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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}}
\centerline{\bf Fall Semester }
\centerline{\bf Final Exam}
\centerline{\bf {December 16, 1991, 1:45 - 3:45 PM, Comp 328}} 

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This Examination contains eight problems worth a total of  150  points 
each problem worth 20 points,except the last which worths 10 points, on 10 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 Exam}":
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\centerline{\bf GOOD LUCK and HAPPY HOLIDAYS}


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1.
i)	Compare the derivative of the function $$f(x) = 1 - x^5 + \frac{x}{1 + x^2} + \sin (x^2 +
3) + \int_{x^2}^{1 + x^4} \sqrt{2 + \cos t}$$

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ii)	Compute the integral
	$$I = \int_0^3 [x] x dx$$

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2.
i)	Let $f(x) = 5x \sin \frac{1}{x}.$  Use the definition of the limit to show that \newline
$\lim_{x \to 0} f(x) = 0.$ 
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ii)	Let $f(x) = \frac{1}{x}, x > 0.$  Use the definition of the derivative to show
that
 $$f(x) = \frac{-1}{x^2}.$$
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3.  Suppose that the function $y = f(x)$ is defined for $- \infty < x < \infty$ and has
first, second, and third derivatives for each $x.$  Use the letters corresponding to the
choices given below to form true statements.  \newline
 i) If $f'(x) < 0$ for each $x$ then
$f$ is  \vrule height .10pt depth 0 pt width 1 in .

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ii)	If $f''(x) > 0$ for each $x$ then $f$ is \vrule height .10pt depth 0 pt width 1 in .

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iii)  If $f'(1) = 0$ and $f''(1) < 0$ then $f$ has a \vrule height .10pt depth 0 pt width 1
in  at $x = 3.$ 

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iv)	If $f''(x) < 0$ for $x < 3$ and $f''(x) > 0$ for $x >3$ then $f$ has a \vrule height .10pt depth 0 pt width 1
in  at $x = 3.$ 

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v)	If $f''(x) = 0$ for all $x$ then the graph of $f$ is \vrule height .10pt depth 0 pt width 1 in .
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\centerline{CHOICES} 
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(a)  concave upwards  \newline
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(b)  concave downwards  \newline
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(c)  relative or local maximum  \newline
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(d)  relative or local minimum \newline
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(e)  point of inflection  \newline
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(f)  straight line  \newline
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(g)  parabola  \newline
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(h)  increasing \newline
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(i)  decreasing 
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4.  The length $l$ of a rectangle is decreasing at the rate of 2 cm/s and the width $w$ is
increasing at the rate of 2 cm/s.  When $l = 12 cm$ and $w = 5 cm$, find the rates of
change of  
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(a)  the area \vrule height .10pt depth 0 pt width 1 in  \newline

(b)  the perimeter \vrule height .10pt depth 0 pt width 1 in  \newline

(c)  the lengths of the diagonals of the rectangle  \vrule height .10pt depth 0 pt width 1 in  \newline
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Which of these quantities are increasing and which are decreasing?  \newline

increasing quantities \vrule height .10pt depth 0 pt width 1 in  \newline

decreasing quantities \vrule height .10pt depth 0 pt width 1 in  \newline

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5)  Find the dimensions $r$ and $h$ of the cone of maximum volume described
symmetrically in a hemisphere of radius 1 as illustrated.  Note:  the volume of a cone
equals one third of its base area times its height.
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6)  Compute the area of the shaded region $A$ illustrated below. \newline
a) The area is \vrule height .10pt depth 0 pt width 1 in  \newline

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b) The volume is  \vrule height .10pt depth 0 pt width 1 in  \newpage

7)  \underbar{Intermediate Value Theorem.}  \newline
i)  If $f$ is continuous on $[a,b]$ and
$y_0$ is a number between $f(a)$ and $f(b)$, $f(a) \neq f(b),$ then there exists an $x_0
\in (a, b) $ such that $f(x_0) = $
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Complete the statement of each of the following theorems: \newline

ii)  {\bf The Mean Value Theorem}  If $y = f(x)$ is continuous at each point of $[a,b]$ and
differentiable at each point of $[a,b],$ then there is at least one number $c$ between $a$
and $b$ for which $\frac{f(b) - f(a)}{b - a} = $ \newline

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iii)  {\bf The Second Fundamental Theorem of Integral Calculus.}  If $f$ is continuous on
$[a,b]$ and $F$ is any antiderivative of $f$ on $[a,b],$ then $$\int_a^b f(x) dx = $$

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iv)  {\bf The First Fundamental Theorem of Integral Calculus.}  If $f$ is continuous on
$[a,b]$ then $$F(x) = \int_a^x f(t) dt$$ 

is differentiable at every point $x$ in $[a,b]$ and

$$\frac{dF}{dx} = \frac{d}{dx} \int_a^x f(t) dt = $$

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v)  {\bf Leibniz's Rule} If $f$ is continuous on $[a,b],$ and $u(x)$ and $v(x)$ are
differentiable functions of $x$ whose values lie in $[a,b],$ then 
$$\frac{d}{dx} \int_{u(x)}^{v(x)} f(t) dt = $$

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8)  Do either (i) or (ii) \newline

(i) Prove any one of the theores we did in class this semester:

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(ii)  Let $f\: [0,2] \rightarrow \Bbb R$ be a function which is continuous at $1.$  If $f(1) >
0$ then show that there is $\delta > 0$ such that 
$f(x) > \frac{1}{2} f(1)$, for any  $x \in (\frac{1}{2} - \delta, \frac{1}{2} +
\delta).$  


  



 



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