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**Unformatted text preview: **abola than
it does a straight line. This suggests that it would make more sense to try and approximate
this function with a function of the form
p(x) = c0 + c1 (x − a) + c2 (x − a)2
DE Math 128 314 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series (You may have noticed that the form for this polynomial looks somewhat unusual. We
write it this way because it will make it easier to properly encode the information we have
about f (x) at x = a).
In constructing the linear approximation, we encoded the value of the function and of its
derivative at the point x = a. We want to again encode this local information, but we
want to do more. If we can add in the second derivative, we might be able to capture the
curvature of the function that was missing in the linear approximation. In summary, we
would like ﬁnd constants c0 , c1 , and c2 , so that
1. p(a) = f (a),
2. p (a) = f (a),
3. p (a) = f (a).
It may not seem immediately obvious that we can ﬁnd such constants. How...

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