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{\bf Math 165: Honors Calculus I}
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{\bf Quiz 5}\quad {\sl Oct.\ 8, 1998}

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\item Find the average value of $f(x) = 4x - x^2$ on the interval $[0,4]$.

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\item Use the addition formulas for $\cos(x)$ to prove that
\[
\sin^2(x) = \frac{1-\cos(2x)}{2}\quad \mbox{and} \quad
\cos^2(x) = \frac{1+\cos(2x)}{2}
\]

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\item Show that if $f(x)$ is integrable and $f(x)\ge0$ on $[a,b]$, then\newline
$F(x) = \int_a^x f(t)\,dt$ is increasing on $[a,b]$

(Hint: For $a \le x_1 \le x_2 \le b$, show that $F(x_2) - F(x_1) \ge 0$.)
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