determine the value of a so that the average rate of change of the function h(x)=x*2+3x+2 on the interval -3 equal sign less than x greater sign equal sign a
To find the average rate of change of a function on an interval, we need to calculate the difference in the function values at the endpoints of the interval and divide it by the difference in the x-values.
For the interval -3 ≤ x ≤ a, the x-values range from -3 to a.
The function h(x) = x^2 + 3x + 2.
At the endpoints:
For x = -3, h(x) = (-3)^2 + 3(-3) + 2 = 4.
For x = a, h(x) = a^2 + 3a + 2.
So, the average rate of change is (h(a) - h(-3)) / ((a) - (-3)).
Plugging in the values, we have:
(h(a) - 4) / (a + 3).
We need to find the value of "a" for which the average rate of change of h(x) on the interval -3 ≤ x ≤ a is equal to a.
So, we set the expression equal to a and solve for "a":
(h(a) - 4) / (a + 3) = a.
Now, we substitute h(a) with the function h(x) = x^2 + 3x + 2:
[(a)^2 + 3(a) + 2 - 4] / (a + 3) = a.
(a^2 + 3a - 2) / (a + 3) = a.
Multiplying both sides by (a + 3):
(a^2 + 3a - 2) = a(a + 3).
Expanding and rearranging terms:
a^2 + 3a - 2 = a^2 + 3a.
Subtracting a^2 and 3a from both sides:
-2 = 0.
This equation has no solutions for "a". Therefore, there is no value of "a" for which the average rate of change of h(x) on the interval -3 ≤ x ≤ a is equal to "a".