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\begin{document}
\begin{center}
\Large{Exercises for Module 10: Differentiation and Integration}
\end{center}


1.  Show that 
    \begin{equation*}
        f(x) = \begin{cases} e^{-\frac{1}{x}} & \text{if } x > 0 \\
        0 & \text{if } x \leq 0
        \end{cases}
    \end{equation*}
    is smooth.

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2. Let $f \in \mathcal{R}([a,b])$ and suppose $|f| \leq M$ for some $M > 0$. Show that $|\int_a^b f(x) dx| \leq M(b-a)$.

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3. Prove the Higher-Order Leibniz product rule, i.e. for $f,g \in C^r([a,b])$ we have 
    \begin{align*}
        (fg)^{(r)}(x) = \sum_{k=0}^r \binom{r}{k} f^{(k)}(x) g^{(r-k)}(x).
    \end{align*}
    You can use properties of the binomial coefficient.


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4. (Challenge Problem) Consider the space of continuous functions on the unit interval, $C([0,1])$. Prove that there exists a unique $f\in C([0,1])$ such that for all $x\in [0,1]$
    \begin{align*}
        f(x) = x + \int_0^x sf(s) \mathrm{d}s.
    \end{align*}
    Hint: You can use that $C([0,1])$ is a complete metric space with respect to the supremums metric $d_\infty (f,g) = \sup_{x\in [0,1]} \vert f(x) -g(x)\vert$ for $f,g \in C([0,1])$.
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