__Taylor Series Experimentation__

`> `
**with(plots):**

**One Variable Case**

**Taylor polynomials of degrees 1 through 4 for f about x=1.**

`> `
**f:=1/(1+x+x^2);
gr0:=plot(f,x=-2..2):
for i from 1 to 5 do
f || i:=mtaylor(f,[x=1],i+1);
od;**

**The first graph compares f with these Taylor polynomials for x between 0 and 2.
The second between -2 and 2.**

**Do you see a difference in the quality of the approximation?
What do you think would happen over [-2,2] with higher degree polynomials?**

**Would they eventually give a good approximation on the interval [-2,2]?**

`> `
**colors:=[red,yellow,green,blue,violet]:
gr0:=plot(f,x=-2..2,color=colors[1]):
for i from 1 to 4 do
gr || i := plot(f || i,x=-2..2,color=colors[i+1]):
od:
display([gr0,gr1,gr2,gr3,gr4],thickness=2,title=`Taylor polynomials about x=1; f is in red`,view=[0 .. 2,0..1]);
display([gr0,gr1,gr2,gr3,gr4],thickness=2,title=`Taylor polynomials about x=1; f is in red`);
**

`> `

**A Basic Two Variable Example**

**The 3 below in the mtaylor call below means degree 3-1=2 Taylor polynomial**
.

`> `
**g:=1/(y^2+x);
g2:=mtaylor(g,[x=1,y=1],3);**

**The first graph shows g and its second Taylor polynomial over the x_range and y_range specified. **

**The second graphs the difference, which may be interpreted as an error. **

**Try rotating the graphs (especially the first) to examine the quality of the approximation.**

**Try some other ranges besides the 0.5 .. 1.5 specified below. **

**Try at least some ranges close to 1 (e.g 0.9 .. 1.1 or 0.75 .. 1.25) **

**as well as some bigger ranges (e.g. 0 ..2 or 0.25 .. 1.75).**

**Use the second graph to compare maximal errors with the width of the x_range.**

**(e.g. compare the largest errors in the cases 0.9 .. 1.1 and 0.95..1.05) **

**How much smaller has the error become as the width of the interval halved?**

**Can you explain approximately why this is so?
Do bigger intervals also work this way when their size is halved?**

`> `
**x_range:=0.5 .. 1.5;
y_range:=x_range;
gr5:=plot3d(g,x=x_range,y=y_range,color=blue):
gr6:=plot3d(g2,x=x_range,y=y_range,color=red):
display([gr5,gr6],axes=normal,title=`g is in blue,p2 in red`);
plot3d(g-g2,x=x_range,y=y_range,axes=normal,title=`error using p2`);**

**Do you see something fundamentally different about the approximation on**

**a big range vs. a small one? You could try changing the degree of the Taylor
approximation above.**

`> `