Final Exam Solutions
1. If (x) = x + [x] and f (x) = x on [0, 5], compute
Solution. From the linear of Riemann Stieljes integral, we have that
x d[x] =
x dx +
x d(x + [x]) =
Since (x) = [x] is a step function, we ha
Quiz 1 Solutions
1. Let (an ) = (1/ n + 1), n N.
(a) Prove that lim(an ) = 0.
(b) Find N N such that 1/ n + 1 < 0.03, for all n N .
Solution. (a) Given > 0, we choose N N such that N > 1/ 2 .
(This is possible due to the Archimedean Property.) Then for al
Quiz 2 Solutions
1. Dene a function of bounded variation on [a, b].
(a) Give an example of a continuous function that is not of bounded
variation on [a, b]. Justify your answer.
(b) Give an example of a function f such that f exists and f is
1. Prove that if (xn ) of real numbers is convergent then (|xn |) is also
convergent. Give an example to show that the convergence of (|xn |)
need not imply the convergence of (xn ).
Solution. Since (xn ) is convergent, given
such that |
I) IMPROPER INTEGRALS OF TYPE 1
Given a continuous function f (x) and a number a, an improper integral of
type one looks like:
f (x)dx or
We say it is improper because and are not numbers you can just