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\title{The Fundamental Theorem of Calculus}
\date{April 25, 2022}

\begin{document}
\ \vspace{-.25in}

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\vspace{-.15in}

\thm{[Fundamental Theorem of Calculus I]
Let $F:[a,b]\to \RR$ be continuous.
If $F$ is differentiable on $(a,b)$, and $f \in \mathcal{R}[a,b]$ such that $f(x) = F'(x)$ for all $x \in (a,b)$, then $\int_a^b f = F(b) - F(a)$.
}

\prob{
Our goal is to prove the theorem.
Let $P=\{x_0, x_1, x_2, \dots, x_n\}$ be a partition of $[a,b]$.
\begin{enumerate}[a)]
\item Apply the MVT to each interval $[x_{i-1}, x_i]$ to find $c_i \in (x_{i-1}, x_i)$.
\item Use the MVT to convert $\sum_{i = 1}^n f(c_i) \Delta x_i$ to a sum involving $F$ only (no derivatives). 
Cancel and simplify where possible.
\item How is $\sum_{i = 1}^n f(c_i) \Delta x_i$ related to $L(P, f)$ and $U(P,f)$?
\item Take the supremum of $L(P, f)$ and infimum of $U(P,f)$ and use the fact that $f$ is integrable to finish the proof.
\end{enumerate}
}

\newpage

\thm{[Fundamental Theorem of Calculus II]
Let $f\in \mathcal{R}[a,b]$. 
Define $F(x) = \int_a^x f$.
Then $F$ is continuous on $[a,b]$ and if $f$ is continuous at $c \in [a,b]$, then $F$ is differentiable at $c$ and $F'(c) = f(c)$.
}

\prob{
Our goal again is to prove the theorem.
\begin{enumerate}[a)]
\item First prove that $F$ is Lipschitz continuous using the fact that $f$ must be bounded.
\item Now suppose that $f$ is continuous at $c$. 
Let $\varepsilon > 0$ be given and let $\delta >0$ be such that $|x-c| < \delta$ implies $|f(x) - f(c)| < \varepsilon$ (definition of continuity).
Rewrite the last inequality with $f(x)$ in the middle.
\item Now use the bounds you found for $f$ to find bounds for $\int_c^x f$ when $x > c$. (Careful with variables here: this $x$ is different from the one before. 
Also watch out for $<$ becoming $\leq$.)
\item Rearrange your inequality to arrive at $-\varepsilon \leq \underline{\hspace{.5in}} \leq \varepsilon$.
\item Now use additivity of integrals ($\int_c^x f = \int_a^x f - \int_a^c f$) and the definition of $F$ to finish proving that $\displaystyle \left\lvert \frac{F(x) - F(c)}{x-c} - f(c)\right\rvert \leq \varepsilon$.
\item Explain why the result follows even though we got $\leq$ instead of $<$.
\end{enumerate}
}


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