Manipulate[
a=0;
b=10;
domain=Table[a+i*h,{i,0,Floor[(b-a)/h]}];
f[y_]:=-y;

record={};
now=1;
record=Append[record,now];
For[i=1,i<Floor[(b-a)]/h,i++,
next=now+h*(f[now]);
now=next;
record=Append[record,now];
];
Show[ListLinePlot[Table[{domain[[i]],record[[i]]},{i,1,Floor[(b-a)/h]-1}]],
Plot[E^(-t),{t,a,b}]]

,{{h,.01},.01,2}]

Manipulate[
a=0;
b=10;
domain=Table[a+i*h,{i,0,Floor[(b-a)/h]}];
f[y_]:=y;

record={};
now=1;
record=Append[record,now];
For[i=1,i<Floor[(b-a)]/h,i++,
next=now+h*(f[now]);
now=next;
record=Append[record,now];
];
Show[ListLinePlot[Table[{domain[[i]],record[[i]]},{i,1,Floor[(b-a)/h]-1}]],
Plot[E^(t),{t,a,b}]]

,{{h,.01},.01,2}]