$$ \newcommand{\cis}{\operatorname{cis}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\sq}[1]{\left[#1\right]} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \renewcommand{\S}{\mathbb{S}} \newcommand{\intr}{{\large\circ}} \newcommand{\limni}[1][n]{\lim_{#1\to\infty}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} $$

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Useful links

Office hours:

  • Mondays 9:00-10:00
  • Wednesdays 14:00-16:00

Email

GSI:

Rahul Dalal
  • M 10:30-12:30
  • TTh 17:30-19:30
  • WF 11:00-13:00

Exams

  1. Suppose $f : [a, b] \to \R$ is integrable, and define $F : [a, b] \to \R$ by $$F(t) = \int_a^t f(x)\,dx.$$ Then $F$ is continuous at $t_0 \in (a, b)$...

    1. ...always.
    2. ...provided that there is some $\epsilon \gt 0$ so that $f$ is continuous on $(t_0-\epsilon, t_0+\epsilon)$.
    3. ...provided that $f$ is continuous at $t_0$.
    4. ...provided that $f$ is differentiable at $t_0$.
  2. Suppose $f : [a, b] \to \R$ is integrable, and define $F : [a, b] \to \R$ by $$F(t) = \int_a^t f(x)\,dx.$$ Then $F$ is differentiable at $t_0 \in (a, b)$...

    1. ...always.
    2. ...provided that there is some $\epsilon \gt 0$ so that $f$ is continuous on $(t_0-\epsilon, t_0+\epsilon)$.
    3. ...provided that $f$ is continuous at $t_0$.
    4. ...provided that $f$ is differentiable at $t_0$.