In Exercises 33–34, find and simplify the difference quotient [f(...

Question

In Exercises 33–34, find and simplify the difference quotient [f(x + h) - f(x)]/h, h =/= 0 for the given function. f(x) = -2x^2 + x + 10

Accepted Answer

Welcome back. I'm so glad you're here. We get to express the difference quotient. So that's going to have F of X plus H. And then that's going to subtract F of X and then it's going to be divided by our little friend H. Down here where age does not equal zero. We don't have to worry about the domain and then it gives us F of X. Which is this F of X equals three X squared plus five X minus two. So now um we're going to start with the F of X plus H. What that means is in this F of X anywhere that there was an X. We're going to now put in X plus H. So instead of three X squared, it's going to be three X plus H squared plus five instead of five X. It's going to be five X plus H and then minus two. Now we get to do a lot of expanding this out. So I've got the three times X plus H times X plus H plus five X plus H minus two. We'll do a quick foil. So first is going to be X squared, then the outside is going to be plus X. H. The inside is going to be plus X. H. And I'm just putting in that order for consistency and the last are going to be a church squared, then we'll bring down everything else three times all of that plus five X plus H minus two. So now we can distribute the three. So we've got three X squared. Well, quick combine these two. Um so that would be a two X. H. So three times two X. H. Is going to be plus six X. H. And then plus three times H squared is going to be plus three H squared. Now we can distribute that five as well and so that's going to be plus five X plus five H minus two. And there are no like terms in there that is a big old expanded thing. But the next thing we're going to do is subtract our original F. Of X. So this is all in the numerator and then we're going to subtract F. Of X. Which is minus three X squared plus five X minus two. And um one really important thing is we don't want to forget to distribute this negative to all three terms in here so we'll bring that down. We've got three X squared plus six X. H. Plus three H squared plus five X plus five H minus two. And then it's going to be a minus three X squared minus because minus times positive five X. And then a negative times a negative is going to be plus two. And now is the really fun part because we get to look for like terms that are going to cancel out. So we've got a three X squared and a minus three X squared. We have got a five X. And a minus five X. And we've got a negative two and a plus two. Now what have we got left? We've got six X. H. Plus three H. Squared Plus five H. We got rid of a bunch. That's great. And now here we don't want to forget that this whole thing this whole numerator the whole time is getting divided by our good friend H. On the bottom here. So can we do any more simplifying? Yeah we can all three of these terms up here have an H. In them so we can factor and H. Out front and so we've got a church goes in front of now six X. Plus three H. Plus five divided by our lovely friend H. And H. S. Cancel out. Goodbye dear friend. And now that just leaves us with this final answer of six X. Plus three H. Plus five. We look at our answer choices and that matches even though it's slightly in a different order. Um That matches our answer choice A. If we were to switch the three H. And put it in front. Well done you've done it we'll catch you on the next one

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