) = (-3)^2 + 3(-3) + 2 = 9 - 9 + 2 = 2
h(g) = g^2 + 3g + 2
Step 2: Find the difference in function values.
The difference in function values is h(g) - h(-3):
h(g) - h(-3) = g^2 + 3g + 2 - 2 = g^2 + 3g
Step 3: Find the difference in input values.
The difference in input values is g - (-3) = g + 3.
Step 4: Set up and solve the equation for the average rate of change.
The average rate of change is the difference in function values divided by the difference in input values:
(g^2 + 3g)/(g + 3) = -1
Step 5: Solve the equation.
To solve for g, we need to multiply both sides of the equation by (g + 3) to eliminate the denominator:
g^2 + 3g = -g - 3
Next, move all the terms to one side to set up a quadratic equation:
g^2 + 3g + g + 3 = 0
g^2 + 4g + 3 = 0
Now, factor the quadratic equation:
(g + 1)(g + 3) = 0
Set each factor equal to zero and solve for g:
g + 1 = 0 or g + 3 = 0
g = -1 or g = -3
Therefore, the value of "g" that makes the average rate of change of h(x) equal to -1 on the interval -3 ≤ x ≤ g is either g = -1 or g = -3.
To find the value of "g" that makes the average rate of change of the function h(x) equal to -1 on the interval -3 ≤ x ≤ g, we can use the formula for average rate of change.
The average rate of change of a function over an interval is given by the formula:
average rate of change = (h(g) - h(-3)) / (g - (-3))
In our case, h(x) = x^2 + 3x + 2, so we need to find the value of "g" that makes the average rate of change -1.
Substituting the function into the formula, we have:
-1 = (g^2 + 3g + 2 - (-3^2 + 3(-3) + 2)) / (g - (-3))
Simplifying the formula:
-1 = (g^2 + 3g + 2 - 16 + 9 + 2) / (g + 3)
-1 = (g^2 + 3g - 3) / (g + 3)
Now, cross-multiplying:
-1 * (g + 3) = g^2 + 3g - 3
-g - 3 = g^2 + 3g - 3
Rearranging and simplifying:
g^2 + 4g = 0
Factoring out g:
g(g + 4) = 0
Therefore, g = 0 or g = -4.
Thus, the possible values of "g" that make the average rate of change of h(x) equal to -1 on the interval -3 ≤ x ≤ g are g = 0 or g = -4.