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{\bf Math 166: Honors Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Exam I}\quad {\sl Mar.\ 2, 1995}

\bigskip
There are 6 questions with a total of 25 parts. Each part is worth 5 points
for a total of 125 points.
\bigskip


\begin{enumerate}
\item Define each of the following functions precisely, including their
domains.
  \begin{enumerate}[a)]
  \item $\log(x)$
  \answerspace

  \item The natural exponential function.
  \answerspace

  \item $a^x$ for $a > 0$.
  \answerspace

  \item $\sinh(x)$
  \answerspace

  \item $\arctan(x)$
  \answerspace

  \item $T_nf(x;a)$
  \answerspace

  \end{enumerate}

\problemspace
\item State each of the following theorems precisely.

  \begin{enumerate}[a)]
  \item The First Fundamental Theorem of Calculus.
  \answerspace

  \item The Theorem on Derivatives of Inverse Functions.
  \answerspace

  \item  Taylor's Theorem with Integral Remainder.
  \answerspace

  \end{enumerate}

\problemspace
\item Compute the following derivatives

  \begin{enumerate}[a)]
  \item $\ddx\left(\int_{x^4}^{4^x} \sqrt{1 + t^4}\,dt\right)$.
  \answerspace

  \item $\ddx\left(\sqrt{x}^{\sqrt{x}}\right)$.
  \answerspace

  \item $\ddx\textrm{arcsec}(x)$ \qquad(Hint: use 2b)
  \answerspace

  \end{enumerate}

\problemspace
\item Compute the following integrals.

  \begin{enumerate}[a)]
  \item $\int \frac{\sin x}{3 + \cos x}\,dx$
  \answerspace

  \item $\int \cos^4(x)\,dx$
  \answerspace

  \item $\int x^2 e^x\, dx$
  \answerspace

\newpage
  \item $\int \log x\, dx$
  \answerspace

  \item $\int \frac{1}{\sqrt{x - x^2}}\,dx$
  \answerspace

  \item $\int \frac{1}{(x-2)(x-3)}\,dx$
  \answerspace
  \end{enumerate}

\problemspace
\item 

  \begin{enumerate}[a)]
  \item Give the partial fraction decomposition of
  $\frac{3x^3 + 4x^2 + 2x + 1}{x^2(x^2+1)}$.
  \answerspace

  \item Convert to an integral involving trignometric functions (do not
  integrate): $\int \frac{x}{\sqrt{x^2 - 4x + 20}}\, dx$.
  \answerspace

  \item Convert to an integral involving rational functions (do not
  integrate): $\int \frac{\sin x}{2 +  \sin x}\, dx$.
  \answerspace
  \end{enumerate}

\problemspace
\item In a)--c) find the Taylor polynomials.

  \begin{enumerate}[a)]
  \item $T_{n+1}\left(\frac{x^2}{(1-x)^2}\right)$  \qquad(Hint: $\ddx
  \frac{1}{1-x} = \frac{1}{(1-x)^2}$).
  \answerspace

  \item $T_n\left(\log\frac{(1-x)^3}{(1+x)^5}\right)$
  \answerspace

\newpage
  \item $T_4(e^x \cos x)$
  \answerspace

  \item Let $f(x) = \cos(x)$. Find $n$ such that $|E_nf(x;0)| \le 10^{-5}$
  for all $x \in [0,\pi/2]$.
  \answerspace

  \end{enumerate}

\end{enumerate}

\end{document}
