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{\bf Math 166: Honors Calculus II}
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{\bf Quiz 1}\quad {\sl Jan. 27, 2000}

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\item 
Let $f$ be continuous on $[a,b]$ and let $A(x) = \int_a^xf(t)\,dt$ for
$x\in[a,b]$. Use the \textsc{Mean Value Theorem} to prove that
$A'(x) = f(x)$ for $x\in(a,b)$.  (Recall that the \textsc{MVT} implies
that if $f$ is continuous on an interval $I$, then for any $x$,
$x+h\in I$, there is a $z$ between $x$ and $x+h$ such that $f(z) =
\frac{1}{h}\int_x^{x+h}f(t)dt$.)

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\item  Compute the following.
  \begin{enumerate}[a)]
  \item 
  $\int \frac{x}{\sqrt{x+1}}\,dx$.
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  \item 
  $\int (x \log(x))^2\,dx$
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  \item 
  $\frac{d}{dx}\int_{3x}^{x^2} \sqrt{\cos(t) + \sin(t)}\,dt$.
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