Taylor Polynomials

Review of the Tangent Line

 Recall that if f(x) is a function, then f '(a) is the slope of the tangent line at x = a.  Hence

        y - f(a) = f '(a)(x - a)  

or

        P1(x) = y = f(a) + f '(a)(x - a) 

is the equation of the tangent line.  We can say that this is the best linear approximation to f(x) near a.

Note:      P1'(x) = f '(a).

Quadratic Approximations

Example:  

Let 

        f(x) = e2x

Find the best quadratic approximation at  x = 0.

Solution  

Note 

        f '(x) = 2e2x  

and 

        f ''(x) = 4e2x

Let 

        P2(x) = a0 + a1x + a2x2  

Note 

         P'2(x) = a1 + 2a2x  

and 

        P''2(x) = 2a
Hence 

        a0 = 1
         

        P2'(0)  =  a1  =  f '(0)  =  2 

Hence

        a1 = 2
        P2''(0) = 2a2 = f ''(0) = 4 

Hence  

        a2 = 2

So

        P2(x) = 1 + 2x + 2x2

        

The Taylor Polynomial


Suppose that we want the best nth degree approximation to f(x) at x = a.  We compare f(x) to

        Pn(x) =  a0 + a1(x - a) + a2(x - a)2  + a3(x - a)3 + ... + an(x - a)n

We make the following observations:

        f(a) =  Pn(a) =  a0  

so that  

        a0 = f(a)

        f '(a) = P'n(a) =  a1 + 2a2(x - a)  + 3a3(x - a)2 + ... + nan(x - a)n-1 at x = a 

so that 

        a1 = f '(a)

        f ''(a) = P''n(a) =   2a2  + (3)(2)a3(x - a)+ ... + n(n - 1)an(x - a)n-2 at x = a 

so that 

        a2 = 1/2 f ''(a)

Note Each time we take a derivative we pick up the next integer in other words
        
                        1
        a3 =                  f '''(a) 
                    (2)(3)

If we define f(k)(a) to mean the kth derivative of f evaluated at a  then

                     1
        ak =              f(k)(a)
                    k!

In General

                         The Taylor Polynomial
The nth degree Taylor polynomial at x = c is

     Pn(x) =  f(c) + f '(c)(x - c) + f ''(c)/2!(x - c)2  + 
                  f (3)(c)/3! (x - c)3 + ... + f (n)(c)/n! (x - c)n

              = S f (k)(c)/k! (x - c)k
where the sum goes from 0 to n.



The special case when a = 0 is called the McLaurin Series
          The McLaurin Polynomial

The McLaurin Polynomial of a differentiable function f(x) is 
          S f (k)(0)/k! xk  

where the sum goes from 0 to n.


Examples:

Find the fifth degree McLaurin Polynomial for sin x

        f(0) = sin(0) = 0

        f '(0) = cos(0) = 1

        f ''(0) = -sin(0) = 0

        f (3)(0) = -cos(0) = -1

        f (4)(0) = sin(0) = 0

        f (5)(0) = cos(0) = 1

So that 

                                1              0               -1              0               5
        P5(x) = 0 +            x +          x2 +           x3 +          x4 +          x5 
                             1!             2!               3!              4!             5!

                     x3             x5
        =  x -             +                       
                    6              120

Taylor's Remainder


Taylor's Remainder Theorem says that any smooth function can be written as an nth degree Taylor polynomial plus a function that is of order n + 1 near x = c.

          Taylor's Remainder Theorem

If f is smooth from a to b, let Pn(x) be the nth degree Taylor polynomial at x = c, then for every x there is a z between x and c with

                                  f (n+1)(z)       
          f(x) = Pn(x) +                    (x - c)n+1   
                                   (n + 1)!


Example

We have

                                 (.1)3           (.1)5        -sin z
        sin(.1) = .1  -              +               +                = .099833416667 + E
                                  6             120            6!

Where 

                   1
        E <           (.1)6  = .0000000014
                  6!

 

Example

Use an 11th degree Taylor polynomial to approximate 

       

Solution

First notice that there is no elementary antiderivative.  Hence, we find the Taylor polynomial and then integrate.  We have

       

Plugging in x2 for x, gives

       

Now integrate to get

       

We ignore all terms after the 11th power term to get

        1.46253

The actual integral up to five decimal places of accuracy is 

        1.46265

 

 

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