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Point) Given the functionf(x) = 2x? + 3x _ 7find the following_ (a) the average rate of change of on [~4,-I]: (b) the average rate of change of f on [x,x + h]:...

Question

Point) Given the functionf(x) = 2x? + 3x _ 7find the following_ (a) the average rate of change of on [~4,-I]: (b) the average rate of change of f on [x,x + h]:

point) Given the function f(x) = 2x? + 3x _ 7 find the following_ (a) the average rate of change of on [~4,-I]: (b) the average rate of change of f on [x,x + h]:



Answers

Compute the average rate of change of the given function over the interval $[x, x+h] .$ Here we assume $[x, x+h]$ is in the domain of the function. . $f(x)=3 x^{2}+2 x-7$

We need to find the average rate of change of the function over the specified interval. So we're going to evaluate the function f at two and subtract the function evaluated at negative four over two -A -4. So we get three times two squared Plus two times 2 -7- in parentheses. three times negative four squared Plus two times -4 -7. All over 2-plus 4. Do we get three times 4 Plus 4 -7- and parentheses three times 16 -8 -7 over six. So we get 12 plus four minus seven. We distribute our one. So negative 16 times three is negative 48 plus eight plus 7/6. So we get 12 plus four minus seven minus 48 plus eight plus seven. Well that's negative 24/6 which reduces to negative four.

They want us to state Samara definitions from what we learned in the section. Well, for the first part, they're asking the the average rate of change between the points and B is a slope of what line between the point's a comma every way and become ever be well, that one is just the slope of the seeking line. That is the line that passes between both points of our graph. And now I might be asking the average rate of change of the linear function between any two points is the slope. So three, they want a G train back from the section as well. We'll be proven in related problems. Always remember, any lion option of the line can be written the, uh, m X speak. And we have it in the form right now so we can tell what soup is always. It's m. You know, in case it was three. Okay,

And part pain. The blank space provided right seek and no for part B for a function F of X is three x plus five. Well, the average so for F of X is three x plus five. As we know, average rate of change is F of B minus F of a over. Do you mind to say so? That's F of B minus F of a over B minus a. So, in this case, half of these three b plus five minus this coos three. A oppose five over B minus A. So what you'll end up with is three three B minus three a over B minus a factor. Three out three and two B minus a over B minus A. So your answer is three. So in the blank space provided boat three.

For this question were using the formula off of being minus half of one over B minus one. So let's plug in off of B minus half of one, so half of one means plug in one into the function over so divided by B minus one. This simplifies to four B squared minus four over B minus swan, which is we can just factor the talk for B squared minus one over B minus one. We can actually further factor the top. Split it up into B plus one times B minus one crescent, and your final solution is four parentheses, B plus one. And because we crossed the soft, this is something called a hole, which another word means that be cannot equal one because we can't divide by one or else we'll end up with undefined, which is impossible.


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