Taylor Polynomial

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Suppose that we have n-th degree polynomial `p(x)=a_0+a_1(x-a)+a_2(x-a)^2+...+a_(n-1)(x-a)^(n-1)+a_n(x-a)^n`, where `a,a_0,a_1,a_2,...,a_n` are constants.

Now, differentiate this polynomial `n` times:

`p'(x)=a_1+2*a_2(x-a)+3*a_3(x-a)^2+...+(n-1)*a_(n-1)(x-a)^(n-2)+n*a_n(x-a)^(n-1)`,

`p''(x)=1*2*a_2+2*3*a_3(x-a)+...+(n-2)(n-1)a_(n-1)(x-a)^(n-3)+(n-1)na_n(x-a)^(n-2)`,

`p'''(x)=1*2*3*a_3+...+(n-3)(n-2)(n-1)a_(n-1)(x-a)^(n-4)+(n-2)(n-1)na_n(x-a)^(n-3)`,

`..............................................................................................`

`p^((n))(x)=1*2*3*...*n*a_n`.

If we now plug `a` instead of `x` in polynomial and all its derivatives we will obtain that

`p(a)=a_0`, `p'(a)=a_1`, `p''(a)=1*2*a_2`, `p'''(a)=1*2*3*a_3`, ..., `p^((n))(a)=1*2*3*...*n*a_n`.

Since factorial of number is `n! =1*2*3*...*n` then

`a_0=p(a)`, `a_1=(p'(a))/(1!)`, `a_2=(p''(a))/(2!)`, ..., `a_n=(p^((n))(a))/(n!)`.

If we now substitute expressions for constants into polynomial we will obtain Taylor Fomula.

Taylor Formula for Polynomials. Polynomial`p(x)=p(a)+(p'(a))/(1!)(x-a)+(p''(a))/(2!)(x-a)^2+...+(p^((n-1))(a))/((n-1)!)(x-a)^(n-1)+(p^((n))(a))/(n!)(x-a)^n`. is called taylor polynomial at `x=a`.

With the help of Taylor formula we can write any polynomial in terms of `(x-a)`.

Example 1. Write polynomial `p(x)=x^3+3x^2-2x+1` in terms of `(x-2)`.

Here `a=2`.

We have that `p'(x)=3x^2+6x-2`, `p''(x)=6x+6`, `p'''(x)=6`. All higher-order derivatives will equal 0.

Now, `p(2)=2^3+3*2^2-2*3+1=17`, `p'(2)=3*2^2+6*2-2=22`, `p''(2)=6*2+6=18`, `p'''(2)=6`.

So,

`p(x)=17+(22)/(1!)(x-2)+(18)/(2!)(x-2)^2+(6)/(3!)(x-2)^3=17+22(x-2)+9(x-2)^2+(x-2)^3`.

So, equivalently polynomial `p(x)=x^3+3x^2-2x+1` can be written as `p(x)=(x-2)^3+9(x-2)^2+22(x-2)+17`.

Thus, Taylor formula for polynomials allows us to rewrite any polynomial in terms of `(x-a)`.

Now, let's see how we can use this idea for any differentiable functions.

Suppose that function `y=f(x)` has finite derivatives up to n-th order at point `a`.

Taylor Formula for any Function. For function `y=f(x)` n-th degree Taylor polynomial at point `x=a` is ` T_n(x)=f(a)+(f'(a))/(1!)(x-a)+(f''(a))/(2!)(x-a)^2+...+(f^((n-1))(a))/((n-1)!)(x-a)^(n-1)+(f^((n))(a))/(n!)(x-a)^n`.

Of course, `T_n(x)!=f(x)`, but as appeared `T_n(x)` is a very good approximation for `f(x)` when `x->a`. And the higher `n` (order of polynomial) the better approximation.

Fact. `T_n(x)~~f(x)` as `x->a`.

This fact allows us to approximate function by polynomial near point `x=a` with any precision we want, by taking high degree polynomial.

Example 2. Find first, third and fifth degree polynomials for function `f(x)=sin(x)` near taylor polynomial`x=0`.

Here `a=0`.

To find fifth degree polynomial we need derivatives up to fifth order.

We have that `f'(x)=cos(x)`, `f''(x)=-sin(x)`, `f'''(x)=-cos(x)`, `f^((4))(x)=sin(x)`, `f^((5))(x)=cos(x).`

Now, `f(0)=0`, `f'(0)=1`, `f''(0)=0`, `f'''(0)=-1`, `f^((4))(0)=0`, `f^((5))(0)=1`.

First degree taylor polynomial is `T_1(x)=f(0)+(f'(0))/(1!)(x-0)=0+1/1 x=x`.

Third degree taylor polynomial is `T_3(x)=f(0)+(f'(0))/(1!)(x-0)+(f''(0))/(2!)(x-0)^2+(f'''(0))/(3!)(x-0)^3=0+1/1 x+0/(2) x^2+(-1)/(6)x^3=x-(x^3)/6`.

Fifth degree taylor polynomial is `T_5(x)=f(0)+(f'(0))/(1!)(x-0)+(f''(0))/(2!)(x-0)^2+(f'''(0))/(3!)(x-0)^3+(f^((4))(0))/(4!)(x-0)^4+(f^((5))(0))/(5!)(x-0)^5=`

`=0+1/1 x+0/(2) x^2+(-1)/(6)x^3+0/(4!)x^4+1/(120)x^5=x-(x^3)/6+(x^5)/(120)`.

So,

`T_1(x)=x`, `T_3(x)=x-(x^3)/6`, `T_5(x)=x-(x^3)/6+(x^5)/120`.

Remember, the higher degree of polynomial, the better precision. Figure illustrates this. Note, that approximation good only near point of expansion, in this case `a=0`.