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.
Sure, I can help you with that!
To find the value of "g" such that the average rate of change of the function h(x) = x^2 + 3x + 2 on the interval -3 ≤ x ≤ g is -1, 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 formula:
Average rate of change = (change in y) / (change in x)
In this case, the change in y is h(g) - h(-3), and the change in x is g - (-3). Since we're given that the average rate of change is -1, we can write the equation as:
-1 = (h(g) - h(-3)) / (g - (-3))
First, let's find h(g) and h(-3) by substituting the values of "g" and "-3" into the function h(x):
h(g) = g^2 + 3g + 2
h(-3) = (-3)^2 + 3(-3) + 2
Now we can substitute these values into the equation and solve for "g":
-1 = (g^2 + 3g + 2 - ((-3)^2 + 3(-3) + 2)) / (g - (-3))
Simplifying the equation gives:
-1 = (g^2 + 3g + 2 - 4) / (g + 3)
-1 = (g^2 + 3g - 2) / (g + 3)
Cross-multiplying gives:
-(g + 3) = g^2 + 3g - 2
Rearranging the equation gives a quadratic equation:
g^2 + 2g - 5 = 0
Now we can solve this quadratic equation to find the values of "g". You can use the quadratic formula:
g = (-2 ± √(2^2 - 4(1)(-5))) / (2(1))
Simplifying gives:
g = (-2 ± √(4 + 20)) / 2
g = (-2 ± √24) / 2
g = (-2 ± 2√6) / 2
Simplifying further gives two possible values for "g":
g1 = -1 + √6
g2 = -1 - √6
Therefore, the possible values of "g" such that the average rate of change of h(x) on the interval -3 ≤ x ≤ g is -1 are g1 = -1 + √6 and g2 = -1 - √6.
I hope this explanation helps! Let me know if you have any further questions.