In Exercises 13-18, find the average rate of change of the functi...

Question

In Exercises 13-18, find the average rate of change of the function from 1 to 2. f(x) = x² + 2x from x₁ = 3 to x2 = 5

Accepted Answer

Welcome back. I'm so glad you're here. We're given this equation F of X equals X to the one third plus X. Were also given these two X coordinates, X. Sub one equals eight and X. Sub two equals 27 were asked to determine the rate of change from X. Sub +12, X sub to what we recall from previous lessons that the rate of change formula is F of X. Sub two minus F of X. Sub one. All divided by X. Sub two minus X. Sub one. And some of these are pretty straightforward to figure out and some of them take a little bit of work. So let's start off with the first one, F of X. Sub two to determine F of X. Sub two. We need to know a couple of things. We need to know what is F of X. And that's given to us F. Of X is X to the one third plus X. And the other thing we need to know is what is X. Sub two and X. Sub two is 27. So that means anywhere that we had an X. Sub two in our original F of X. We are going to substitute A 27 so not X to the one third, but now 27 raised to the one third plus not X but 27. Now we'll do that. Simplification. 27 to the one third is the same as the cubed root of 27 then bring down the plus 27. The cubed root of 27 is +33 plus, 27 equals 30. So F of X. Sub two is 30. Great. That's one down now. Let's work on F of X. Sub one. Well when we've got F of X sub one we need to know a couple of things like the other one. We need to know F of X. I'll rewrite that over here. So F of X equals X to the one third plus X. The other thing we need to know is what is X. Sub one and X sub one is eight. That means anywhere we had an X. In our original F of X equation, we're going to substitute an eight so no longer X to the one third. It's now eight to the one third plus no longer X. But now eight doing the simplification here. Eight to the one third is the same as the cubed root of eight. Bringer plus eight down. The cubed root of eight is two plus +82 plus eight is 10. So F of X. Sub one is 10, write that up here. Well the X. Up to and the ex of one are a little bit easier to identify. They're given to us except to is 27 X. Sub one is eight. Now let's copy this down in the numerator, we've got 30 minus 10 and then the denominator we've got 27 minus 8, 30 minus 10 is, 2027 minus eight is 19. That is as simplified as that one is going to get. So our rate of change from X sub 12 X sub two is 39th. We look at our answer choices and that matches with answer choice. D. Well done, we'll catch you on the next one.

In Exercises 13-18, find the average rate of change of the function from 1 to 2. f(x) = x² + 2x from x₁ = 3 to x2 = 5

Key Concepts

Average Rate of Change

Function Evaluation

Quadratic Functions

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