page 1 of 3
The theory of Taylor polynomials and Taylor series rests upon once crucial insight: in order to approximate a function, it is often enough to approximate its value and its derivatives (first, second, third, and so on) at one point. We will see why this is true in the next section; for now, we content ourselves with figuring out how to accomplish it.
Let us first examine polynomials themselves. Setting
|p(x) = anxn + ... + a1x + a0|
we see that
|p'(x)||=||nanxn-1 + ... +2a2x + a1|
|p''(x)||=||(n)(n - 1)anxn-2 + ... + (3)(2)a3x + 2a2|
|p(3)(x)||=||(n)(n - 1)(n - 2)anxn-3 + ... + (4)(3)(2)a4x + (3)(2)a3|
Substituting 0 for x in all of these functions yields
Indeed, we see a pattern emerging. If we set p(0) = p, then we may write
|p(i)(0) = i!ai|
for i = 0, 1, , n. For i > n, it is easy to see that p(i)(0) = 0.
Take a Study Break!