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{\bf Name:}  \vrule height .4pt depth 0 pt width 2.0
in  \hskip .5 in {\bf Professor:} A.Alexandrou Himonas
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 \centerline{\bf {Mathematics 165}}
\centerline{\bf {Honors Calculus 1}}
\centerline{\bf Fall Semester }
\centerline{\bf Exam 2}
\centerline{\bf {November 11, 1991}} 

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This Examination contains five problems worth a total of  100  points, 
each problem worth 20 points, on (7) sheets of paper including the
front cover and one extra sheet on the back.  Do all your work in this
booklet and show your computations.  Calculators, books and notes are
not allowed.
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{\bf Sign the pledge:}  \lq \lq {On my honor, 
I have neither given nor received unauthorized aid on this Test}":
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\centerline{\bf GOOD LUCK}



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1a.)  Compute the limit
$$\lim_{x\to 0}(\frac{1}{x \sin x} - \frac{1}{x^2})$$

where $R$ is the radius of the earth and $g$ is the gravitational acceleration at sea level.

Find the linear approximation of $W(x)$ at $x=0$ (sea level). 
 

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1b.)  The weight of a body of mass $m$ at altitude $x$ above sea level is given by
$$w(x) = \frac{mgR^2}{(R + x)^2}$$ 
where $R$ is the radius of the earth and $g$ is the gravitational acceleration at sea level.   

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2a.)  Let $f(x) = 4x^5 - 5x^4 + 2.$  Sketch the graph of $f$ by using the information
given by $f'(x)$ and $f''(x).$   

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2b.)Show that $f(x)$ has exactly one real root which is negative.   

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3).  Among all right circular cylinders with a given volume, find the one with the
least surface area.   

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4a.)  Let the set $A$ be defined by
$A = \left \{ \frac{1}{1 + x^2}  :  x \in \Bbb R \right \}$

Find sup$A$ and inf$A$ and examine if they belong in $A$.

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5. {\bf Bonus} \, \,  Find a function $f: \Bbb R \longrightarrow \Bbb R$ such that it is
continuous only at $0$. 

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1a.)   

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1b.   

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