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

{\bf Final Exam}\quad {\sl May 9, 1995}

\bigskip
There are 13 problems worth a total of 180 pts.

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\begin{enumerate}
\problemspace
\item (10 pts) State and prove the First Fundamental Theorem of Calculus.


\newpage

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

  \item Compute $\int_{-2}^0 \frac{x}{\sqrt{x+4}}\,dx$

  \answerspace

  \item Integrate $\int e^{-x}\sin(x)\,dx$

  \answerspace

  \end{enumerate}

\vfil

\item (15 pts) Give complete definitions of the following
functions, including domains and ranges.

  \begin{enumerate}[a)]

  \item $\log(x)$

  \answerspace

  \item $\log_a(x)$ for $a > 0$

  \answerspace

  \item $\exp(x)$

  \answerspace

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

  \answerspace

  \item $\tanh(x)$

  \answerspace

  \item $\arcsin(x)$

  \answerspace

  \end{enumerate}

\newpage

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

  \item Derive $\frac{d}{dx}\arctan(x) = \frac{1}{1+x^2}$.

  \answerspace

\vfil

  \item Let $f(x) = \frac{e^x +1}{e^x - 1}$. Determine the interval(s)
  on which $f(x)$ has an inverse and derive a formula for
  $f^{-1}(x)$.

  \answerspace

  \end{enumerate}

\newpage

\item (10 pts) Compute the following derivatives.

  \begin{enumerate}[a)]

  \item $\frac{d}{dx} \int_{\sin(x)}^{\cos(x)}\frac{1}{\sqrt{e^x + x}}\,dx$.

  \answerspace

\vfil

  \item $\frac{d}{dx} \log(x)^x$

  \end{enumerate}


\newpage


\item (15 pts)

  \begin{enumerate}[a)]

  \item Compute the partial fraction decomposition of
  $\frac{3x^3 - x^2 -1}{(x^3+x^2+x)(2x-1)}$.
  
  \answerspace

\vfil

  \item Use an appropriate substitution to transform the integral
  $$\int \frac{1}{2 + \sin(\theta)}\,d\theta$$
  into an integral of a rational function (do not evaluate the integral).

  \answerspace

\vfil

  \item Integrate $\int\frac{1}{\sqrt{2x^2 + 4x +1}}\,dx$.

  \answerspace

  \end{enumerate}

\newpage

\item (15 pts)

  \begin{enumerate}[a)]

  \item State Taylor's Formula with Lagrange Remainder.

  \answerspace

\vfil

  \item Find $T_{2n+1}\left(\frac{x}{x^2 + 2}\right)$.

  \answerspace

\vfil

  \item Find $T_6(\log(\cos(x)))$.

  \answerspace

  \end{enumerate}


\newpage

\item (20 pts) Compute the following limits.

  \begin{enumerate}[a)]

  \item $\lim_{x\to 0}\frac{\sin(x^4)}{x^2(e^x - 1)^2}$

  \answerspace

  \item $\lim_{x\to 0} \frac{2^x - 3^x}{x}$

  \answerspace

  \item $\lim_{x\to 0} \frac{1}{x} - \frac{1}{e^x-1}$

  \answerspace

  \item $\lim_{x\to \infty} \sqrt{x}\sin(\frac{1}{x})$

  \answerspace

  \item $\lim_{n\to \infty} \frac{\sqrt[n]{n^2 + (-1)^n}}{\sqrt{n^3 + n}}$
  
  \answerspace

  \end{enumerate}


\newpage

\item (10 pts) State and prove the Monotone Convergence Theorem.

\newpage

\item (15 pts) Compute the value of the following infinite series.

  \begin{enumerate}[a)]

  \item $\sum_{n=1}^\infty \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+2}}$

  \answerspace

\vfil

  \item $\sum_{n=17} \frac{2+(-1)^n}{5^n}$

  \answerspace

\vfil

  \item $\sum_{n=1}^\infty 2n x^{2n+1}$ for $|x|<1$.

  \answerspace

  \end{enumerate}


\newpage

\item (20 pts) Test whether the following series converge
absolutely, conditionally, or diverge. Justify your answers.

  \begin{enumerate}[a)]

  \item $\sum \frac{(-1)^n}{n\log(n)^s}$ for $s > 1$.

  \answerspace

  \item $\sum_{n=1}^\infty (-1)^n\left(1+\frac{1}{n}\right)^{1/n}$

  \answerspace

  \item $\sum_{n=1}^\infty (-1)^n\frac{\log(n+1)}{\sqrt{n(n+1)}}$

  \answerspace

  \item $\sum_{n=1}^\infty \frac{(-2)^n n!}{n^n}$

  \answerspace

  \end{enumerate}


\newpage 

\item (10 pts) Determine the interval of convergence for the following power
series.

  \begin{enumerate}[a)]

  \item $\sum_{n=0}^\infty (1 + (-3)^n)x^n$

  \answerspace

\vfil

  \item $\sum_{n=0}^\infty \frac{(x+1)^n}{(n+2)3^n}$

  \answerspace

\vfil

  \end{enumerate}

\newpage 

\item (20 pts) Compute the following integrals.
Justify your answers.

  \begin{enumerate}[a)]

  \item $\int_0^\infty x e^{-x}\,dx$

  \answerspace

  \item $\int_{0^+}^1\log(x)^2\,dx$

  \answerspace

  \item $\int_{0}^2\frac{1}{\sqrt[3]{(x-1)^2}}\,dx$

  \answerspace

  \item $\int_{0^+}^{1^-}\frac{\log(1-x)}{x}\,dx$ (Test for convergence
  only.)

  \end{enumerate}



\end{enumerate}

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