$$ \newcommand{\cis}{\operatorname{cis}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\sq}[1]{\left[#1\right]} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \renewcommand{\S}{\mathbb{S}} \newcommand{\intr}{{\large\circ}} \newcommand{\limni}[1][n]{\lim_{#1\to\infty}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bI}{\mathbb{I}} \newcommand{\bJ}{\mathbb{J}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bL}{\mathbb{L}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bS}{\mathbb{S}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bU}{\mathbb{U}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bW}{\mathbb{W}} \newcommand{\bX}{\mathbb{X}} \newcommand{\bY}{\mathbb{Y}} \newcommand{\bZ}{\mathbb{Z}} $$

Useful links

Office hours:

  • Mondays 18:00-19:00
  • Wednesdays 14:00-16:00

Email

GSI:

Nima Moini
  • Mondays 10:00-14:00
  • Tuesdays 10:00-14:00
  • Wednesdays 10:00-12:00

Exams

  1. Suppose $f : [a, b] \to \R$, and $\mathcal{P}, \mathcal{Q}$ are partitions of $[a, b]$. Then $L(f, \mathcal P) \leq U(f, \mathcal Q)$...

    1. ...always.
    2. ...provided either $\mathcal{P}$ refines $\mathcal{Q}$ or $\mathcal{Q}$ refines $\mathcal{P}$.
    3. ...provided $\mathcal{P} = \mathcal{Q}$.
    4. ...provided $f$ is integrable.

  2. Suppose $f, g : [a, b] \to \R$ are differentiable on $(a, b)$, that $f(a) \leq g(a)$, and that $f'(t) \leq g'(t)$ for all $t \in (a, b)$. Then $f(b) \leq g(b)$...

    1. ...always
    2. ...provided $f$ is continuous at $a$ and $b$.
    3. ...provided that $f$ and $g$ are continuous at $a$ and $b$.
    4. ...provided $f$ and $g$ are continuous at $a$ and $b$, and $f'(t) = g'(t)$ at only finitely many $t \in (a, b)$.

  3. Suppose that $f : [a, b] \to \R$. Which of the following is true, and could concievably be useful in a proof?

    1. For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \lt U(f) - \epsilon$.
    2. For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \lt U(f) + \epsilon$.
    3. For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \gt U(f) - \epsilon$.
    4. For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \gt U(f) + \epsilon$.

    (Also, consider: of the other answers, which are false and which are true but useless?)

  4. Suppose that $f, g : [a, b] \to \R$ and we wish to prove that $U(f+g) \leq U(f) + U(g)$. Which inequality is true and will be useful in the proof?

    1. For any set non-empty $I \subseteq[a, b]$, $\sup_{x \in I}(f+g)(x) \leq \sup_{x\in I}f(x) + \sup_{y\in I}g(y)$.
    2. For any set non-empty $I \subseteq[a, b]$, $\sup_{x \in I}(f+g)(x) \geq \sup_{x\in I}f(x) + \sup_{y\in I}g(y)$.
    3. For any partition $\mathcal{P}$, we have $U(f, \mathcal{P}) \leq U(f+g, \mathcal{P})$.
    4. For any partition $\mathcal{P}$, there is a partition $\mathcal{Q}$ refining $\mathcal{P}$ so that $U(f + g, \mathcal{P}) \leq \frac12U(f, \mathcal{Q})$.