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{\bf Math 165: Honors Calculus I}
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{\bf Exam II}\quad {\sl Dec.\ 1, 1994}

\bigskip
There are 9 problems worth from 5 to 20 points for a total of 120 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 $f$ is continuous at $p$.
  \answerspace

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

  \item The span of a bounded function $f$ on $[a,b]$.
  \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

  \item \textsc{Chain Rule}.
  \answerspace

  \end{enumerate}

\problemspace
\item (20 pts) Compute the following limits or prove they do not exist.

  \begin{enumerate}[a)]
  \item $\lim_{x\to a}\frac{x^2-a^2}{x^3-a^3}$.
  \answerspace

  \item $\lim_{x\to 0} x \sin \frac{1}{x}$.
  \answerspace

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

  \item $\lim_{x\to 3^-} x - [x]$.
  \answerspace

  \item $\lim_{x\to 0} x\sqrt{1 + \frac{1}{x^2}}$.
  \answerspace

  \end{enumerate}

\problemspace
\item (20 pts) Compute the derivatives of the following functions. (Do
not simplify.)

  \begin{enumerate}[a)]
  \item $f(x) = x^{2/3} + \frac{1}{x}$.
  \answerspace

  \item $f(x) = (5x^2 - 3)(x^3 + 1)^4$
  \answerspace


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

  \item $f(x) = \sin^3(x^2-1)$
  \answerspace

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


\item (5 pts) Determine the equation of the line tangent to the graph
of the function $f(x) = \frac{x}{\sqrt{1 + x^2}}$ at the point
$(1,1/\sqrt{2})$.
\answerspace


\problemspace
\item (10 pts)
  \begin{enumerate}[a)]
  \item Prove that if $f$ is differentiable at $p$ then $f$ is
  continuous at $p$.
  \answerspace

  \item Give an example of a function that is continuous $p$ but not
  differentiable at $p$. (Give a proof of this).
  \answerspace

  \end{enumerate}

\problemspace
\item (10 pts) Let $f(x) = \sqrt{x}$ 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 $f'(p) =
  \frac{1}{2\sqrt{p}}$.
  \answerspace

  \end{enumerate}
\end{enumerate}

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