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MATH 155/GRACEY
EXAM 1/CH. 6.1-6.3 & REVIEW
100 POINTS POSSIBLE
SHOW ALL WORK FOR FULL CREDIT/EXACT ANSWERS ONLY UNLESS
OTHERWISE INDICATED
1.
(24 POINTS) Find a general solution of the differential
equation.
c. y'=x.Jl-x
2.
(8 POINTS) Differentiate
a.
Y
= ln x'
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c
b.~
ax
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f
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the following equations with respect to x.
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= )0¥ lu) t\in
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MATH ISS/GRACEY
3.
(10 POINTS) Sketch a slope field for the differential
equation F(x,y)
=y
- x
,
for the points (-2,4), (-1,4), (0.4), (1,4),and (24)
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4.
(10 POINTS) Geologists have determined that Crater Lake in Oregon was
formed by a volcanic eruption. Chemical analysis of a wood chip that is
assumed to be from a tree that died during the eruption has shown that it
contains approximately 25'10 of its original carbon-14. Determine how long ago
the volcanic eruption occurred. Use 5715 years as the half-life of carbon-14.
Round to the nearest year.
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MATH ISS/GRACEY
(10 POINTS) When an object is removed from a furnace and placed in an
environment with a constant temperature of 80°F, its core temperature is
1500°F. One hour after it is removed, the core temperature is 1120°F. Find
the core temperature 5 hours after the object is removed from the furnace.
(HINT: Newton's Law of Cooling)
5.
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f
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= 2y
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0--1
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(10 POINTS) Find the equation of the graph that passes through the point
(8,2) and has slope y'
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MATH ISS/GRACEY
7.
(12 POINTS) Find the particular solution that satisfies the initial condition.
a.
b.
du
.
-=uvsmv
2
dv
u(O}=l
du f
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ll-
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e.
1.
-~tpS'
\A.. =
t((
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a.
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6dx
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2
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c.
,,( 1 .
J.
l\
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s: (A('CS I() ~
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,
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b.
\
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-)
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)
x"l-+l
dX.
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-x
x
(X
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=f x_~
y
t
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0ita
r"l)(
-i
CJ
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...;l-x
t.;~
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:=
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e,:
l/"Z-- -l.co<\
lr\ =e,
CO S G 1...
- Vz.. -----'
(24 POINTS) Integrate.
8.
I
t c.
Vz..
e-
1...::.c"- s \_
-z..
...

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