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The Taylor Series

Approximating Functions With Polynomials

Terms

Approximating Functions With Polynomials, page 2

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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 .

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