Definition of Taylor polynomials 

>	p[n](x) = sum((D@@i)(f)(a)/i!*(x-a)^i,i=0..n);

 

(1.1)

 

The Taylor remainder function is defined as  

(1.2)

 

>	p11 := taylor(sin(x),x=0,12);

 

(1.3)

 

p11 is not actually a polynomial because of the term at the end which is used to signal which polynomial is represented (in case some of the coefficients are 0). We can convert to a polynomial. 

>	p11 := convert(p11,polynom);

 

(1.4)

 

The Taylor polynomials are usually good approximations to the function near a. Let's plot the first few polynomials for the sin function at x =0.  

>	sinplot := plot(sin,-Pi..2*Pi,thickness=2):

 

>	tays:= plots[display](sinplot): for i from 1 by 2 to 11 do tpl := convert(taylor(sin(x), x=0,i),polynom): tays := tays,plots[display]([sinplot,plot(tpl,x=-Pi..2*Pi,y=-2..2, color=black,title=convert(tpl,string))]) od:

 

>	plots[display]([tays],view=[-Pi..2*Pi,-2..2]);

 

Animation of Taylor series convergence 

The Taylor Polynomials gradually converge to the Taylor Series which is a representation of the original function in some interval of convergence. In this section, we'll see with our own eyes how this convergence takes place in an animation. 

First we define a function and a generic Taylor polynomial. Then we define some constant so that our graph displays for a =< x =< b, and c =<y =< d.  

>	restart; with(plots):

 

>	f := x -> sin(x);

 

(2.1)

 

>	on := x -> piecewise( x <0, 0, x < 1, 1,1);

 

(2.2)

 

>	a := -6: b := 6: c := -3: d:= 4:

 

>	g := (x,k) -> convert(series(f(x), x, k), polynom):

 

>

display( plot( f(x), x = a..b, y = c..d, thickness = 3, color = blue), animate( on(t-1)*g(x,2) , x = a..b, t = 0..7, view = c..d, color = cyan), animate( on(t-2)*g(x,4) , x = a..b, t = 0..7, view = c..d, color = coral),animate( on(t-3)*g(x,6) , x = a..b, t = 0..7, view = c..d, color = green), animate( on(t-4)*g(x,8) , x = a..b, t = 0..7, view = c..d, color = violet), animate( on(t-5)*g(x,10) , x = a..b, t = 0..7, view = c..d, color = red), animate( on(t-6)*g(x,12) , x = a..b, t = 0..7, view = c..d, color = coral) );

 

Taylor remainder theorem 

Theorem: (Taylor's remainder theorem) If the (n+1)st derivative of f is defined and bounded in absolute value by a number M in the interval from a to x, then  

This theorem is essential when you are using Taylor polynomials to approximate functions, because it gives a way of deciding which polynomial to use. Here's an example. 

Problem Find the 2nd Taylor polynomial p[2] of at . Plot both the polynomial and f on the interval [.5,1.5]. Determine the maximum error in using p[2] to approximate ln(x) in this interval.  

Solution: 

>	f := x -> ln(x)*sin(exp(x))+1;

 

(3.1)

 

>	fplot := plot(f,.5..1.5,thickness = 2):

 

>	p[2] := x -> sum((D@@i)(f)(1.)/i!*(x-1.)^i,i=0..2);

 

(3.2)

 

>

p[2](x);

 

(3.3)

 

>	t2 := unapply( convert(taylor(f(x),x=1,3),polynom),x);

 

(3.4)

 

>	tplot := plot(t2,1..1.5,color=black):

 

>	plots[display]([fplot,tplot]);

 

>	plot((D@@3)(f),.5..1.5) ;

 

We could use M = 75. 

>

M := 75;

 

(3.5)

 

So the remainder is bounded by  

>	M/3!*(1.5-1)^3;

 

(3.6)

 

We can see from the plot of f and the polynomial that the actual error is never more than about .1 on the interval [.5,1.5].  

Another example:  

Which Taylor polynomial would you use to approximate the sin function on the interval from -Pi to Pi to within 1/10^6? 

Solution:  

Well, 1 is a bound on any derivative of the sin on any interval. So we need to solve the inequality  

>	ineq := 1/n!*Pi^n <= 1/10^6;

 

(3.7)

 

for n. Solve will not be much help here because of the factorial, but we can find the smallest n by running through a loop. 

>	n := 1: while evalf(1/n!*Pi^n) > 1/10^6 do n := n+1 od: print (`take n to be `,n);

 

(3.8)

 

>	(seq(evalf( 1/n!*Pi^n) ,n=15..20));

 

(3.9)

 

>

restart;

 

>	t17 := convert(taylor(sin(x),x=0,18),polynom);

 

(3.10)

 

>	plot(t17,x=-Pi..Pi);

 

Looks pretty much like the sin function.