Taylor๎series๎expansion๎of๎f=cos(x)๎about๎x=0.๎๎
(3) (4)
'(0) sin(0) 0, "(0) cos(0) 1
(0) sin(0) 0, (0) cos(0) 1, .
ff
f
fetc
=โ = =โ =โ
== = =๎
So๎Taylor๎series๎expansion๎is๎(as๎given๎in๎Problem๎4.10)๎
2468
cos( ) 1 2! 4! 6! 8!
xxxx
x=โ + โ + +"
๎
An๎mโfile๎that๎calculates๎this๎approximation๎with๎n๎terms๎is๎
function๎apx=costaylor(x,n)๎
%Calculates๎the๎Maclaurin๎series๎approximaton๎to๎cos(x)๎using๎the๎first๎n๎
%terms๎in๎the๎expansion.๎
apx=0;๎
for๎i=0:nโ1๎
๎๎๎๎apx=apx+(โ1)^i*x^(2*i)/factorial(2*i);๎
end๎
Problem๎4.10๎asks๎us๎to๎increment๎n๎from๎1๎until๎the๎approximate๎error๎indicates๎
that๎we๎have๎accuracy๎to๎two๎significant๎digits๎for๎x=pi/3.๎
We๎start๎with๎๎
>>๎a1=costaylor(pi/3,1)๎
a1๎=๎
๎๎๎๎๎1๎
>>๎a2=costaylor(pi/3,2)๎
a2๎=๎
๎๎๎๎0.4517๎
With๎cos(pi/3)=0.5,๎the๎true๎error๎in๎a1๎is๎100%๎and๎in๎a2๎it๎is๎๎
>>๎true2=(a2โ0.5)/0.5*100๎
