%% Math 131, Fall, 2003
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\noindent\textbf{Name}:\ \vbox{\hrule width 3in}

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\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Quiz \#6 Solutions} \\
Professor Hwang \\
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\be
\item Let $f(x) = \ds\frac{x^2+1}{x}$. Find $f'(x)$.

  \soln Breaking up the fraction, we have
  \[
  f(x) = \frac{x^2}{x}+\frac{1}{x} = x + x^{-1}.
  \]
  Using the power rule, we find that
  \[
  f'(x) = 1-x^{-2} = 1-\frac{1}{x^2}.
  \]

\vfil
  
\item Find the equation of the line tangent to the parabola $y=x^2$ at
  the point~$(1,1)$.

  \soln Let $f(x)=x^2$. The derivative is $f'(x)=2x$, and the slope of
  the tangent line at $x=1$ is $f'(1)=2$. Using the point-slope form
  of the equation of a line, we have
  \[
  y-1=2(x-1),\qquad\text{or}\quad y=2x-1.
  \]

\ee
\vfil\eject

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