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{\bf Math 165: Honors Calculus I}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Final Exam}\quad {\sl Dec.\ 15, 1995}

\bigskip
There are 15 problems worth a total of 165 points.
\bigskip


\begin{enumerate}
\item (15 pts) Give complete definitions for the following concepts.
  \begin{enumerate}[a)]
  \item $\sum_{i=1}^n a_i$
  \answerspace

  \item A function $f$ is integrable on $[a,b]$.
  \answerspace

  \item $\lim_{x\to p} f(x) = A$.
  \answerspace

  \item A function $f$ is differentiable at $p$.
  \answerspace

  \item A function $f$ is convex on $[a,b]$.
  \answerspace

  \end{enumerate}


\problemspace
\item (15 pts) State the following theorems precisely.
  \begin{enumerate}[a)]
  \item \textsc{Linearity Theorem for Integrals}.
  \answerspace

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

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

  \item \textsc{Chain Rule}.
  \answerspace

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

  \end{enumerate}


\problemspace
\item (15 pts) Calculate the following.
  \begin{enumerate}[a)]
  \item $\int_0^2 [x^2]\, dx$ where $[u]$ is the greatest integer
  ${}\le u$.
  \answerspace
 
  \item $\int_1^x t^3 + (t-1)^{1/3}\, dt$, $x \ge 1$.
  \answerspace

  \item The average value of the function $f(x) = x(4-x^2)$ on the interval
  $[0,2]$.
  \answerspace

  \item $\int_{-4}^{4} \frac{x^3+x}{x^8 + x^2 + 1}\,dx$
  \answerspace

  \end{enumerate}


\problemspace
\item (15 pts) Compute the following limits or prove they do not exist.
  \begin{enumerate}[a)]
  \item $\lim_{x\to \pi/4} \frac{\sin(x)-\cos(x)}{x-\pi/4}$ (Hint:
  $x = (x-\pi/4) + \pi/4$).
  \answerspace

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

  \item $\lim_{x\to 0} \frac{(1 + x)^n - (1 - x)^n}{x}$ where $n$
  is a fixed positive integer.
  \answerspace

  \item $\lim_{x\to 1} (x^2-x)f(x)$ where $f(x) =
  \cases{x&$x>1$\cr-x&$x<1$\cr}$.
  \answerspace

  \end{enumerate}


\problemspace
\item (15 pts) Compute the following derivatives.

  \begin{enumerate}[a)]
  \item $\frac{d}{dx}\left[x^3(1 - x^2)^{50}\right]$.
  \answerspace

  \item $\frac{d}{dx}\sqrt{1 + x^2}$
  \answerspace

  \item $\frac{d}{dx}\cos(\sin(x^3))$
  \answerspace

  \item $\frac{dy}{dx}$ where $y$ is defined implicitly by the equation
  $y^4 + x^2 y^3 + 1 = 0$.
  \answerspace

  \end{enumerate}

\problemspace
\item (10 pts) Consider the sequence $\frac{5}{1}$, $\frac{19}{5}$,
$\frac{65}{19}$, $\frac{211}{65}$,\ldots, defined recursively by
\[
  a_1 = 5,\quad a_{n+1} = 5 - \frac{6}{a_n},\ n\ge1
\]
Use induction to prove that 
\[
  a_n = \frac{3^{n+1}-2^{n+1}}{3^n-2^n},\ n\ge1
\]
\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 $x$.
Sketch the graph of $f$ on $I$ and the graph of $f^{-1}$ on its domain.
\answerspace
 

\problemspace
\item (5 pts) Find an expression for the area between the graphs of the
functions $f(x) = x^2 - 2x$ and $g(x) = 1 - x$ on the interval $[0,3]$.
Write the answer as a sum of integrals without absolute values---do not
evaluate the integrals.
\answerspace


\problemspace
\item (10 pts)  Use the \emph{definition of a limit} to prove that 
$\lim_{x\to1} x^2 = 1$.
\answerspace


\problemspace
\item (10 pts) Prove that if a function $f$ is differentiable at $p$, then
$f$ is continuous at~$p$.  
\answerspace


\problemspace
\item (10 pts) Let
\[
  f(x) = \cases{6\sqrt{x} - 5, & for $x < 1$\cr x^3, & for $x\ge 1$\cr}
\]
Prove that $f'(1) = 3$. You must use the definition of a derivative and
should examine one-sided limits.
\answerspace


\problemspace
\item (5 pts) Find the equation of the line tangent to the curve
defined by $y = x^4 - x^3 + x^2 - x + 1$ at the point $(1,1)$.
\answerspace


\problemspace
\item (10 pts) If $a$ and $b$ are legs of a right triangle whose hypotenuse
is 1, find the largest value of $2a + b$.
\answerspace


\problemspace
\item (10 pts) Let $f(x) = 3x^4 + 4x^3 - 12 x^2$.
  \begin{enumerate}[a)]
  \item Determine the intervals on which $f$ is increasing and decreasing.
  \answerspace

  \item Determine the relative extrema of $f$.
  \answerspace

  \item Determine the intervals on which $f$ is convex and concave.
  \answerspace

  \item Sketch the graph of $f$.
  \answerspace\answerspace

  \end{enumerate}


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

  \item Use \textsc{Bolzano's Theorem} and \textsc{Rolle's Theorem} to
  prove that the function $f(x) = x^3 - x^2 + 1$ has \emph{exactly one}
  root in the interval $[-1,0]$.
  \answerspace

  \end{enumerate}

\end{enumerate}

\end{document}

