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) = a _{n} x ^{n} + ^{ ... } + a _{1} x + a _{0} |
we see that
p'(x) | = | na _{n} x ^{n-1} + ^{ ... } +2a _{2} x + a _{1} | |
p''(x) | = | (n)(n - 1)a _{n} x ^{n-2} + ^{ ... } + (3)(2)a _{3} x + 2a _{2} | |
p ^{(3)}(x) | = | (n)(n - 1)(n - 2)a _{n} x ^{n-3} + ^{ ... } + (4)(3)(2)a _{4} x + (3)(2)a _{3} |
Substituting 0 for x in all of these functions yields
p(0) | = a _{0} | ||
p’(0) | = a _{1} | ||
p’’(0) | = 2a _{2} | ||
p ^{(3)}(0) | = 6a _{3} |
Indeed, we see a pattern emerging. If we set p ^{(0)} = p , then we may write
p ^{(i)}(0) = i!a _{i} |
for i = 0, 1,…, n . For i > n , it is easy to see that p ^{(i)}(0) = 0 .