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{\bf Math 165: Honors Calculus I}
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{\bf Exam II}\quad {\sl November\ 11, 1999}

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There are 7 problems worth a total of 110 points.
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\begin{enumerate}

\item (25 pts) Define the following precisely.

  \begin{enumerate}[a)]

  \item The area between the graphs of two integrable functions on~$[a,b]$.

  \answerspace

  \item The functions $\sin(x)$ and $\cos(x)$.

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  \item The limit of a function $f(x)$ as $x$ approaches~$p$.

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  \item A one-to-one function.

  \answerspace

  \item The average value of an integrable function on~$[a,b]$.

  \answerspace

  \end{enumerate}

\problemspace

\item (25 pts) State the following theorems.

  \begin{enumerate}[a)]

  \item \textsc{Basic Limit Theorems}

  \answerspace

  \item \textsc{The Squeezing Principle}

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  \item \textsc{Bolzano's Theorem}

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  \item \textsc{Intermediate Value Theorem}

  \answerspace

  \item \textsc{Extreme Value Theorem} 

  \answerspace

  \end{enumerate}

\problemspace

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

  \begin{enumerate}[a)]

  \item $\lim_{x\to 1} \frac{\sin(x^2-1)}{x-1}$

  \answerspace

  \item $\lim_{x\to 0} \frac{\sqrt{x^4+1}-1}{x^4}$

  \answerspace

  \item $\lim_{x\to2}\sqrt{x^3 - \frac{1}{x^3}}$

  \answerspace

  \item $\lim_{x\to 0^+} x^2\Big(1 + \left[\frac{1}{x}\right]\Big)$. (As
  usual, $[a]$ denotes the greatest integer $\le a$.)

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  \end{enumerate}

\problemspace

\item (10 pts) Assume $\lim_{x\to p} f(x) = A$ and $\lim_{x\to p} g(x)
= B$. Using the definition of limit, prove that $\lim_{x\to p} \big(f(x)
- g(x)\big) = A-B$.

\answerspace

\problemspace

\item (10 pts) Let $f$ be continuous at $p$ and let $g$ be continuous at $q
= f(p)$. Prove that $g\circ f$ is continuous at~$p$.

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\item (10 pts)
Let $f$ be integrable and positive on~$[a,b]$. Show that the function
$F(x) = \int_a^x f(t)\,dt$ is increasing on $[a,b]$.

\answerspace

\problemspace

\item (10 pts) Suppose the inverse of a function is $f^{-1}(x) =
\sqrt{\frac{1}{\sqrt{x}}-1}$ for $0 < x \le 1$. Determine the original function,
$f(x)$, including its domain.

\end{enumerate}

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