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\noindent \underline{MATH 4590 -- Study for exam 3} \\

\noindent \underline{\textbf{Definitions}} \\
partition pair, mesh of a partition, Riemann sum, Riemann integral, upper sum, lower sum, upper integral, lower integral, Darboux integrable, refinement of a partition, zero set, antiderivative, indefinite integral, pointwise convergence of sequence of functions, uniform convergence of sequence of functions

\noindent \underline{\textbf{Stuff from homework}} \\
\textbf{(HW8)}: be able to compute Riemann sums and upper and lower sums on the picture of a graph (a calculator will be provided if this comes up), be able to handle proofs involving the definition of Riemann or Darboux integrability (such as those in numbers 4 and 5)\\
\textbf{(HW9)}: induction proof about sums, drawing piecewise functions and using them to compute integrals, proofs that a described set is a zero set, proofs with the antiderivative theorem and fundamental theorem of calculus, simple proofs involving uniform convergence \\
\textbf{(HW10)}: calculations showing the necessity of uniform convergence as an assumption on a sequence that ``preserves nice properties" (such as numbers 1,3,5), finding a counterexample (like number 2) \\

\noindent\underline{\textbf{Theorems to know}}: If $f$ is Riemann integrable, then $f$ is bounded. The integral is linear, i.e.
$$\displaystyle\int \alpha f(x) + \beta g(x) \mathrm{d}x = \alpha \displaystyle\int f(x) \mathrm{d}x + \beta \displaystyle\int g(x) \mathrm{d}x.$$
If $f(x) \leq g(x)$ for all $x$, then $\displaystyle\int f(x) \mathrm{d}x \leq \displaystyle\int g(x) \mathrm{d}x$. If for all $x$, $|f(x)| \leq M$, then $\left| \displaystyle\int_a^b f(x) \mathrm{d}x \right| \leq M(b-a)$. The ``refinement principle": refining a partition causes the lower integral to increase and the upper integral to decrease. Being ``Riemann integrable" and being ``Darboux integrable" are equivalent. Riemann integrability criterion: A bounded function is Riemann-integrable if and only if $\forall \epsilon > 0 \exists P$ such that $U(f,P) - L(f,P) < \epsilon$. 
$$\displaystyle\int_a^b f(x) \mathrm{d}x = \displaystyle\int_a^c f(x) \mathrm{d}x + \displaystyle\int_c^d f(x) \mathrm{d}x$$
The following kinds of functions are Riemann-integrable: continuous, bounded piecewise-continuous, monotone, product of Riemann-integrable functions, composition of Riemann-integrable functions, the absolute value of a Riemann-integrable function.  The fundamental theorem of calculus: if $f \colon [a,b] \rightarrow \mathbb{R}$ is Riemann-integrable, then its indefinite integral $F \colon [a,b] \rightarrow \mathbb{R}$ is continuous and $F'(x)=f(x)$. Antiderivative theorem: if $f \colon [a,b] \rightarrow \mathbb{R}$ is Riemann-integrable and $G$ is any antiderivative of $f$ (i.e. $G'(x)=f(x)$), then there is a constant $C$ so that 
$$G(x) = \displaystyle\int_a^x f(t) \mathrm{d}t + C.$$ 
Cauchy Criterion for uniform convergence: $f_n \rightrightarrows f$ if and only if $\forall \epsilon > 0 \exists N \in \mathbb{N}$ so that if $m, n \geq N$ and $x \in [a,b]$, then $|f_n(x)-f_m(x)|<\epsilon$. If $f_n \rightrightarrows f$, $\displaystyle\lim_{n \rightarrow \infty} f_n(x)=A_n$, and $A_n$ is a convergent sequence, then 
$$\displaystyle\lim_{t \rightarrow x} f(t) = \displaystyle\lim_{n \rightarrow \infty} A_n.$$
If $f_n \rightrightarrows f$ and each $f_n$ is Riemann-integrable, then so is $f$ and moreover
$$\displaystyle\int_a^b f(x) \mathrm{d}x = \displaystyle\lim_{n \rightarrow \infty} \displaystyle\int_a^b f_n(x) \mathrm{d}x.$$
Suppose $f_n$ is a sequence of differentiable functions and there exists some $x_0$ so that $f_n(x_0)$ converges as $n \rightarrow \infty$. If $f_n'$ converges uniformly to a function, then $f_n$ converges uniformly to a function $f$, and moreover,
$$f'(x) = \displaystyle\lim_{n \rightarrow \infty} f_n'(x).$$
\noindent \underline{\textbf{Stuff from quizzes}} \\
\textbf{Quiz 7}: proving that uniform convergence implies pointwise convergence \\
\textbf{Quiz 8}: calculations that justify the definition of uniform convergence (from slides) \\
\textbf{Quiz 9}:  calculations that justify the definition of uniform convergence (from slides) \\
\textbf{Quiz 10}: calculations that justify the definition of uniform convergence (from slides) \\
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