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In the last several videos we learned how we can approximate an arbitrary function

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but a function that is differentiable and twice and thrice differentiable and all off the rest

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how we can approximate a function around x is equal to zero using a polynomial. If we have a zeroth degree poynomial

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which is a constant, you can approximate it with a horizontal line that just goes through that point. Not a great approximation.

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If you have a first degree polynomial, you can at least get the slope right at that point.

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If you get to a second degree polynomial, you get something that hugs the function a little longer

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If you go a third degree polynomial, something that hugs the function even a little bit longer than that

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But all of that was focused on approximating the function around x = 0

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and that is why we call it the McCloran series or the Taylor series at x is equal to zero

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What I want to do now is expand it a little bit -- generalize it a little bit and focus on the Taylor

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expansion at x equals anything

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So let's say we want to approximate this functiono

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when x--this is our x-axis-- is equal to c

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So we could do it the exact same thing.

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We could say "Look out first approximation at c should be equal to a function. Whatever the function

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equals at c