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

{\bf Exam I}\quad {\sl Mar.\ 2, 2000}

\bigskip
There are 8 questions, most with several parts, worth a total of 110 points.
Be sure to show all your work and justify all steps.
\bigskip

\begin{enumerate}

\item (25 pts) Define each of the following completely and precisely.

  \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 $\sinh(x)$ and $\cosh(x)$
  \answerspace

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

  \end{enumerate}

\problemspace

\item (10 pts)

  \begin{enumerate}[a)]

  \item State the Fundamental Theorem of Calculus.
  \answerspace

  \item Prove this theorem for a continuous function on $[a,b]$ using the
  Mean Value Theorem.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item (10 pts)
  \begin{enumerate}[a)]

  \item State Taylor's Formula with the remainder (or error) term expressed
  as an integral.
  \answerspace

  \item Prove the error term $E_nf(x)$ has the form stated above by
  considering the function $F(t) = T_nf(x;t)$. (Show that $E_nf(x) =
  F(x) - F(a) = \dots$)
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace

\item (15 pts) Compute the following derivatives.

  \begin{enumerate}[a)]

  \item $\ddx \int_{1/x^2}^{x^2} \log(\cos(\sqrt{t}))\,dt$
  \answerspace

  \item $\ddx \big(1 + x^3\big)^{(1+x^5)}$
  \answerspace

  \item $(f^{-1})'(1)$ if $f(x) = x + e^x$.
  \answerspace

  \end{enumerate}

\problemspace

\item (15 pts) Compute the following integrals.

  \begin{enumerate}[a)]

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

  \item $\int (x\cos(x))^2\,dx$
  \answerspace

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

  \end{enumerate}

\problemspace

\item (10 pts)

  \begin{enumerate}[a)]

  \item Give the partial fraction decomposition of $\frac{1}{x^3-1}$.\newline
  (Solve for the constants.)
  \answerspace

  \item Use an appropriate substitution to transform
  \[ \int \frac{(\sin(x)+1)^2}{\cos(x)+3}\,dx \]
  into the integral of a rational function of $u$.\newline
  (Simplify as much as possible, but do not integrate.)
  \answerspace

\end{enumerate}

\problemspace

\item (15 pts) Find the Taylor polynomials.

  \begin{enumerate}[a)]

  \item $T_{3}\tan(x)$
  \answerspace

  \item $T_{2n}(\log(1-x^2))$
  \answerspace

  \item $T_{3n}\Big(\frac{1+x^n}{1+x^{2n}}\Big)$
  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts)

  Use the Taylor polynomial of degree 3 for $e^x$ to find an
  approximate value for the integral
  \[ \int_{0.5}^{1.0} \frac{e^x}{x}\,dx \]
  and give an estimate for the error.

\end{enumerate}

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