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{\bf Math 166: Honors Calculus II}
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{\bf Exam I}\quad {\sl Feb.\ 18, 1999}

\bigskip
There are 6 questions, each with several parts, worth a total of 110 points.
Be sure to show all your work and justify all steps.
\bigskip


\begin{enumerate}
\item (30 pts) Define each of the following functions precisely, including
their domains.

  \begin{enumerate}[a)]

  \item The natural logarithm function, $\log(x)$.
  \answerspace

  \item The natural exponential function, $\exp(x)$.
  \answerspace

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

  \item $\tanh(x)$
  \answerspace

  \item $\arccos(x)$
  \answerspace

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

  \end{enumerate}

\problemspace
\item (10 pts) 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

  \end{enumerate}

\problemspace
\item (15 pts)

  \begin{enumerate}[a)]

  \item Compute $\frac{d}{dx}\int_{-\log(x)}^{\log(x)} \sin(e^t)\,dt$
  \answerspace

  \item Compute $\ddx \cos(x)^{\sin(x)}$.
  \answerspace

  \item Prove $\ddx\arcsin(x) = \frac{1}{\sqrt{1-x^2}}$.
  \answerspace

  \end{enumerate}

\problemspace
\item (30 pts) Compute the following integrals.

  \begin{enumerate}[a)]

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

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

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

\newpage

  \item $\int \frac{1}{(x^2+1)^2}\,dx$
  \answerspace

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

  \item $\int \frac{1}{x^2-5x+7}\,dx$
  \answerspace
  \end{enumerate}

\problemspace
\item (15 pts)

  \begin{enumerate}[a)]
  \item Give the partial fraction decomposition of
  $\frac{4x^3 - x^2 - 4x + 5}{(x-1)^2(x^2+1)}$\newline
  (Solve for the constants).
  \answerspace

  \item Use a trignometric substitution to integrate
  $\int \frac{x}{\sqrt{x^2 - 6x}}\, dx$.
  \answerspace

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

\problemspace
\item (10 pts) Find the Taylor polynomials.

  \begin{enumerate}[a)]

  \item $T_{4n+1}\left(\frac{x}{1+x^4}\right)$
  \answerspace

  \item $T_{2n}(\cos^2(x))$ \qquad[Hint: use a trig identity first.]
  \answerspace

  \end{enumerate}

\end{enumerate}

\end{document}
