Find the average rate of change of f(x) = x^2 - 4x from x_1 = 5 t...

Question

Find the average rate of change of f(x) = x^2 - 4x from x_1 = 5 to x_2 = 9.

Accepted Answer

Hello today we're going to find the average rate of change of a given function. So the function given to us is F. Of X. Is equal to negative X squared plus nine X. And we are told that we want to find the average rate of change between the intervals of of two and of six. So I'm gonna go ahead and label my first interval as A. And we're going to set that equal to two. And my second interval I'm gonna label that as B. And set that equal to six. So how do we find the rate of change between these two intervals? Well let's go ahead and recall the rate of change. The rate of change states that we plug in our second interval B subtract that from our first interval A. And that's going to be divided by b minus A. So what we are doing is we're gonna plug in our second interval into our function and plug in our first interval into our function. And we're gonna take the difference of those two values and divided by the value of our second interval minus the value of our first interval. So in order to start this let's go ahead and figure out what F. Of B is Well B is equal to six. So what we need to solve is f. of six. And in order to do that I'm gonna go ahead and plug in six for every X. Inside of our function. What that does is give me negative six squared plus nine times six Simplifying this is going to give me negative 36 plus 54 and that's going to give me the value Of eight positive 18. So positive 18 is going to be our f of B. So let's just go ahead and circle that and now we need to go ahead and figure out what F. Of A. Is well F. Of A is equal to two. So we're gonna solve for F. Of two. So again I'm gonna go ahead and plug in two for every X in R function. Doing so is going to give me negative two squared plus nine times two. And simplifying this is going to give me negative four plus 18, negative four plus 18 is going to give me positive 14 and that's going to be equal to R. F. Of A. Now we already know what our B and A. Is. So now we want to take these four pieces of information and directly plug them in to our rate of change. So doing so is going to be F of B which is minus F. Of A, which is 14. And that's going to be divided by b minus A. Or in this case here six minus two. So 18 -14 is going to give me positive four and 6 -2 is going to give me four. So four divided by four gives me one which is going to be our average rate of change and this is going to be our solution. That also means that the solution to our problem is going to be D. So I hope this video helps you in understanding how to find the average rate of change, and I'll go ahead and see you all in the next video.

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