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critical_points(expr,var):= block([numer:true,f:diff(expr,var)],
realroots(num(f)*denom(f),10^-7));
function_max(expr,var,a,b,critical_points):=
block([y:max(at(expr,var=a),at(expr,var=b)) ],
for v in critical_points
do (v:at(x,v),
if a<v and v<b and at(expr,var=v) > y then y:at(expr,var=v)),
y);
upper_sum(expr,var,partition):=
block([crit:critical_points(expr,var)],
partition:sort(partition),
sum(function_max(expr,var,partition[i],partition[i+1],crit)*
(partition[i+1]-partition[i]),i,1,length(partition)-1));
/* this will be part of maxima in a little while .. */
graph_riemann_sum(up,expr,var,partition):=
block([lis,numer:true,crit:critical_points(expr,var),n:length(partition),xx,
interval,display2d:false,sgn,min:100000,max:-10000000],
sgn: if up= upper then 1 else -1,
partition:sort(partition),
interval:partition[n]-partition[1],
lis:[],
for i:0 thru 50 do (xx:partition[1]+i*interval/50,
lis:cons(val:at(expr,var=xx),lis),
lis:cons(xx,lis),
if val < min then min:val,
if val > max then max:val),
sprint("{plot2d ", "{xrange " ,partition[1]-.5,partition[n]+.5,"} {yrange" ,
min-.5,max+.5, "} {xversusy "),
tcl_output(lis,1),
tcl_output(lis,2),
sprint( "}"),
sprint(" {xversusy {"),
for i:1 thru n-1
do(
sprint(partition[i],partition[i],partition[i+1],partition[i+1],
partition[i])),
sprint(" } { "),
for i:1 thru n-1
do(xx:sgn*function_max(sgn*expr,var,partition[i],partition[i+1],crit),
sprint(0.0,xx,xx,0.0,0.0)),
sprint("}}}"),
""
);
upper_and_lower_sums(expr,var,partition):=
block([up:upper_sum(expr,var,partition),
low:-upper_sum(-expr,var,partition)],
[up,low,up-low]);
make_partition(a,b,n):=makelist(a+(b-a)*i/n,i,0,n);
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