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Approximating Functions With Polynomials

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Approximating Functions With Polynomials

Approximating Functions With Polynomials

Approximating Functions With Polynomials

Approximating Functions With Polynomials

Approximating Functions With Polynomials

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 .