MATH 136 -- AP Calculus

Taylor polynomial examples

December 1, 2003

Taylor polynomials for f(x) = sin(x)

>    p3:=convert(taylor(sin(x),x=0,4),polynom);

p3 := x-1/6*x^3

>    p5:=convert(taylor(sin(x),x=0,6),polynom);

p5 := x-1/6*x^3+1/120*x^5

>    p7:=convert(taylor(sin(x),x=0,8),polynom);

p7 := x-1/6*x^3+1/120*x^5-1/5040*x^7

>    plot([p3,sin(x)],x=-2..2,color=[blue,red]);

[Maple Plot]

>    plot([p5,sin(x)],x=-3..3,color=[blue,red]);

[Maple Plot]

>    plot([p7,sin(x)],x=-4..4,color=[blue,red]);

[Maple Plot]

For this function, as the degree of the Taylor polynomial increases,

we get a better and better approximation to the function, on a larger

and larger interval of   x - values.