Maclaurin and Taylor  Pollynomials

tan(x), esp(x), ln(x)

>	with(plots):

Warning, the name changecoords has been redefined

tan(x)

>	evalf(tan(0.2));

>	evalf(0.2+0.2^3/3);

>	evalf(tan(Pi/4));

>	evalf(Pi/4+(Pi/4)^3/3);

>	Order:=10;

>	t:=taylor(tan(x),x=0);

Convert to a polynomial and then a function.

>	p:=convert(t,polynom);

>

f:=p;

>	eval(f,x=Pi/4);

>

evalf(%);

Here I graph tan(x) and the third and ninth Maclaurin polynomials.

>	P3:=plot(x+x^3/3,x=-Pi/2..Pi/2,y=-10..10,thickness=3,color=blue,labels=[x,y]):

>	P9:=plot(f,x=-Pi/2..Pi/2,y=-10..10,thickness=3,color=green,labels=[x,y]):

>	TanGraph:=plot(tan(x),x=-Pi/2..Pi/2,y=-10..10,thickness=3,color=red,labels=[x,y]):

>	display(P3,P9,TanGraph);

Construct and graph the tenth Maclaurin polynomial for exp(x).

>	Order:=11;

>	te10:=taylor(exp(x),x=0);

>	pe10:=convert(te10,polynom);

>	pe10Graph:=plot(pe10,x=-10..10,y=-10..10000,thickness=3,color=blue,labels=[x,y]):

>	ExpGraph:=plot(exp(x),x=-10..10,y=-10..10000,thickness=3,color=red,labels=[x,y]):

>	display(pe10Graph,ExpGraph);

Below we see exp(2) being approximated by 10th and 20th Maclaurin polynomials.

>

f:=pe10;

>	eval(f,x=2);

>

evalf(%);

>	evalf(exp(2));

>	Order:=21;

>	t20:=taylor(exp(x),x=0);

>	pe20:=convert(t20,polynom);

>

f:=pe20;

>	eval(f,x=2);

>

evalf(%);

>	evalf(exp(2));

Here I graph ln(x) and the 10th Taylor polynomial for ln(x) centered at 1 and approximate ln(1.2) using the 10th Taylor polynomial centered at 1.

>	Order:=11;

>	tln10:=taylor(ln(x),x=1);

>	pln10:=convert(tln10,polynom);

>	lnGraph:=plot(ln(x),x=0..2,thickness=3,color=red,labels=[x,y]):

>	pln10Graph:=plot(pln10,x=0..2,thickness=3,color=blue,labels=[x,y]):

>	display(lnGraph,pln10Graph);

>	fln10:=pln10;

>	eval(fln10,x=1.2);

>	eval(ln(1.2));

>