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{\bf Math 165: Honors Calculus I}
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{\bf Exam II}\quad {\sl Nov.\ 19, 1998}

\bigskip
There are 8 problems worth a total of 110 points.
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\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 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{Basic Limit Theorems}.
  \answerspace

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

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

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

  \end{enumerate}

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

  \begin{enumerate}[a)]
  \item $\lim_{x\to 1} (x-1) \sin \frac{x}{x-1}$
  \answerspace

  \item $\lim_{x\to 0} \frac{\cos(x)-1}{x^2}$
  \answerspace

  \item $\lim_{x\to 2^-} x^3 - [x^3]$
  \answerspace

  \item $\lim_{x\to 0} x\Big|1 + \frac{1}{x}\Big|$
  \answerspace

  \end{enumerate}

\problemspace
\item (15 pts)

  \begin{enumerate}[a)]
  \item Let $f(x) = x^{2/3} + \frac{1}{x}$, $x>0$. Compute $f'(x)$.
  \answerspace

  \item Let $a$ and $b$ be constants, not both zero, and let $f(x) =
  \frac{ax+b}{a-bx}$. Show that $f'(x) > 0$ for $x \ne a/b$.

  \answerspace

  \item Let $f(x) = (ax+b)\cos(x) + (cx+d)\sin(x)$. Determine values of the
  constants $a$, $b$, $c$, $d$, such that $f'(x) = x\sin(x)$.
  \answerspace

  \end{enumerate}

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

\item (5 pts)
Let $n\ge2$ be a positive integer. Prove that the polynomial
$f(x) = x^n - nx + 1$ has a root in $[0,1]$.
\answerspace

\problemspace
\item (10 pts) Find the largest interval $I$ containing $x = 1$ on which
the function $f(x) = \frac{1}{1+x^2}$ has an inverse. Give a formula for
the corresponding inverse function, $f^{-1}(x)$, as a function of $x$ and
determine its domain.

\problemspace
\item (10 pts)
  \begin{enumerate}[a)]
  \item Using the definition of continuity, prove that $f(x) = x^2$ is
  continuous at any real number~$p$.
  \answerspace

  \item Using the definition of the derivative, prove that the derivative
  of $f(x) = \sqrt{x}$ is $f'(x) = \frac{1}{2\sqrt{x}}$.
  \answerspace

  \end{enumerate}
\end{enumerate}

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