Thomas Calculus 13th [Solutions]

368 Chapter 5 Integration
95-102. Example CAS commands:
Maple:
with( Student[Calculus1] );
f : x - sin(x);
a : 0;
b : Pi;
plot( f(x), x a..b, title "#95(a) (Section 5.3)" );
N : [ 100, 200,1000 ];
for n in N do
Xlist : [ a 1.*(b-a)/n*i $ i 0..n ];
Ylist : map( f, Xlist );
end do:
for n in N do
Avg[n] : evalf(add(y,y Ylist)/nops(Ylist));
end do;
avg : FunctionAverage( f(x), x a..b, output value );
evalf( avg );
FunctionAverage(f(x),x a..b, output plot);
fsolve( f(x) avg, x 0.5 );
fsolve( f(x) avg, x 2.5 );
fsolve( f(x) Avg[1000], x 0.5 );
fsolve( f(x) Avg[1000], x 2.5 );
# (b)
# (c)
# (d)
89-102. Example CAS commands:
Mathematica: (assigned function and values for a, b, and n may vary)
Sums of rectangles evaluated at left-hand endpoints can be represented and evaluated by this set of commands
Clear[x, f, a, b, n]
{a, b} {0, }; n 10; dx (b a)/n;
2
f Sin[x] ;
xvals Table[N[x],{x, a, b dx, dx}];
yvals f /.x xvals;
boxes MapThread[Line[ {{# 1, 0},{ # 1, # 3},{ # 2, # 3},{ # 2, 0}] &,{xvals, xvals dx, yvals}];
Plot[f, {x, a, b}, Epilog boxes];
Sum[yvals[[i]] dx, {i, 1, Length[yvals]}]//N
Sums of rectangles evaluated at right-hand endpoints can be represented and evaluated by this set of
commands.
Clear[x, f, a, b, n]
{a, b} {0, }; n 10; dx (b a)/n;
2
f Sin[x] ;
xvals Table[N[x], {x, a dx, b, dx}];
yvals f /.x xvals;
boxes MapThread[Line[ {{# 1, 0},{ # 1, # 3},{ # 2, # 3},{ # 2, 0}] &,{xvals, dx,xvals, yvals}];
Plot[f, {x, a, b}, Epilog boxes];
Sum[yvals[[i]] dx, {i, 1, Length[yvals]}]//N
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