a0=Normal[Series[Sin[x],{x,0,1}]]
a1=Normal[Series[Sin[x],{x,0,3}]]
a2=Normal[Series[Sin[x],{x,0,5}]]
a3=Normal[Series[Sin[x],{x,0,7}]]
a4=Normal[Series[Sin[x],{x,0,9}]]
a5=Normal[Series[Sin[x],{x,0,11}]]
a6=Normal[Series[Sin[x],{x,0,13}]]
a7=Normal[Series[Sin[x],{x,0,15}]]
Plot[{a0, a1,a2,a3, a4, a5, a6, a7}, {x,-5,5}]

Plot[{Sin[x]-a1, Sin[x]-a2, Sin[x]-a3, Sin[x]-a4, Sin[x]-a5, Sin[x]-a6, Sin[x]-a7}, {x,-5,5}]

restart
DSolve[y'[x]==ax+b,y,x]

a=1
b=1

Plot[x^2+x,{x, 0, 1},Filling->Bottom]

x0=0

prval=0
dt=0.01

f[x_]:=a*x+b

f[0]

xi2[x_]:=f[x]dt+prval

s=0

For[i=0,i<1,
x0=i;
  s=s+f[x]dt;
i=i+dt]
Print[FullSimplify[s]]

ff=FullSimplify[s]