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\vskip20pt
\head {\bf MATH 165 \rm} \endhead
\head {\bf TEST 2 - PART A - \;\;\;(12/02/02)}\endhead
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\noindent NAME: 
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\noindent 1) Write a proof of the following theorem (state the theorem here):
\vskip100pt


\noindent 2) Write a short essay on the axiomatic approach to the concept of volume, including its relation to Cavalieri's principle.  

\vskip20pt
\head {\bf MATH 165 \rm} \endhead
\head {\bf TEST 2 - PART B - \;\;\;(12/04/02)}\endhead
\vskip50pt
\noindent NAME: 
\vskip20pt

\noindent 3) Write TRUE or FALSE for each of the following statements, in each case justifying your answer:
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\noindent a) If $f:[a,b]\to \Bbb R$ is such that $f^2$ is continuous, then $f$ itself must be continuous.
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\noindent b) If $f:[a,b]\to \Bbb R$ is bounded and integrable, then the function $F:[a,b]\to \Bbb R$ given by $F(x)=\int_a^xf(t)dt$ is continuous.
\vskip10pt
\noindent c) An application of Bolzano's theorem allows us to conclude that the equation $ x^2-x^4=-1$ has a solution in the interval
$[0,{3 \over 2}]$.
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\noindent d) If $f$ is a continuous function on an \it open \rm interval $J$, then the equation $f(x)=0$ has only finitely many solutions in $J$.
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\noindent e) If $f$ is a continuous function on a \it closed \rm interval $J$, then the equation $f(x)=0$ has only finitely many solutions in $J$.
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\noindent f) The stipulation that $f:\Bbb R\to \Bbb R$ is continuous at the point $a\in \Bbb R$ means, in symbols:
$$\forall \epsilon>0 \;\;\exists \delta>0 : \;\;|x-a|\leq \epsilon \implies |f(x)-f(a)|\leq \delta.$$
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\noindent g) Consider a function $f:[a,b]\to \Bbb R$ having the following property:  
$$ (\dagger) \;\;\;\;\;\;\;\hbox{For all} \;\;n\in \Bbb Z  \;\;\hbox {the equation} \;\;f(x)=n \;\;\hbox{has
at least one solution}.$$
Then $f$ must be discontinuous at some point of $[a,b]$. 
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\noindent h) One has $$\lim_{x\to 0} |x|^{1 \over 10000} \sin ({1\over x})=0. $$
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\noindent 4) Using the double-angle formula, compute
$$\int _0^{\pi\over 2}
\sin^2x dx.$$

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\noindent 5) Compute
$$\lim_{t\to 0} {\sin(\tan t) \over {\sin t}}.$$
\vskip10pt

  
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