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{\bf Math 165: Honors Calculus I}
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{\bf Exam II}\quad {\sl Nov.\ 16, 1995}

\bigskip
There are 8 problems worth a total of 110 points.
\bigskip


\begin{enumerate}
\item (20 pts) Define the following concepts precisely.
  \begin{enumerate}[a)]
  \item $\lim_{x\to p} f(x) = A$.
  \answerspace

  \item A function $f(x)$ is continuous at $p$.
  \answerspace

  \item The inverse of a 1-to-1 function $f(x)$.
  \answerspace

  \item The derivative $f'(x)$ of a function $f(x)$.
  \answerspace

  \end{enumerate}

\problemspace
\item (20 pts) State the following theorems precisely.
  \begin{enumerate}[a)]
  \item \textsc{Intermediate Value Theorem}.
  \answerspace

  \item \textsc{Mean Value Theorem for Integrals}.
  \answerspace

  \item \textsc{Extreme Value Theorem}.
  \answerspace

  \item \textsc{Squeezing Principle}.
  \answerspace

  \end{enumerate}

\problemspace
\item (20 pts) Evaluate the following limits (justify your answers!) or
prove they do not exist.

  \begin{enumerate}[a)]
  \item $\lim_{x\to 1}\frac{\sqrt{x^2-2x+5}-2}{(x-1)^2}$.
  \answerspace

  \item $\lim_{x\to 2}(x-2)\cos\bigl(\frac{1}{x-2}\bigr)$.
  \answerspace

  \item $\lim_{x\to 0} \frac{\tan(3x)}{\tan(2x)}$.
  \answerspace

  \item $\lim_{x\to 4^-} x - [x]$.
  \answerspace

  \item $\lim_{x\to 0} x\left|\frac{1}{x}-1\right|$.
  \answerspace

  \end{enumerate}

\problemspace
\item (10 pts) Compute the derivatives of the following functions. (Do
not simplify.)

  \begin{enumerate}[a)]
  \item $f(x) = x^{4/5} + \frac{1}{x^2}$.
  \answerspace

  \item $f(x) = \frac{x-3}{(x-1)(x-2)}$
  \answerspace

  \item $f(x) = x\cos(x)\sin(x)$
  \answerspace

  \end{enumerate}

\problemspace
\item (10 pts) Let $f$ be an integrable function on $[a,b]$ and let
$F(x) = \int_a^x f(t)\,dt$. Prove that $F$ is continuous at $c \in
[a,b]$.
\answerspace

\problemspace
\item (10 pts)
  \begin{enumerate}[a)]
  \item State Bolzano's Theorem.
  \answerspace

  \item Prove that the polynomial $f(x) = x^5 - 5x + 1$ has a
  root in $[0,1]$.
  \answerspace

  \end{enumerate}


\problemspace
\item (10 pts)
  \begin{enumerate}[a)]
  \item State the \textsc{Small Span Theorem}.
  \answerspace

  \item Use a) to prove that if a function $f$ is continuous
  on $[a,b]$ then $f$ is integrable on $[a,b]$.
  \answerspace

  \end{enumerate}


\problemspace
\item (10 pts) Let $f(x) = \frac{1}{x}$ and let $p > 0$.
  \begin{enumerate}[a)]
  \item Using the definition of continuity, prove that $f(x)$ is
  continuous at~$p$.
  \answerspace

  \item Using the definition of the derivative, prove that $f'(p) =
  -\frac{1}{p^2}$.
  \answerspace

  \end{enumerate}
\end{enumerate}

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