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\begin{center}
{\Large Final

Math 129 - R5}
\end{center}
\vspace{.1in}
You have 2 hours to complete the test.
Show all relevant work (or else no credit).  
Make your work legible, if I can't read it, I can't grade it.  
Each problem is worth 10 points, unless otherwise noted.
Use back of pages if necessary.
For the 'essay' questions, I am looking for clear sentences 
that convey a sense of understanding.

REMEMBER: Use all the differentiation rules when needed,
and don't interchange integral and derivative!

\vspace{.1in}

Name \rule{5in}{.2pt}

\vspace{.2in}

Code Name (optional) \rule{4in}{.2pt}

\begin{enumerate}

\item  There were three new concepts introduced in this class.
How would you explain to someone who doesn't know calculus the following ideas?
\begin{enumerate}
\item  Limit
\vspace{2.1in}
\item  Derivative
\vspace{2.1in}
\item  Integral
%\vspace{2in}
\end{enumerate}
\vspace{2in}
\newpage

\item  Can you fill in the question mark (?) to make the following function 
continuous?
If you can, fill in the (?) and tell me why that makes the function
continuous.
If not, explain why not.
\[
g(x)=\left\{
\begin{array}{cr}
\frac{6x^2-x-15}{2x-5} & x \neq \frac{5}{2} \\
? & x = \frac{5}{2} \\
\end{array}
\right.
\]
%\newpage
\vspace{4.5in}

\item  What is the (limit) definition of derivative?
%\vspace{2in}
\newpage

\item  The percentage of U.S. workers in farm occupations can be
modeled by the function
\[
F(t)=\frac{-4000t+1200000)}{t^2-60t+15000}
\]
Where $t$ is the number of years past 1800.
\begin{enumerate}
\item  Find $F(200)$ and explain what this number means to someone who
doesn't understand calculus.
\vspace{2in}
\item  Find $F'(200)$ and explain what this number means to someone who
doesn't understand calculus.
(Hint: what rule should you use?)
\end{enumerate}
%\vspace{2in}
\newpage

\item  The percentage of high school seniors who have tried pot
can be modeled by the formula:
\[
P(t)=-.228t^2 +37.1t - 1440
\]
where $t$ is the number of years past 1900.
In what year did the highest percentage of seniors try pot,
and what was this maximum percentage?
\newpage

\item  Draw a graph of a function that matches the data below:
(label all local maximums and minimums and all inflection points)
\[
\begin{array}{ll}
f'(x)>0 & \mbox{ if } x<0 \mbox{ or } 0<x<2 \\
f'(x)<0 & \mbox{ if } 2<x \\
f''(x)<0 & \mbox{ if } 0<x<1 \\
f''(x)>0 & \mbox{ if } x<0 \mbox{ or } 1<x \\
\end{array}
\]
\newpage

\item  Find $\frac{dy}{dx}$ for the curve:
(Hint: don't forget your rules!)
\[
x^4+2x^3 y^2+\frac{x}{y+1}=x-y^3
\]
\vspace{3.3in}

\item  Find the integral
\[
\int \frac{x}{(3x^2-8)^\frac{4}{3}} \, dx
\]
%\vspace{3in}
\newpage

\item  You are helping Detective Colombo on a homicide case.
Colombo found the dead body of Mrs.\ Greene at 9 pm, 
quickly found that the temperature of her body at this time was $70^\circ$ F.
Now, he needs to determine when she was murdered.
The temperature, $B$ in $^\circ$F, of a body, after being killed satisfies the
following differential equation:
\[
\frac{dB}{dt}= -\frac{1}{2}(B-60)
\]
were $t$ is in hours.
What time was Mrs.\ Greene murdered?
(Hint: it will be necessary to use that fact that the temperature of
a living body is $98.6^\circ$F.)
%\vspace{3in}
\newpage

\item  
\begin{enumerate}
\item   Approximate the following integral using 3 rectangles(3 subintervals):
(Use either a left or right hand rule.)
\[
\int_1^9 x^2+2x \, dx
\]
\vspace{4.9in}

\item  Evaluate the integral and tell me why this number is not 
the same as in the previous problem
\[
\int_1^9 x^2+2x \, dx
\]
\end{enumerate}
\newpage

\item  What is the Fundamental Theorem of Calculus, 
and what makes it fundamental?
%\newpage
\vspace{3in}

\item  Evaluate the integral.
\[
\int_1^2 \frac{-x^2+5x-3}{x} \, dx
\]
%\vspace{3.5in}
\newpage

\item  Find the area enclosed between the two graphs
\begin{eqnarray*}
y = x-x^2 \\
y = x^3-x^2 \\
\end{eqnarray*}
\vspace{4.2in}

\item  Evaluate the integral.
\[
\int \frac{\ln x}{x^3} \, dx
\]
%\vspace{3in}
\newpage

\item  Evaluate the integral
\[
\int_4^{\infty} \frac{1}{x^{\frac{7}{2}}} \, dx
\]
\vspace{3.9in}
%\newpage



\end{enumerate}


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