CAS MA 124 Study Guide - Midterm Guide: Ion, Horse Length, Partial Fraction Decomposition
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10 Oct 2018
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MA 124 - Fall ’12 - Exam 1A - Answers
GENERAL INSTRUCTIONS: Show all your work. A correct answer without justifi-
cation receives no credit. Check your algebra and arithmetic carefully.
1) (16 points) When given as a medication, iodine has a half-life in the bloodstream
of 8 days.
a) A patient is given an initial dose of 80 millicuries. Find the function that gives the
amount in the body after tdays.
b) How long does it take for the amount of iodine to reach 20% of the original dose ?
ANSWER: a)
A(t)=A0e−kt = 80e−kt
Need k
Using the formula
k=ln 2
t1/2
=ln 2
8
which can be obtained by
A(8) = 40 = 80e−k8⇒1
2=e−k8
ln 1
2=lne−k8=−k8⇒k=−ln 1
2
8=ln 2
8
So
A(t) = 80e−
ln 2
8t
b) Solve
.2(80) = 16 = 80e−
ln 2
8t
I.e.
.2=e−
ln 2
8t
ln(.2) = ln e−
ln 2
8t=−ln 2
8t
So
t=−8ln(.2)
ln 2 =−8 ln(1/5)
ln 2 =8ln5
ln 2
2) a) (8 points) Find
�e
1
xln xdx
ANSWER a) Use integration by parts with
u=lnxdv=xdx
du =1
xdx v =�xdx=1
2x2
�udv=uv −�vdu
gives
�xln xdx=1
2x2ln x−1
2�xdx
=1
2x2ln x−1
2·1
2x2
=1
2x2ln x−1
4x2
So �e
1
xln xdx=1
2e2ln e−1
4e2−(1
2ln 1 −1
4)
=1
2e2−1
4e2−(−1
4)
=1
4e2+1
4
=1
4(e2+ 1)
