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\topmatter


 \centerline{\bf Mathematics 108,  Review 1}
\centerline{\bf Spring Semester 1998}
\medskip
\noindent
{\bf 1.} Find  a function  $y=y(x)$ such that:
$ y'=e^{-2x}-4x^3+1$, and $y(0)=12$.
\medskip
\noindent
{\bf 2.} Compute the  indefinite integral
${\int} te^{-.25t} dt$.
\medskip
\noindent
{\bf 3.} Calculate the  integral
${\int^1_0}xe^{x/3}  dx. $
\medskip
\noindent
{\bf 4.}  Compute the area of the region enclosed by the curves
 $y= 7-x^2$ and $y= -1+x^2$.
\medskip
\noindent
{\bf 5.}  Find the function whose graph has tangent of slope $x + lnx$
 for each positive value of $x$, and  passes through the
point $(1, 8)$.

\medskip
\noindent
{\bf 6.} The marginal cost function for producing $ x $ units
of a certain product is given by the function
$MC(x) = \frac{1}{4} x + 5,000.$
If the cost  for producing 20 units is \$50,000 then
determine the cost function  $C(x)$.
Recall, marginal cost is by definition the derivative
of the cost  function.
\medskip
\noindent
{\bf 7.} The marginal profit function for a company
producing and selling a certain product is
$MP(x) = -0.6x + 3,000.$
If 40 units are produced and sold then the company's profit is \$25,000.
Determine the profit function $P(x)$.
Find the sales amount $x$ which maximizes the profit.
Explain  in economic terms the meaning of the negative
coefficient of $x$ in $MP(x)$.
\medskip
\noindent
{\bf 8.} If $y=y(x)$ is the solution to the initial value problem:
$y'+\frac{1}{x}=x^5$, $ y(1)=3$, then find $y(e)$.
\medskip
\noindent
{\bf 9.}  Let $ f(x), 2\le x \le 4,  $ be a continuous function with
$f(2)=10$, $f(2.5)=14$, $f(3)=12$, and $f(3.5)=8$.
By computing the  Riemann sum with 4 subintervals, and using
the value of the function at the left endpoints,
 one finds that
   $ \int_{2}^4 f(x) dx$ is approximately  equal to:
\smallskip
A.  44 \hskip 0.7 in B. -22 \hskip 0.7 in C. 22 \hskip 0.7 in D. 21 \hskip
0.7 in  E. 4
\medskip
\noindent
{\bf 10.}  The area between the curves $y= 2x^{2} -4x +6$ and $y= -x^{2}
-2x +1$
from $x=1$ to $x=2$ is
\smallskip
A. 1   \hskip 0.7 in B. 14 \hskip 0.7 in
  C. 5 \hskip 0.7 in D. 4 \hskip 0.7 in E. 9
\medskip
\noindent
{\bf11.} The area of the region between  the curve $y=x^5$ and the
$x$-axis, from $x=-1$ to $x=1$,  is equal to:
\smallskip
A. 0   \hskip 0.7 in B. 1/6  \hskip 0.7 in C. 1/3
\hskip 0.7 in D. -1/6 \hskip 0.7 in E. 1/5
\medskip
\noindent
{\bf 12.}  $\int x^7 e^{x^8}$ is  equal to:
\smallskip
A. $\frac 17 e^{x^7}+c$   \hskip 0.3 in B. $\frac{ e^{x^8}}{x^8}+c$  \hskip
0.3 in
C. $\frac 18 e^{x^8}$
\hskip 0.3 in D. $\ln x^8 +c$ \hskip 0.3 in E. $\frac 18 e^{x^8}+c$
\medskip
\noindent
{\bf 13.} Which of the followings is an antiderivative of
 $\displaystyle{\frac{e^{3x} - e^{-3x}}{e^{3x} + e^{-3x}}}$?
\smallskip
A. $\frac{e^{3x} + e^{-3x}}{e^{3x} -e^{-3x}}$
\smallskip
B. $\ln(e^{3x} + e^{-3x})$
\smallskip
C. $(e^{3x} - e^{-3x})^{-1}$
\smallskip
D. $ \frac{1}{3} \ln(e^{3x} + e^{-3x})$
\smallskip
E. $(e^{x} + e^{-x})^{-\frac{3}{2}}$
\medskip
\noindent
{\bf14.}  $\int x^5\ln x dx$ is  equal to:
\smallskip
A. $(\frac 16 \ln x -\frac{1}{36})x^6 +c$   \quad
B. $(\frac 16 \ln x +\frac{1}{36})x^6 +c$  \quad C. $x^5e^x+c$

 D. $\ln x +c$ \quad E. $(\frac 16 \ln x -\frac{1}{36})x^6.$
\medskip
\noindent
{\bf 15.}  Estimate the
integral
$\int_1^3 f(x) dx $
by using a  uniform partition of 4 subintervals
 and by evaluating the function at the RightSide
of each subinterval, when the following values of the function are given
$$\vbox{\offinterlineskip
\halign{
\strut \vrule # & \vrule # &\vrule # &\vrule #
&\vrule # \vrule  height .2 in depth .1 in width .3pt \cr
\noalign{\hrule}
$\ \ x$ &\ \  1.5 &\ \  2 &\ \ 2.5 &\ \ 3 \cr
\noalign{\hrule}
$f(x)$  &\ \ 0.5 &\ \ -1\  &\ \ -2
&\ \ -1.5 \cr \noalign{\hrule}
}}.$$
\medskip
\noindent
{\bf 16.}
Suppose $F(x)$ is an antiderivative of a  continuous function
$f(x)$. If $F(2)=8$ and $F(4)=-2$ then compute
$\int_2^4 f(x)dx$.
\medskip
\noindent
{\bf 17.}   Estimate the area  of the region between the
two curves and the $p$-axis  shown below.
$$\epsfxsize=3.3in\epsfbox{Rev-1-dem-supl.eps}$$
\medskip
\noindent
{\bf 18.} In the above picture
 the curves  are given by
$D(q) =20e^{-0.002q}$ and  by  $S(q) = 0.02q+ 1$.
Compute the area of the region between the
two curves and the $p$-axis.
\medskip
\noindent
{\bf 19.} Write the Riemann sum for the function
$f(x) \ =\ {1\over x^3}$
over the interval $[0,1]$ if the interval is partitioned into 10 equal segments
and the right-hand endpoint is used as the point $x_j$ in the $j$-th
interval. Do not compute it.
\medskip
\noindent
{\bf 20.}
Suppose $A(t)$ is the area under the graph of
$y = {1\over{3+x^4}}$
 from $0$ to $t$, for any positive
number $t$.
Compute is $F'(t)$.


\newpage

\centerline{\bf {Mathematics 108, Exam 1, Spring 1997}}
\bigskip
\noindent
{\bf 1.} (Partial Credit)
\smallskip
\noindent
{\bf a.} The marginal profit  of a company
producing and selling a certain product is given by
$MP(x) = -0.2x + 50$,
where  $x$ is the number of units sold per day.
When 100 units are produced and sold,  the company's
 profit is \$1,500. Find the profit function $P(x)$.
\smallskip
\noindent
{\bf b.} Find the sales amount $x$ which maximizes the profit.
\medskip
\noindent
{\bf 2.} (Partial Credit) (Not done yet)
\smallskip
\noindent
{\bf a.}
 An Individual Retirement Account (IRA) is opened with
an initial investment of \$1,000.
Then \$2,000 per year is deposited uniformly and
continuously throughout the year. Assume the interest rate is $10\%$
compounded continuously. Find the initial condition,
 and the differential equation
for the amount of money, $M(t)$, in the IRA at any time t.
{\bf DO NOT SOLVE the differential equation}.
\medskip
\noindent
$\boxed{\text{Differential Equation:}\qquad \qquad \quad
\qquad \qquad \qquad .},
\quad $\boxed{\text{Initial Condition:}\qquad \quad\quad   .}
\medskip
\noindent
{\bf b.} Solve the initial value problem:
$y + y^\prime = x^2e^{-x }$, $  y(0) = 1$
\medskip
\noindent
{\bf 3.} (Partial Credit) ((a) Not done yet)
\smallskip
\noindent
{\bf a.}  The demand and supply curves for a product are as shown in
the following picture. Shade the region whose area
represents   the Consumer  Surplus.
Then give your best estimate for
the Consumer  Surplus.

$\epsfxsize=3.3in\epsfbox{Ex1-dem-sup.eps}$
\medskip
\noindent
{\bf b.}
 Compute the area of the region enclosed by the curves
 $y= x^2$ and $y= x+2$.  Draw the region first, and
 beware  that the {\bf area is a  positive number!}
\medskip
\noindent
{\bf 4.} (No Partial Credit)
\smallskip
\noindent
 Compute the following integrals.  Show your work.
\medskip
\noindent
\quad {\bf a.}  \quad $\displaystyle \int x e^{0.5x} dx$
\medskip
\noindent
 \quad {\bf b.}  \quad $\displaystyle \int_0^{-1+e} \frac{1}{1+t} dt$
\medskip
\noindent
 \quad {\bf c.}  \quad $\displaystyle \int (x+1) \left(2 x^2 +
4x \right ) ^{99} dx$
\medskip
\noindent
{\bf 5.} (No Partial Credit) ((b) and (c) Not done yet)
\smallskip
\noindent
{\bf a.} A continuous function $f(x)$ defined on the interval
$[1,6] $ is known to have the following values:
$$\vbox{\offinterlineskip
\halign{
\strut \vrule # & \vrule # &\vrule #&\vrule #&\vrule #
&\vrule # \vrule  height .2 in depth .1 in width .3pt \cr
\noalign{\hrule}
$\ \ x$  &\ \  3/2 &\ \ 5/2 &\ \ 7/2 &\ \ 9/2 &\ \ 11/2 \cr
\noalign{\hrule}
$f(x)$   &\ \  0.2 \  &\ \  0.7 &\ \  0.1 &\ \ -0.3  &\ \ -0.3 \cr
\noalign{\hrule}
}}.$$
If the interval $[1,6] $ is subdivided into 5 equal
subintervals find the value of the Riemann sum of $f$ arising from the
above data.
Show your work.
\medskip
\noindent
{\bf b.}  An apartment in Manhattan produces a perpetual
income stream flowing continuously and uniformly at a rate of
 \$50,000 per year. This income is continuously and uniformly
invested at a rate of  5\%. Find the present value of this stream.
Show your work.
\medskip
\noindent
{\bf c.} A certain sum of money is invested at an interest rate
of 14\% compounded continuously. How many years will it take
for the capital to double? (You may take  $\ln 2 = 0.7$).



\bigskip
\hrule
\bigskip
\centerline{\bf EXAM 1 IS ON FEBRUARY 3, TUESDAY, 8:00-9:15 AM }
\centerline{\bf  Section 02 in NIEW 127}
\bigskip
\hrule
\bigskip
\noindent
{\bf REVIEW SESSIONS: Open to all Math 108 Students}
\smallskip
\noindent
$\bullet$ Sunday, February 1, from 2:30 to 3:30pm
in Cushing 117,  by Prof. Alex.  Himonas.
\smallskip
\noindent
$\bullet$ Monday, February 2, from 7:00 to 8:00pm
in   DBRT 102, by Alan Howard.
\bigskip
\hrule
\bigskip
\noindent
{\bf Format:} The  Math 108, Exam 1, Spring 1996, will be
done by the Teachers in all sections on Monday's class.
Therefore the Review sessions will not cover the old Exam 1
but rather only answer questions and problems asked by the
students. The Teachers of Math 108
{\bf STRONGLY} recommend that the Students  do/try as many
of the review problems as they can before attending the
REVIEW SESSIONS. 


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