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% abbreviation for reals

\def\Cx{\mathbb{C}}
\def\Nl{\mathbb{N}}
\def\Ra{\mathbb{Q}}
\def\Rl{\mathbb{R}}
\def\Ts{\mathbb{T}}
\def\Ir{\mathbb{Z}}
\def\DD{\mathcal{D}}

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\newcommand{\eps}{\varepsilon}
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\begin{document}

\centerline{\textbf{Sample Questions}}
\centerline{\textbf{Midterm II}}
\centerline{\textbf{Math 127B}. \textbf{Winter, 2005}}
\centerline{\emph{Closed Book. No calculators.}}
\centerline{\emph{Except in Question 1, give complete proofs of all your answers.}}
\centerline{\emph{You can use any standard theorem provided you state it carefully.}}

\bigskip\bigskip
\noindent
\textbf{1.} For each of the following statements, say if it is true or false. (No explanation is required.)

\smallskip\noindent
(a) If $f$ is differentiable and $f^\prime > 0$, then $f$ is strictly increasing.

\smallskip\noindent
(b) If $f$ is strictly increasing and differentiable, then $f^\prime > 0$.


\smallskip\noindent
(c) If $f$ is the sum of a convergent Taylor series
in an open interval containing the origin, then $f$ is infinitely differentiable.


\smallskip\noindent
(d) If $f$ is infinitely differentiable in an open interval containing the origin, then the Taylor
series of $f$ converges.

\smallskip\noindent
(e) There exists $0<x<1$ such that
$e^x \sin 1 = \cos x\left(e-1\right)$.




\bigskip
\noindent
\textbf{2.} Define the derivative. Consider
\[
f(x) = \cases{x^a& for $x$ irrational,\cr
0 & for $x$ rational.}
\]
For what values of $a > 0$ is $f$ differentiable at $0$? Is $f$ differentiable at $x\ne 0$?

\bigskip
\noindent
\textbf{3.} 
State Taylor's theorem. Prove that
\[
\log(1+x) < x
\]
for all $x > 0$.




\bigskip
\noindent
\textbf{4.} Carefully state a version of L'Hospital's rule that applies
to the following limit. Use it to prove that the limit exists, and find its value:
\[
\lim_{x\to 0} \frac{1-\cos x}{x^2}.
\]




\newpage
\noindent
\textbf{5.} Define the hyperbolic sine
\[
\sinh x = \frac{e^x - e^{-x}}{2}.
\]
Prove that $\sinh x$ is strictly increasing on $\Rl$ and hence has an inverse.
Prove that the inverse is differentiable and compute its derivative.


\bigskip
\noindent
\textbf{6.} A function $f$ has a jump discontinuity at $x_0$ if
both the left and right limits
\[
\lim_{x\to x_0^+} f(x), \qquad \lim_{x\to x_0^-} f(x)
\]
exist but have different values. Suppose that $f : (a,b) \to \Rl$ is differentiable in
$(a,b)$. Prove that $f^\prime$ does not have a jump discontinuity in $(a,b)$.


\end{document}
