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<=x<=a is -1
h(a) = a^2 + 3 a + 2
h(-3)= 9 - 9 + 2
h(a) -h(-3) = a^2 + 3 a
so (a^2 + 3 a) /(a+3) = -1
a^2 + 3 a = -a - 3
a^2 + 4 a + 4 = 0
(a+2)(a+2) = 0
a = -2
Awesome Thanks!!
You are welcome :)
a^2 + 4 a = -a -3
a^2 + 4 a + 3 = 0
a = [ -4 +/- sqrt (16-12)]/2
a = -2 +/- 1
a = -1 or -3
Got it! thanks for the correction.
Well, to determine the value of 'a', we need to find when the average rate of change of the function h(x) is -1 on the interval from -3 to 'a'.
So let's start by finding the average rate of change of h(x) on this interval. The formula for average rate of change is (h(a) - h(-3))/(a - (-3)) = -1.
Plugging in the expression for h(x) into the formula, we get ((a^2 + 3a + 2) - ((-3)^2 + 3(-3) + 2))/(a - (-3)) = -1.
Simplifying the numerator and denominator gives us (a^2 + 3a + 2 - 16 + 9 + 2)/(a + 3) = -1.
Combining like terms in the numerator gives (a^2 + 3a - 3 + 2)/(a + 3) = -1.
Simplifying the numerator further gives (a^2 + 3a - 1)/(a + 3) = -1.
Now, if we multiply both sides of the equation by (a + 3), we get a^2 + 3a - 1 = -a - 3.
Combining like terms gives a^2 + 4a + 2 = 0.
Now, we can solve this quadratic equation using the quadratic formula.
The solutions are a = (-4 ± √(4^2 - 4(1)(2)))/(2(1)).
Simplifying gives a = (-4 ± √(16 - 8))/2.
Further simplifying gives a = (-4 ± √8)/2.
Finally, simplifying gives a = (-4 ± 2√2)/2.
So, the two possible values of 'a' are (-4 + 2√2)/2 and (-4 - 2√2)/2.
Now, here's where the humor comes in... unfortunately, I can't simplify it any further because I'm just a silly Clown Bot and I can't do square roots. But hey, at least we got this far with the math! Keep solving those equations and remember to find the funny in everything!
To determine the value of 'a' such that the average rate of change of the function h(x) on the given interval is -1, we need to find the average rate of change and set it equal to -1, and then solve for 'a'.
The average rate of change of a function on an interval [a, b] is given by the formula:
Average rate of change = (f(b) - f(a))/(b - a)
In this case, our interval is -3 <= x <= a, so our starting point is -3 and our ending point is 'a'. The function h(x) = x^2 + 3x + 2.
Let's calculate the average rate of change:
Average rate of change = (h(a) - h(-3))/(a - (-3))
Substituting h(x) = x^2 + 3x + 2:
Average rate of change = ((a^2 + 3a + 2) - ((-3)^2 + 3(-3) + 2))/(a - (-3))
Simplifying:
Average rate of change = (a^2 + 3a + 2 - 16)/(a + 3)
Average rate of change = (a^2 + 3a - 14)/(a + 3)
Now, set the average rate of change equal to -1:
(a^2 + 3a - 14)/(a + 3) = -1
Multiply both sides by (a + 3) to eliminate the denominator:
a^2 + 3a - 14 = -a - 3
Rearranging the equation:
a^2 + 4a - 11 = 0
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring does not yield integer solutions, so let's use the quadratic formula:
a = (-4 +/- sqrt(4^2 - 4(1)(-11))) / (2*1)
Simplifying:
a = (-4 +/- sqrt(16 + 44)) / 2
a = (-4 +/- sqrt(60)) / 2
a = (-4 +/- 2*sqrt(15)) / 2
Simplifying further:
a = -2 +/- sqrt(15)
So, the two possible values for 'a' are:
a = -2 + sqrt(15)
a = -2 - sqrt(15)
These are the values of 'a' such that the average rate of change of the function h(x) = x^2 + 3x + 2 on the interval -3 <= x <= a is -1.