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Exam 2 Nov. 10, 2005 \hfil SHOW ALL WORK \hfil
\vskip -10pt
Math 25 Calculus I \hfil Either circle your answers or place on answer line. \hfil

[16] 1.) If $f'(x) = 3x^4 + 2 + 4x^{-1}$ and $f(1) = 8$, find $f$   

\vfill
\centerline{Answer 1.) $\underline{\hskip 4in}$}
\vskip 10pt
[16] 2.)$lim_{x \rightarrow 0^+} [x^{x^3}]$~= $\underline{\hskip 2.5in}$
 
%%$lim_{x \rightarrow 0^+} (x^2 - x)^{x^3}$= $\underline{\hskip 2.5in}$


\vfill



\eject



[10] 3a.)  Given 
$x^2 + 2xy + y^3 - x - 3 = 0$, 
then $y'= 
\underline{\hskip 2in}$

\vfill

[6] 3b.) Find the equation of the tangent line to the curve 
$x^2 + 2xy + y^3 - x - 3 = 0$, 
at the point (-2, 1).

\vskip 1in
\centerline{Answer 3b.) $\underline{\hskip 4in}$}
\vskip 10pt

[16] 4.) 
If $g(3) = 4$ and $g'(x) \leq 2$, how large can $g(8)$ be? $\underline{\hskip 1.4in}$

\vfill \eject [16] 5.)  A tank is in the form of an inverted cone having 
a height of 16m and a diameter at the top of 8 m.  Water is flowing into 
the 
tank at the rate of 2 m$^3$/min.  How fast is the water level rising when 
the water is 5m deep? (Volume of cone = ${1 \over 3}\pi r^2 h$)



\vfill
\centerline{Answer 5.) $\underline{\hskip 4in}$}


\eject

6.) Find the following for $f(x) = {x  \over (x-1)^2}$ (if 
they 
exist; if they don't exist, state so).  Use this information to graph $f$.

Note $f'(x) = {-x-1 \over (x-1)^3}$ and $f''(x) = {2(x+2) \over (x-1)^4}$
 
%%Is $f$ even, odd, periodic?  What is the domain and range of $f$?
\vskip -2pt

[1] 6a.) critical numbers: $\underline{\hskip 1.5in}$
		

[1.5] 6b.) local maximum(s) occur at $x = \underline{\hskip 1.5in}$

[1.5] 6c.) local minimum(s) occur at $x = \underline{\hskip 1.5in}$

[1.5] 6d.)  The global maximum of $f$ on the interval [0, 5] is 
$\underline{\hskip .5in}$ and 
occurs at $x = \underline{\hskip 1in}$

[1.5] 6e.)  The global minimum of $f$ on the interval [0, 5] is 
$\underline{\hskip .5in}$ and 
occurs at $x = \underline{\hskip 1in}$

[1.5] 6f.) Inflection point(s) occur at $x = \underline{\hskip 1.5in}$


[1] 6g.)  $f$ increasing on the intervals $\underline{\hskip 1.5in}$

[1] 6h.)  $f$ decreasing on the intervals $\underline{\hskip 1.5in}$

[1.5] 6i.)  $f$ is concave up on the intervals $\underline{\hskip 1.5in}$

[1.5] 6j.)  $f$ is concave down on the intervals$\underline{\hskip 1.5in}$

[1.5] 6k.)  Equation(s) of vertical asymptote(s)$\underline{\hskip 1.5in}$

[4] 6l.)  Equation(s) of horizontal and/or 
slant asymptote(s)$\underline{\hskip 1.5in}$


\vskip -4pt
[4] 6m.)  Graph $f$
\vskip -8pt
\includegraphics[width=48ex]{grapht}

\end

\end{document}


3.7--4.)  Find $f'''(x)

6.) 3.11)  1 or 23, 27, 31


9.) 4.9

7, 3, 8, 5


3.8--2.)  If $f(x) = [sin(x)]^2$, then $f'(x) = \underline{\hskip 4in}$
Simplify answer



\vskip 10pt
[15] 2.) Find the derivative of $f(x) = \sqrt{x}$ by using the definition
of derivative.


\vfill
\centerline{Answer 2.) $\underline{\hskip 4in}$}
\eject

Find the following derivatives

\vskip 10pt

[15]~  3.) ${d \over dx}[{e^x(x^2 - x + 3) \over cos(2x)}]$

\vfil
\centerline{Answer 3.) $\underline{\hskip 4in}$}
\vskip 10pt

[15]~  4.) ${d \over dx}[2sin(e^{x^3} + 4)]$

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

\eject


5.)  Answer the following questions based on the graph of $f$ given below.

\centerline{\includegraphics[width=36ex]{graph25a}}


%%\centerline{\epsfbox{graph25a.eps}}

[2] 5a.)  domain of $f = \underline{\hskip 0.7in}$
\hfil
[2] 5b.)  range of $f = \underline{\hskip 0.7in}$
\vskip 15pt
[1] 5c.)  Is $f$ one-to-one? $\underline{\hskip 0.7in}$
\hfil
[2] 5d.)  Does $f^{-1}$ exist? $\underline{\hskip 0.7in}$

\vskip 15pt

[1] 5e.)  $f(1) = \underline{\hskip 0.7in}$
\hfil
[2] 5g.)  $f\hskip 1pt'(-1) = \underline{\hskip 0.7in}$
\vskip 15pt
[2] 5f.)  Solve $f(x) = 1: \underline{\hskip 0.7in}$


\vskip 15pt

[2] 5i.)  $lim_{x \rightarrow +\infty} f(x) = \underline{\hskip 0.7in}$
\hfil
[2] 5j.)  $lim_{x \rightarrow -\infty} f(x) = \underline{\hskip 0.7in}$
\vskip 15pt

[2] 5k.)  $lim_{x \rightarrow 3^+} f(x) = \underline{\hskip 0.7in}$
\hfil
[2] 5l.)  $lim_{x \rightarrow 3^-} f(x) = \underline{\hskip 0.7in}$
\vskip 15pt

[2] 5m.)  State all points where $f$ is not continuous: $\underline{\hskip 1.7in}$

\vskip 15pt

[2] 5n.)  State all points where $f$ is not differentiable: $\underline{\hskip 1.7in}$

%%\centerline{\epsfbox{graph25.eps,width=6truein}}

\includegraphics[width=83ex]{graph25}

\end


6.)  Given the graph of $y = f(x)$ below, draw the following graphs:


[4] 6a)  y =







\hfil





\end
Review Problems

Ch 3 p.270-274

Concept check:  1, 3

TF Quiz:  1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13

Exercises:  1 - 14, 17, 19, 21-24, 26, 30, 32, 35, 37, 39, 41, 42, 44, 55-57, 59, 61, 63, 67, 68, 84, [85-87 (but ignore acceleration)], 88, 100, 101, 102


Ch 2 p. 176 - 179

Concept check:  2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14

TF Quiz:  All

Exercises:  [1, 2], 3-12, [13], 14, [15-21], 25, 26, [31, 32], [35], 37, [39], 40, 41ab, [44-46], [47ab], 49

Ch 1:

Concept check:  1, 3, 4, 6, 7, 8, 9, 11, 12, 13

Exercises:  1, 2, 5-8, 10, 19, 24, 25, 26

Tuesday Quiz:  Proof + 3.6

  Prove 

$(f + g)' = f' + g' $

$(cf)' = c(f')$ 

$f$ differentiable implies $f$ continuous:

Show $lim_{x \rightarrow a} f(x) = f(a)$ 

Show $lim_{x \rightarrow a} [f(x) - f(a)] = 0$ 

$f$ differentiable at $a$ implies $f'(a)$ exists and $lim_{x \rightarrow a} {f(x) - f(a) \over x - a} = f'(a)$

Hence $lim_{x \rightarrow a} [f(x) - f(a)] = lim_{x \rightarrow a} {f(x) - f(a) \over x - a} (x-a) = lim_{x \rightarrow a} f'(a) (x - a)   
f'(a) lim_{x \rightarrow a} (x - a) = f'(a) (0) = 0$.

 $lim_{x \rightarrow a} f(x) = lim_{x \rightarrow a} [f(x) - f(a) + f(a)] = lim_{x \rightarrow a} [f(x) - f(a)] + lim_{x \rightarrow a} f(a) =   
0 + f(a)$

\end