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\newtheorem{theorem}{Theorem}
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\begin{document}





\def\tT{Math126, Test I}
\def\ddd{February 9, 1999}
\def\rightanswers{%
\ll 1 3
\ll 2 2
\ll 3 5
\ll 4 2
\ll 5 1
\ll 6 4
\ll 7 2
\ll 8 1
\ll 9 3
\ll {10} 5
}
\input 126Header.tex
\exam{
\centerline{\bf Multiple Choice}


\pr(5pts) Let $f(x)=e^x+\ln x$ for $x>0$.
Find $\frac{df^{-1}}{dx}(e)$.\vskip10pt
\ans{$\frac 1e$}{$e$}{$\frac 1{e+1}$}{$f$ is not one to one}
{$\bigl(e^e+\frac 1e\bigr)^{-1}$.} %%(c)

\problemskip
\pr(5pts)
Compute
$\qquad\displaystyle \int^{\textstyle e^\pi}_{\textstyle 1}\frac{\sin(\ln x)}{x}\>dx$\vskip10pt
\ans{$1$}{$2$}{$-2$}{$0$}{$\pi$}%%(b)

\newpage
\pr(5pts)
Find the critical point(s) of $f(x)=x e^x$ and determine whether they
are local minima, maxima or neither.\vskip10pt
\threetwo{$x=0$, local max}{no critical points}{$x=e$ neither}
{$x=1$, local max}{$x=-1$, local min}
%%(e)

\problemskip
\pr(5pts)
Find all solutions of the equation
$\log_4(1+2x)=\log_2(3)$\vskip10pt
\ans{no solutions}{$4$}{$2$}{$1$}{$-2$}%%(b)

\newpage

\pr(5pts)
For $x>0$ let $f(x)=\Int^{x^2}_1\frac 1t\>dt$.
Find $\frac{df}{dx}$.\vskip10pt
\ans{$\frac 2x$}
{$2x\ln x$}
{$f$ is not differentiable}
{$\frac{2x}{\ln x}$}
{$2x$}%%(a)

\problemskip

\pr(5pts)
Calculate\quad $\Lim_{x\to 3}\>\frac{x^2+9}{x^3+18}$.\vskip10pt
\ans{$\frac 15$}{$\frac 29$}{$\frac13$}{$\frac 25$}{$\frac23$}
%%(d)

\newpage
\pr(5pts)
Calculate\quad $\Lim_{x\to 3}\>\frac{e^x-e^3}{\ln(4-x)}$.\vskip10pt

\ans{$-\infty$}
{$-e^3$}
{$e^{-3}$}
{$\infty$}
{$0$}
%%(b)
\problemskip

\pr(5pts)
Calculate $\Int^1_0\frac{e^x}{e^{2x}+1}\>dx$.\vskip10pt
\ans{$\arctan(e)-\frac \pi4$}
{$\ln(e^2+1)-\ln(2)$}
{$1$}
{$2$}
{$\cot(e^2+1)-\cot(2)$}%%(a)
\newpage

\pr(5pts)
Calculate $\frac{d\ }{dx}\>\arcsin(x^2)$\vskip10pt
\ans{$\frac{2}{x\sqrt{1-x^4}}$}
{$\frac{x^2}{{1+x^2}}$}
{$\frac{2x}{\sqrt{1-x^4}}$}
{$\frac{1}{\sqrt{1-x^4}}$}
{$\frac{2x}{\sqrt{1-x^2}}$}%%(c)

\problemskip

\pr(5pts)
Calculate $\Int^{-2}_{-3}\frac{dx}{x\sqrt{x^2-1}}$.\vskip10pt
\threetwo{$7$}{$\arcsin(3)-\arcsin(2)$}
{$\arctan(-2)-\arctan(-3)$}{$e^{-2}-e^{-3}$}{$\arcsec(2)-\arcsec(3)$}%%(e)
\newpage

\centerline{\bf Partial Credit}

\pr(10pts)
Consider the function $f(x)=\sqrt{1+x^2}$.
\item{a)} Show that $f$ is one to one on the domain $(0,\infty)$.
\item{b)} Find the slope of the tangent line to the graph of
the inverse function $f^{-1}$ at the point $(\sqrt{10},3)$.


\newpage

\pr(10pts)
Find the derivative of the function
$$f(x)=\root{\textstyle 3}\of{\frac{(x^2-1)^4e^{\sin x}}{(x+1)^5}}\quad .$$

\newpage
\pr(10pts)
A certain bacteria culture, undergoing natural growth, doubles
in size after 10 minutes.
If there were 100 specimens at time $t=0$, when will the number have
increased to 3000 specimens?
Express your answer using $\ln$'s where needed.

\newpage
\pr(10pts)
Calculate $\Lim_{x\to-\infty}\bigl(\>1+\smfrac{x^2}{2}\>\bigr)^{^{\frac 1{x^2}}}$.

\newpage
\pr(10pts)
Students sometimes feel that functions described by expressions like
\hfill\par\noindent
$F(x)=\cases{0&$x\geq0$\cr x^2&$x<0$\cr}$\quad are not very natural.
Consider the function
$$f(t)=\arcsin\biggl(\frac{t^2-1}{t^2+1}\biggr) - 2 \arctan t\ .$$
\item{a)} Show that $f(t)$ is constant for $t\geq 0$.
\item{b)} Show $f(t)$ is not constant for $t<0$.

\noindent{\bf Hint:}
Compute $\frac{df}{dt}$ carefully.
\vfill\eject}

%\print{Bulllwinkle}
\print{Cao}
\print{Alber}
\print{Taylor}
\print{Nollet}
\print{Patrizio \quad{\sl Section \#:}}
\marked{}
\end

