EXAM 3 SOLUTIONS

> COMPARISONS OF ASYMPTOTIC APPROXIMATIONS WITH
NUMERICAL QUADRATURE RESULTS

> restart:
approx:=x->evalf(exp(x^2)/(2*x));

> num:=x->evalf(Int(exp(t^2),t=0..x));

> test:=proc(x)
  local a,n,rel_err;
  a:=approx(x);
  n:=num(x);
  rel_err:=(a-n)/n;
  return [a,n,rel_err];
end proc:
test(1);
test(5);
test(10);
test(20);

> restart:
approx:=lambda->evalf((sqrt(Pi))*( exp(lambda)/sqrt(lambda)));

> num:=lambda->evalf(Int(cos(t/2)*exp(lambda*sin(t)),t=0..Pi));

> p:=lambda->plot(cos(t/2)*exp(lambda*sin(t)),t=0..Pi);
p(50);
test:=proc(lambda)
  local a,n,rel_err;
  a:=approx(lambda);
  n:=num(lambda);
  rel_err:=(a-n)/n;
  return [a,n,rel_err];
end proc:
test(1);
test(10);
test(20);
test(50);

Animations of solution here