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Final Exam  Dec. 13, 2006 \hfil SHOW ALL WORK \hfil
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Math 25 Calculus I \hfil Either circle your answers or place on answer line.


1.) Suppose $f(x) = (x - 3)^2 $ and $g(x) = -x^2 + 8x - 11 $.


\l\l

[3]~ 1a.)  Set up, {\bf but do NOT evaluate}, an integral for the area of the 
region enclosed by $f$ and $g$.

\l

[4]~ 1b.) Set up, {\bf but do NOT evaluate}, an integral for the volume of the solid obtained by rotating the region bounded by the curves $f$ and $g$ about the line $x = 1$ (hint:  use cylindrical shells).

\l

[4]~ 1c.) Set up, {\bf but do NOT evaluate}, an integral for the volume of the solid obtained by rotating the region bounded by the curves $f$ and $g$ about the line $y = -3$ (hint:  use washers).
\l

\eject




[10]~ 2.)  Find the derivative of $g(x) = \sqrt{ln(x^3 + 1)}$


\vfill
\centerline{Answer 3.) $\underline{\hskip 4in}$}

[4]~ 3.) State the mean value theorem.

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[4]~ 4.) Express the following integral as a limit of Riemann sums.  Do not evaluate the limit:  $\int_0^1 x^2 dx$.

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[3]~ 5.) Is $f: {\cal R} \rightarrow {\cal R}$, $f(x) = x^2$ one-to-one? $\underline{\hskip .5in}$.  Explain your answer. 

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[3] ~ 6.) The domain of $f(x) = ln(x + 3)$ is $\underline{\hskip 1.7in}$.  
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\eject


[10] 7a.)  If $y = ln(x)$ find the differential $dy$ and evaluate 
$dy$ when $x = 1$ 
and $dx = 0.5$

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\vfill


7b)  Find the linearization of $f(x) = ln(x)$ at $x = 1$.
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\vfill

7c.)  Use the linearization (or differential) to estimate $ln(1.5)$.  Is this an over-estimate or an under-estimate?
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\eject

[10]~ 8.) Given $xy = sin(x) + cos(y)$,
find $y''$.  
 You do NOT need to
simplify your answer,
and you can
leave your answer in terms of $x$ and $y$
($y'$ should not appear in your final answer).

\eject

9.)  Find the following integrals (SIMPLIFY your answer):

[10]~ a.) $\int_0^6 {4x + 1 \over \sqrt{2x + 4}} dx$ $= \underline{\hskip 2in}$


\vfill

[3]~ b.)  $\int {2 \over 1 + x^2} dx$ $= \underline{\hskip 2in}$



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\eject
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10.) Find the following for $f(x) = {5x^2 \over x^2 + 3}$
 (if
they
exist; if they don't exist, state so).  Use this information to graph $f$.



Note $f'(x) = {30x \over (x^2 + 3)^2}$
 and $f''(x) = {90(1-x)(1+x) \over (x^2 + 3)^3}$

%%Is $f$ even, odd, periodic?  What is the domain and range of $f$?
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[1] a.) critical numbers: $\underline{\hskip 1.5in}$


[1] b.) local maximum(s) occur at $x = \underline{\hskip 1.5in}$

[1] c.) local minimum(s) occur at $x = \underline{\hskip 1.5in}$

[1] d.)  The global maximum of $f$ on the interval [0, 5] is
$\underline{\hskip .5in}$ and
occurs at $x = \underline{\hskip 1in}$

[1] e.)  The global minimum of $f$ on the interval [0, 5] is
$\underline{\hskip .5in}$ and
occurs at $x = \underline{\hskip 1in}$



[1] f.) Inflection point(s) occur at $x = \underline{\hskip 1.5in}$


[1] g.)  $f$ increasing on the intervals $\underline{\hskip 1.5in}$

[1] h.)  $f$ decreasing on the intervals $\underline{\hskip 1.5in}$

[1] i.)  $f$ is concave up on the intervals $\underline{\hskip 1.5in}$

[1] j.)  $f$ is concave down on the intervals$\underline{\hskip 1.5in}$


[2] k.)  Equation(s) of horizontal asymptote(s)$\underline{\hskip 1.5in}$


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[4] 6m.)  Graph $f$
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\eject



Choose 2 out of the following 4 problems. 
{\bf Clearly indicate which 2 problems you choose.}  Each problem is worth 10 points
You may do more than 2 problems for up to five points extra credit.

I have chosen the following 2 problems: $\underline{\hskip 3in}$

A.)  A rectangular field is bounded on one side by a river and on the 
other three sides by a fence.  Find the dimensions of the rectangular 
field that will maximize the enclosed area if the fence has total length 
100m.  
How do you know that these dimensions correspond to the enclosed field 
with maximum area?

\vfill
\centerline{Answer A.) $\underline{\hskip 4in}$}

\eject


B.) At noon, ship A is 100 miles west of ship B.  Ship A is sailing west at 40 mph 
and ship B is sailing north at 30mph.  How fast is the distance between the ships changing at 3:00pm?
\vfill
\vskip 70pt
\centerline{Answer B.) $\underline{\hskip 4.5in}$}
\eject

C.) Use the mean value theorem to show that $f(x) = x^3 + 3x + 5$ has at most one real root.
\vfill


D.)  State the $\epsilon$, $\delta$ definition of limit and use this definition to prove $lim_{x \rightarrow 2} 3x + 1 = 7$




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Final Exam Part C, Dec., 2005 \hfil SHOW ALL WORK \hfil
\vskip -10pt
Math 25 Calculus I \hfil Either circle your answers or place on answer line.


Note the proof problems on this page are completely optional.  You may choose to
prove one (and only one) of the following statements.  If you choose to do one of
the
following problems, it can replace your lowest point problem in the optional
section
OR 80\% of your lowest point problem in the required section.  If you chose to
do one
of the following problems, clearly indicate your choice.

I have chosen the following problem: $\underline{\hskip 3in}$



I.)  Prove 
$f$ differentiable implies $f$ continuous.



II.)  State and prove the Extreme Value theorem (you may use the phrase ``other case is similar'' in your proof).



\end