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{\bf Math 166: Honors Calculus II}
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{\bf Exam I}\quad {\sl Feb.\ 29, 1996}

\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 $\cosh(x)$
  \answerspace

  \item $\sin^{-1}(x)$
  \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

  \begin{enumerate}[a)]
  \item Show $\frac{d}{dx}\int_{\cos(x)}^{\sin(x)} \sqrt{1 - t^2}\,dt = 1$
  \answerspace


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

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

  \end{enumerate}

\problemspace
\item Compute the following integrals.

  \begin{enumerate}[a)]
  \item $\int \frac{x-1}{\sqrt{x-3}}\,dx$
  \answerspace

  \item $\int \sin^3(x)\,dx$
  \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 $\int \frac{1}{x^2-3x+3}\,dx$
  \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): $\int \frac{\cos x}{2 +  \cos x}\, dx$.
  \answerspace
  \end{enumerate}

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

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

  \item $T_n\left(\log\frac{(x-1)^2}{(x+2)}\right)$
  \answerspace

\newpage
  \item $T_5(e^{x} \sin x)$
  \answerspace

  \item Let $f(x) = e^x$. Find $n$ such that $|E_nf(x;0)| \le 10^{-5}$
  for all $x \in [-1,1]$.
  \answerspace

  \end{enumerate}

\end{enumerate}

\end{document}
