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\vspace{-1in}
\begin{center}
 \large\bf \m Math 366,  Test \# 1
$\left.\hspace{2em}\oval(80, 40)\right/ 100$
 \end{center}
\begin{flushright}
February 12,  1997\\
Instructor: Bei Hu
\end{flushright}

\hspace{.5\textwidth} Name:\hrulefill

\vspace*{2ex}

1. (30 points). True or False. (No proof or counter example needed).

(a). Let $\alpha$ be monotonically increasing and continuous on
$[a,b]$.  Suppose $f$ is bounded and monotone, 
then $f$ is Riemann-Stieltjes integrable
with respect to $\alpha$ over $[a,b]$.
\vspace{.8in}

(b). If $\gamma(t)$ is continuously differentiable on $t \in [a,b]$,
then $\gamma$ is rectifiable.
\vspace{.8in}

(c). The limit function of a sequence of continuous functions
is always continuous.
\vspace{.8in}

(d). If $\{f_n(x)\}$ and $\{g_n(x)\}$ converge uniformly on a set $E$,
then their product $\{f_n g_n\}$ also converges uniformly on $E$.
\vspace{.8in}

(e). Suppose $f_n(x)$ are continuous functions,
$f_n(x)$ converges to $f(x)$ pointwise on $[a,b]$ and 
$x_n$ converges to $c\in (a,b)$. Then 
$$ \lim_{n\rightarrow \infty} f_n(x_n) = f(c). $$
\vspace{.6in}

(f). If $\{f_n(x)\}$ converges to $f(x)$ pointwise on $[a,b]$
and $f_n \in {\cal R}(\alpha)$, then $f\in {\cal R}(\alpha)$.

\newpage
2. (20 points).  

(a). State PRECISELY the definition of a Riemann-Stieltjes integral,
using Partition.
\vspace{2in}

(b). State PRECISELY the definition of a Rectifiable curve.
\vspace{2in}

(c). State PRECISELY the $M$-test for uniform convergence of a series.
\vspace{2in}

(d). State PRECISELY the definition of equi-continuity of a 
family of functions.
\vspace{2in}
\newpage

3. (10 points). Suppose that $p>0$, $q>0$ and 
$$ \frac1p+\frac1q =1. $$
For $u \ge 0$, $v \ge 0$, prove that
$$
 uv \le \frac{u^p}{p} + \frac{v^q}{q}.
$$

\vspace*{4in}
\newpage

4. (10 points). Let
$$ f_n(x) = \frac{x}{1+ n^2 x^2}. $$
It is clear that $f_n(x)$ converges to $0$ pointwise.

Does $\{f_n\}$ converge uniformly over the interval 
$[0,1]$?  Prove your claim.
\newpage

5. (10 points). Suppose that $\dis \int_a^{a+1} e^{u(x)} dx = 1$,
and that $u(a) = u(a+1) = 0$.
Evaluate the integral
$$
 \int_a^{a+1} x e^{u(x)} u'(x) dx 
$$

\newpage
6. (10 points). 
Let  $f(x) = 1$ for $0<x\le 1$, and $f(0)=0$.

Can you find a sequence of continuous functions $f_n(x)$ defined 
on $[0, 1]$ such that, \\ \ \\

(a) $f_n(x)$ converges to $f(x)$ pointwise on $[0,1]$.
Prove your answer.
 \\ \ \\

(b) $f_n(x)$ converges to $f(x)$ uniformly on $[0,1]$.
Prove your answer.


\newpage


7. (10 points). Suppose that $f_n$ and $f$ are continuous
functions. $f_n $ converges to $f$ pointwise on $[0,1]$.
Assume that $f_n(x) \le f_{n+1}(x)$ ($n=1,2,3\cdots)$.  Prove that
$$
 \lim_{n\rightarrow \infty}\int_0^1 f_n(x) dx = \int_0^1 f(x) dx.
$$




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