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\head {\bf MATH 165 \rm} \endhead
\head {\bf TEST 1 (10/17/02)}\endhead
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\noindent NAME: 
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\noindent 1) Write TRUE or FALSE for each of the following statements, in each case justifying your answer:
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\noindent a) The number $2 \sqrt 2$ is irrational.
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\noindent b) Let $A,\;B$ be any two sets which are dense, in the sense that any (non-empty) open interval in $\Bbb R$ contains points from both $A$ and
$B$. Then the intersection of
$A$ and $B$ is non-empty.
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\noindent c) The set  $$S=\{x\in \Bbb R | \;|2x-1|\leq 2,\;\; |x+1|<3\}$$ is empty.
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\noindent d) If a function $f:[a,b]\to (0,\infty)$ is increasing, then the function $g=f^2$ is integrable.
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\noindent e) Every bounded function $f:[a,b]\to \Bbb R$ is integrable.
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\noindent 2) Find the area of the region between the graphs of  $f(x)=x^7$ and $g(x)=x^5$.
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\noindent 3) State and prove a theorem whose proof explicitly relies on the completeness axiom.
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\noindent 4) Prove the binomial formula 
$$\big (a+b\big)^n=\Sigma^n_{k=0} {n\choose k}a^k b^{n-k}.$$

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