WEBVTT
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all right, we're going to evaluate this difference quotient
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for the function F of X equals X cubed.
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And this kind of problem is really good practice for
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something that's coming up in another chapter or two.
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So we're going to need to find f of A
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plus h. Let's do that off to the side
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F of a plus age would be a plus age
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quantity cubed. We're also going to need to find
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1/2 of a after they would be a cubed.
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So let's go ahead and substitute those into the quotient
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. And we have a plus age quantity cubed minus
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a cubed over h. Okay, now the bulk
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of the work is going to be figuring out how
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to expand a plus h quantity. Cute. There
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is a shortcut for this. If you happen to
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know the binomial theorem and you know how to use
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Pascal's triangle as your coefficients, you'll get one A
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to the third power plus three, a squared age
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plus three, a age squared, plus one age
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cubed. If you don't know that binomial theorem shortcut
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, what you're going to do is multiply a plus
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h times a plus h times a plus H,
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and you can do that by multiplying two of them
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together using the foil method. And when you're done
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with that, then you multiply it by the third
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, so you get a squared plus two a h
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plus H squared. Now you multiply it by the
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3rd 1 And when you're done with all of that
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and you have combined the like terms, it's going
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to end up to be equivalent to what I have
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above. Okay, so expanding the try the binomial
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cubed gives us a cube plus three a squared age
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plus three a age squared minus a plus. H
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Q. Excuse me and then we still have minus
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a cubed at the end, and that's all over
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. H notice that we haven't a cube Jenna minus
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a cubed. Let's cancel those and let's look at
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what we have left. All three terms in the
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numerator have a factor of H, so if we
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factor h out of them, the other factor will
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be three. A squared plus three a h plus
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h squared, and that's all over H. So
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we can cancel the age from the top of the
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bottom. And what we have now is three a
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squared plus three age plus age squared