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\begin{document}

\hbox to \hsize{\textbf{Math 331\hfill Homework 7}}
\nointerlineskip
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\medskip

\centerline{\textit{This homework is due by 11:59 PM on Tuesday, November 3.}}

\bigskip

\def\ds{\displaystyle}


\begin{problem}  % problem 1
We showed that if $f$ is integrable on $[a,b]$, then $|f|$ is also integrable on $[a,b]$.
Now, suppose we know that $|g|$ is integrable on $[a,b]$.  Is it necessarily true
that $g$ is integrable on $[a,b]$?  [Hint: Consider a simple modification of the
Dirichlet function.]
\end{problem}

\begin{answer}
%You can put your answer here for problem 1, 2 points
\end{answer}



\begin{problem}  %problem 2
Suppose $f$ is a continuous function on $[a,b]$ and $f(x)>0$ for $x\in[a,b]$.
Define $F(x)=\int_a^xf$.  Prove that $F$ is strictly increasing on $[a,b]$.
[Hint: This is trivial, using two facts that we have proved.]
\end{problem}

\begin{answer}
%You can put your answer here for problem 2, 2 points
\end{answer}




\begin{problem}  % problem 3
Suppose that $f$ and $g$ are continuously differentiable functions on $[a,b]$.  So,
$f$, $g$, $f'$ and $g'$ are all continuous.  Prove the \textit{Integration by Parts}
formula $$\int_a^b f(x)'(x)\,dx = f(x)g(x)\bigg|_a^b - \int_a^bf'(x)g(x)\,dx$$
[Hint: One way to do this is to define, for $x\in[a,b]$,  $P(x) = \int_a^xf(t)g'(t)dt$
and $Q(x) = f(t)g(t)\big|_a^x - \int_a^x f'(t)g(t)dt = f(x)g(x)- f(a)g(a) - \int_a^xf'(t)g(t)dt$. 
Show that $P'(x) = Q'(x)$ and $P(a)=Q(a)$, and explain why this means $P(x) = Q(x)$
for all $x\in[a,b]$.  Finally, use $P(b) = Q(b)$.] 
\end{problem}

\begin{answer}
%You can put your answer here for problem 3, 3 points
\end{answer}




\begin{problem}  % problem 4
Let $f$ be the polynomial $f(x)=3 + 2x^2 -5x^3+x^4$.  Use Taylor's Theorem to write $f$ as a
polynomial in powers of $(x-1)$.  (That is, find the Taylor polynomial, $p_{4,1}(x)$, of degree 4 at $1$ for $f$.)
\end{problem}

\begin{answer}
%You can put your answer here for problem 4, 3 points
\end{answer}




\begin{problem}  % problem 5
Find the general Taylor polynomial at 0, $p_{n,0}(x)$, for the function $\ln(x+1)$.
\end{problem}

\begin{answer}
%You can put your answer here for problem 5, 4 points
\end{answer}




\begin{problem}[from 4.2.14 from the textbook]    % problem 6
We have shown that the $n^{th}$ Taylor polynomial for $e^x$ at 0 is
$p_{n,0}(x)=\sum_{k=1}^n\frac1{n!}x^n$.
Show that $e$ is irrational by using proof by contradiction.  Suppose,
for the sake of contradiction, that $e=\frac pq$ for some integers $p$ and $q$.

\smallskip
\textbf{(a)}\ \ Use the Lagrange form of the remainder term from Taylor's Theorem to show that there is a $c\in[0,1]$
such that $\frac pq-\big(\frac1{0!}+\frac1{1!}+\cdots+\frac1{n!}\big)=\frac{e^c}{(n+1)!}$.

\smallskip
\textbf{(b)}\ \ Multiply both sides of the equation in (a) by $n!$, and show that left side of the
resulting equation is an integer when $n\ge q$.

\smallskip
\textbf{(c)}\ \  Show that the right side of the equation that
you got in part (b) is not an integer when $n>e$.  Conclude that $e$ is irrational.
\end{problem}

\begin{answer}
%You can put your answer here for problem 6, 4 points
\end{answer}






\end{document}
