#pkg load symbolic;

#1.a
syms x;

f=exp(x)

t1=taylor(f,x,0,'order',1);
t2=taylor(f,x,0,'order',2);
t3=taylor(f,x,0,'order',3);
t4=taylor(f,x,0,'order',4);
t10 = taylor(f,x,0,'order',10);

#ezplot(t1)
hold on
ezplot(t2)
hold on
ezplot(t3)
hold on
ezplot(t4)
xlim([-3,3])

#1.b
vpa(exp(1),7)

vpa(subs(t10,x,1),7)

#2.a

syms x;
f=sin(x)

t3=taylor(f,x,0,'order',3);
t5=taylor(f,x,0,'order',5);
hold off
ezplot(f)
hold on
ezplot(t3)
hold on
ezplot(t5)
xlim([-pi,pi])
ylim([-2,2])

#2.b
t10=taylor(f,x,0,'order',10);

vpa(sin(pi/5),5)
vpa(subs(t10,x,sym(pi)/5),5)

vpa(sin(10*pi/3),5)
vpa(subs(t10,x,10*sym(pi)/3),5) #it's not precise enought, increase n for Taylor series or move x0=0 to x0= 10pi/3


#3.a

syms x;
f=log(1+x);
t2= taylor(f,x,0,'order',2);
t5= taylor(f,x,0,'order',5);

hold off
ezplot(f)
hold on
ezplot(t2)
hold on
ezplot(t5)

xlim([-0.9,1])
ylim([-1,1])
t= taylor(f,x,0,'order',10);
vpa(log(2),5)
vpa(subs(t,x,1),5)

syms x;
g=log(1-x);
t2 = taylor(g,x,0,'order',10)

t-t2

vpa(subs(t,x,sym(0.999,'f')) - subs(g,x,sym(0.999,'f')),5)