Taylor Polynomials Demo. 

The following worksheet demonstrates some of Maple's abilities to work with Taylor polynomials.  

To begin we need to include the following Maple packages... 

>	restart; with(Student[Calculus1]): with(Optimization):

 

Let's work with a "Bell Curve" like function...  

>	f := x -> exp(-x^2): 'f(x)'=f(x);

 

(1)

 

The "Student[Calculus1]" package has a function called "TaylorApproximation" which will do pretty much everything we want. 

Here is the (fifth order) Taylor polynomial for centered at =1... 

>	TaylorApproximation(f(x),x=1,order=5);

 

(2)

 

This function will also happily churn out a whole list of Taylor polynomials.  

For example here are and  ... 

>	TaylorApproximation(f(x),x=1,order=1..3);

 

(3)

 

"TaylorApproximation" will also produce plots. Here is a plot of the first 3 Taylor polynomials (in blue) 

along with the original function (in red). I have set the view window to and ... 

>	TaylorApproximation(f(x),x=1,order=1..3,output=plot,view=[-2..4,-0.5..1.5]);

 

Another nice feature is that this function will output animations (a series of plots shown over time). 

Here are plots of the first 25 Taylor polynomials.  

Note: To play this animation, click on the plot and then click on the play button appearing in the toolbar at the top of this window. 

>	TaylorApproximation(f(x),x=1,order=1..25,output=animation,view=[-2..4,-0.5..1.5]);

 

Let's compute the actual maximum error for the 9th order Taylor polynomial. 

>	P9 := TaylorApproximation(f(x),x=1,order=9);

 

(4)

 

Next, we use the "Maximize" function again to find the max error... 

>	ActualMaxError := Maximize(abs(f(x)-P9),x=0..2)[1];

 

(5)

 

So if we stay with of the base point , our Taylor polynomial never gives an answer more than off of our actual function. 

A second example: Let's consider and base at x = 0 this time. 

>	f := x -> sin(x^2)+sin(x): 'f(x)'=f(x);

 

(6)

 

The Student[Calculus1] package has a function called "TaylorApproximation" which will do pretty much 

everything we want. 

Here is the (fifth order) Taylor polynomial for   centered at  =0... 

>	TaylorApproximation(f(x),x=0,order=5);

 

(7)

 

Here are   and   ... 

(8)

 

>	TaylorApproximation(f(x),x=0,order=1..3);

 

(9)

 

Here is a plot of the first 3 Taylor polynomials (in blue) 

along with the original function (in red). I have set the view window to and . 

>	TaylorApproximation(f(x),x=0,order=1..3,output=plot,view=[-4..4,-4..4]);

 

Here are plots of the first 25 Taylor polynomials.  

>	TaylorApproximation(f(x),x=0,order=1..25,output=animation,view=[-4..4,-4..4]);

 

Let's compute the actual maximum error for the 9th order Taylor polynomial. 

>	P9 := TaylorApproximation(f(x),x=0,order=9);

 

(10)

 

Next, we use the "Maximize" function again to find the max error... 

>	ActualMaxError := Maximize(abs(f(x)-P9),x=-1..1)[1];

 

(11)

 

So if we stay with of the base point , our Taylor polynomial never gives an answer more than off of our actual function.