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)x<(equal sign ) a is -1 units/unit

1 answer

The average rate of change of a function h(x) on an interval [x1, x2] is given by the formula:

average rate of change = (h(x2) - h(x1))/(x2 - x1)

In this case, h(x) = x^2 + 3x + 2, and the interval is -3 < x < a. We are given that the average rate of change is -1 units/unit. Plugging in these values into the formula, we get:

-1 = (h(a) - h(-3))/(a - (-3))

Now, let's substitute h(a) and h(-3) with their corresponding values:

-1 = ((a^2 + 3a + 2) - ((-3)^2 + 3(-3) + 2))/(a + 3)

Simplifying this expression:

-1 = (a^2 + 3a + 2 - 9 + 9 + 2)/(a + 3)
-1 = (a^2 + 3a - 5)/(a + 3)

Now, let's solve this equation for a. Multiply both sides by (a + 3):

-(a + 3) = a^2 + 3a - 5
-a - 3 = a^2 + 3a - 5
a^2 + 4a - 2 = 0

To solve this quadratic equation, we can use the quadratic formula:

a = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 1, b = 4, and c = -2. Plugging in these values into the quadratic formula:

a = (-(4) ± √((4)^2 - 4(1)(-2)))/(2(1))
a = (-4 ± √(16 + 8))/2
a = (-4 ± √24)/2
a = (-4 ± 2√6)/2
a = -2 ± √6

Therefore, the possible values of a that will give an average rate of change of -1 units/unit on the interval -3 < x < a are -2 + √6 and -2 - √6.