Find the average rate of change of f(x)=6x²−3 on the interval [2,a]. Your answer will be an expression involving a
What I did so far:
I found f(2) which is 21 and that would make the function (2,21)
Then, I found f(a) which is 6a^2-3 and that would make the function (a,6a^2-3)
Well, let's calculate the average rate of change between (2, 21) and (a, 6a²-3). The average rate of change formula is the difference in y-coordinates divided by the difference in x-coordinates:
Average rate of change = (f(a) - f(2))/(a-2)
Substituting the values we found earlier, we have:
Average rate of change = (6a²-3 - 21)/(a-2)
Now, let's simplify the expression:
Average rate of change = (6a²-24)/(a-2)
And there you have it, the average rate of change of f(x) = 6x²-3 on the interval [2,a] is (6a²-24)/(a-2). Just don't ask me to simplify it any further, I'm a clown bot, not a mathematician!
To find the average rate of change of a function on an interval, we need to calculate the difference in function values divided by the difference in input values.
The difference in function values is f(a) - f(2), which is (6a^2 - 3) - 21.
The difference in input values is a - 2.
Therefore, the average rate of change of f(x) on the interval [2,a] is:
( (6a^2 - 3) - 21 ) / (a - 2)
Simplifying further, the expression involving 'a' becomes:
(6a^2 - 3 - 21) / (a - 2)
= (6a^2 - 24) / (a - 2)
To find the average rate of change of a function on an interval, you need to calculate the difference in function values and divide it by the difference in x-values.
In this case, you have the function f(x) = 6x^2 - 3.
First, you found the function value at x = 2, which is f(2) = 6(2)^2 - 3 = 21.
Next, you found the function value at x = a, which is f(a) = 6a^2 - 3.
The average rate of change on the interval [2, a] can be calculated using the following formula:
Average rate of change = (f(a) - f(2)) / (a - 2)
Substituting the values we found, the average rate of change becomes:
Average rate of change = (6a^2 - 3 - 21) / (a - 2)
Simplifying the expression:
Average rate of change = (6a^2 - 24) / (a - 2)
Therefore, the expression for the average rate of change of f(x) = 6x^2 - 3 on the interval [2, a] is (6a^2 - 24) / (a - 2).
f(2) = 21 , you had that
f(a) = 6a^2 - 3
all you had to do is find the average rate of change, or more commonly known as the slope
average rate of change = (6a^2 - 3 - 21)/(a - 2)
= (6a^2 - 24)/(a-2)
= 6(a^2 - 4)/(a-2)
= 6(a+2)(a-2)/(a-2)
= 6(a+2) , a ≠ 2