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\leftline{Mathematics 134 -- Intensive Calculus for Science 2}
\leftline{Makeup Exam I -- February 20, 2006}
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{\it Directions}.  Do all work in the blue exam booklet.  You
may use a calculator on any of the following problems.  There are
100 possible points.
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I.  The given graph shows $y = f(x)$. 
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\item{A)} (10)  Use the graph to determine $\int_0^4 f(x)\ dx$.
\item{B)}  (5) Let $F(x)$ be the antiderivative of $f(x)$
with $F(0) = 1$.  What is $F(4)$?
\item{C)}  (15) Give a qualitative sketch of $y = F(x)$
from part B.  Your graph should show where $F$ is 
increasing/decreasing, and where $F$ is concave up/concave down.
\item{D)} (5) The function $G(x)$ is $G(x) = \int_0^x f(t)\ dt$
using the function $f$ in the graph above.  What is $G'(1)$?
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II.  A car travels down a straightline road with acceleration
$a(t) = 4 + t/2$ m/sec${}^2$. 
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\item{A)} (10) Solve the equation ${dv/dt} = 4 + t/2$ to 
find the car's velocity $v(t)$, assuming $v(0) = 2$ m/sec.
\item{B)} (10) Use your answer $v(t)$ from part A.  If the 
car's initial position is $s(0) = 0$, find the position
as a function of time by solving ${ds\over dt} = v(t)$
\item{C)}  (10)  What is the car's average velocity 
over the time $t = 0$ to $t = 3$ sec?
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III.  Compute each of the integrals below using basic rules
and/or substitution.  You must show all work
for full credit. 
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\item{A)} (10) 
$$\int 2x^3 - {4\over \sqrt{x}} + 5 x\sqrt{x} + {2\over x}\ dx$$
\item{B)} (15)
$$\int \sin(3x) + \cos(5x)\ dx$$
\item{C)} (10) 
$$\int_0^1 x e^{2x^2 - 3}\ dx$$
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