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{\bf Math 166: Honors Calculus II}
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{\bf Final Exam}\quad {\sl May 6, 1999}

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There are 12 pages of problems worth a total of 165 points.
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\begin{enumerate}

\item (25 pts) Give complete definitions of the following
functions, including domains and ranges.

  \begin{enumerate}[a)]

  \item $\log(x)$

  \answerspace

  \item $\exp(x)$

  \answerspace

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

  \answerspace

  \item $\sinh(x)$

  \answerspace

  \item $\arcsin(x)$

  \answerspace

  \end{enumerate}

\newpage

\item (15 pts) State and prove the First Fundamental Theorem of Calculus.

\newpage

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

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

  \answerspace

  \item Let $f(x) = \frac{x +1}{x - 1}$. 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)

  \begin{enumerate}[a)]

  \item Compute $\frac{d}{dx} \int_{\sin(x)}^{\cos(x)}\exp(1-t^2)\,dt$.

  \answerspace

  \item Find the maximum value of the function $f(x) =  x^{-x}$, $x > 0$.

  \answerspace

  \end{enumerate}

\newpage

\item (25 pts) Compute the integrals

  \begin{enumerate}[a)]

  \item $\int \frac{x}{\sqrt{x+4}}\,dx$

  \answerspace

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

  \item $\int\frac{1}{\sqrt{3-2x-x^2}}\,dx$.

  \answerspace

\newpage

  \item $\int\frac{1}{2 + \cos(\theta)}\,d\theta$

  \answerspace

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

  \answerspace

  \end{enumerate}

\newpage

\item (15 pts)

  \begin{enumerate}[a)]

  \item State Taylor's Formula with Lagrange Remainder.

  \answerspace

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

  \answerspace

  \item Find $T_7[\log(1+x^2) - \log(1+x^3)]$.

  \answerspace

  \end{enumerate}

\newpage

\item (20 pts) Compute the following limits.

  \begin{enumerate}[a)]

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

  \answerspace

  \item $\lim_{x\to 0^+}\frac{1}{x}
  \Big(\frac{1}{\sqrt{1-x}} - \frac{1}{\sqrt{1+x}}\Big)$

  \answerspace

  \item $\lim_{x\to \infty} \frac{\sinh(1+x)}{\cosh(x)}$

  \answerspace

  \item $\lim_{n\to \infty} \Big(n^{100} + (-1)^n\Big)^{1/n}$
  
  \answerspace

  \end{enumerate}

\newpage

\item (20 pts) Compute the value of the following infinite series.

  \begin{enumerate}[a)]

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

  \answerspace

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

  \answerspace

\newpage

  \item $2x^3 + 4x^5 + 6x^7 + 8x^9 + \cdots$\quad for $|x|<1$

  \answerspace

  \item $\frac{1}{2} - \frac{1}{4} + \frac{1}{6} - \frac{1}{8} +
  \cdots$\quad[Hint: Abel's Theorem]

  \answerspace

  \end{enumerate}

\newpage

\item (15 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 $\sum_{n=1}^\infty (-1)^n\Big(1+\frac{1}{n}\Big)^n$

  \answerspace

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

  \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

  \item $\sum_{n=1}^\infty \frac{(x+1)^{2n}}{n3^n}$

  \answerspace

  \end{enumerate}

\end{enumerate}

\end{document}
