\documentclass[oneside]{amsart} % Boilerplate to put your name on every page % This makes grading easier \usepackage{authoraftertitle} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhead{} \fancyfoot{} \lhead{\MyAuthor} \rhead{\thepage} % Use letters in enumerated lists \usepackage{enumitem} \setenumerate[0]{label=(\alph*)} % You can define abbreviations here \newcommand{\problem}[1]{\par\noindent\textbf{Problem~#1}} \newcommand{\answer}[1]{{\par\medskip\sffamily#1\par\medskip}} \newcommand{\cl}{\mathop{\mathrm{cl}}\nolimits} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \begin{document} % Put your title, name, and the date here \title{Homework 11} \author{Your Name Here} \date{April 30, 2020} \maketitle \problem{3.2.1.} Prove the Fundamental Theorem of Calculus when the point $c$ is the left endpoint $a$ of $[a,b]$. \answer{Type your answer here.} \clearpage \problem{3.2.7.} If $f$ is infinitely differentiable on $[a,b]$ and $n\geq 1$, show that \[f(b)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(b-a)^k + \frac{1}{n!}\int_a^b f^{(n+1)}(x)\,(b-x)^n\,dx.\] % \, in math mode produces a thin space, suitable for separating different pieces of an equation or an integrand from a dx. \answer{Type your answer here.} \clearpage \problem{3.3.3.} Let $f:(0,\infty)\to\R$ be a continuous function that satisfies $f(x)-f(y)=f(x/y)$ for all $x$, $y\in(0,\infty)$. Show that if $f$ is differentiable at $x=1$ with $f'(1)=1$, then $f$ is differentiable everywhere in $(0,\infty)$ and $f(x)=\log x$ for all $x\in (0,\infty)$. \answer{Type your answer here.} \end{document}