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MATH 110
EXAM 3 Review
Jeopardy
Oh Rats
Show me the
Money
The Big “e”
Who are
those guys?
Famous Log
Cabins
Potpourri
100
100
100
100
100
100
200
200
200
200
200
200
300
300
300
300
300
300
400
400
400
400
400
400
500
500
500
500
500
500
Oh Rats 100
• Find the domain for the function:
x2
f x 2
x 5 x 14
• Answer: x ≠ 7, -2
Oh Rats 200
• Find all asymptotes and intercepts to the
function:
5 4x
f x 2
7 x 15x 2
2
• Answer: V.a. at x = -2, (-1/7). H.a. y = (4/7). X-int at x = ± √(5/4). Y-int at (0,2.5)
Oh Rats 300
• Find the equation of the slant asymptote
for the function
7 x 15x 2
f x
x2
2
• Answer: y = 7x + 29
Oh Rats 400
• Find the equation for a rational function
with: (1) vertical asymptotes of x = 2, -3
(2) x – intercept (4,0) and a (3) horizontal
asymptote of y = 0.
x4
• Answer: f x
x 2x 3
•
Oh Rats 500
Find the equation of the rational function shown:
y
x
•
Answer:
x 12 x 3
x 1x 3
y
Show Me the Money 100
• How much money should you invest at 6%
compounded quarterly so that you have
$20,000 after 9 years?
• Answer:$11,701.79
Show Me the Money 200
• Suppose $10,000 was invested in a trust
fund for a child. After 20 years, the fund
has matured to a value of $27,126.40.
What is the interest rate of the fund if
interest has been compounded
continuously?
• Answer: 4.99%
Show Me the Money 300
• Suppose you wish to invest $6000 at 7.8%
interest for 3 years. What is the difference in the
value of your investment after 3 years if you
invest it compounded continuously as opposed
to compounded weekly?
• Answer: The continuous compounding is $1.33
higher than the weekly compounding.
Show Me the Money 400
• How long does it take for a $10,000 US
Treasury Bond earning interest at 2.4%
compounded monthly to mature to
$15,000?
• Answer: Around 17 years
Show Me the Money 500
• How long does it take for an investment of
$5000 to double if it is invested at a rate of
7.5% compounded quarterly?
• Answer: Around 9 years
The Big “e” 100
• Plutonium-240 is a radioactive material
whose decay is modeled according to the
exponential function:
Q Q0e
0.00011t
• t is in years
• What is the half-life of plutonium-240?
• Answer: 6301 years
The Big “e” 200
• One hundred kilograms of a certain radioactive
substance decays to 40 kilograms after 10
years. If the amount of the substance is
rt
Q
Q
e
modeled by the exponential function
,
0
what is the rate of decay?
• Answer: r =-0.09163 or the substance decays at
a continuous rate of 9.163% annually.
The Big “e” 300
• The air in a factory is being filtered so that
the quantity of a pollutant P (mg/liter) is
decreasing according to the exponential
function P P0e kt where t is in hours. If
10% of the pollution is removed in the first
5 hours, how long until the pollution is 50%
of its original value?
• Answer: 33 hours
The Big “e” 400
• When a murder is committed, the body,
originally 37 degrees C, cools to 35
degrees C after two hours. Suppose that
the police find the body at 4 pm and find
that the body temperature is 30 degrees
C. If the temperature of the surrounding
air is a constant 22 degrees C, determine
when the murder was committed?
• Answer: 7:13am
The Big “e” 500
• Radium has a half-life of 1,620 years. If
1000 grams are initially present and the
amount of Radium can be modeled by the
exponential function Q Q0e rt, how much
of the substance would remain after 1000
years?
• Answer: 652 grams
Who are those guys? 100
•
Which of the following represents a oneto-one function?
(1) f x 3x 2
(2) The function f which assigns each U of A
student his/her student I.D. number
(3)
2
y
x
Who are those guys? 200
• Find the inverse for the function below:
y 4 x 3 5
• Answer:
log 4 x 5 3
Who are those guys? 300
• Suppose the function L(x) gives a person’s
life expectancy when they are born x years
after 1950. Find a formula for the inverse
and explain what this function represents.
x 66.94
L x
0.01x 1
• Answer: The inverse tells you the year
you were born given your life expectancy.
66.94 x
L x
0.01x 1
1
Who are those guys? 400
• The graph below shows the equation of a
logarithmic function. Find its equation.
y
• Answer: y log 3 x 4
x
Who are those guys? 500
• Find the inverse of the function
f x 2 ln 2 x 3
• Answer: f 1 x 1 e 2 x 3
2
2
Famous Log Cabins 100
• Simplify each expression:
e
1
ln
2
e
2
log1 4 x
2
1
2
5e 5 A2
5eln A
2 ln e 3 ln e
12
ln A2
2
A
ln
B
2 ln e A 3 ln e B 2 A 3B
log1 4 x
1
2
2
4x
Famous Log Cabins 200
• Expand the following logarithm completely:
10
log 5 2
ab
10
log 5 2 log 5 10 log 5 a 2 log 5 b
ab
Famous Log Cabins 300
• Write the following as a single logarithm:
1
2 ln x ln y ln 8
3
x3 y3
1
2 ln x ln y ln 8 ln
3
2
2
1
Famous Log Cabins 400
• Which of the following is/are true?
ln 2e
ln 2e ln e ln 2 ln e ln e ln 2
ln e
2
ln ln 2 ln e ln 2 1
e
1c
ln M
ln M
c
Famous Log Cabins 500
• Solve for x: log 2 3 x log 2 x 2
• Answer: x = -1
Potpourri 100
• A population of lemmings declined from
100,000 to 75,000 in 3 months. Assuming
an exponential decline, find a model for
the lemming population as a function time.
What is the decay rate each month?
• Answer: 9.1% decline each year
Potpourri 200
• Without calculating any values or
graphing, which of the following
exponential functions should be on top, in
the middle, and on the bottom in the first
quadrant? What about for the second
quadrant?
1st
Quadrant
Middle
y 10001.5
Bottom
y 10001.1
Top
y 10001.8
x
x
Middle
Top
x
Bottom
2nd
Quadrant
Potpourri 300
• Before kerosene can be used as jet fuel, federal
regulations require that the pollutants in it be
removed by passing the kerosene through clay.
Suppose that the clay is in a pipe and that each
foot of pipe removes 20% of the pollutants that
enter it. Write an exponential model of the form
x
Q Q0a to show how much pollutant would be
left from an initial quantity of 100 grams.
• Answer:
Q 1000.80
x
Potpourri 400
• An almanac lists the world’s population in
1980 as 4.478 billion people and in 1994
as 5.642 billion people. What is the
doubling time of the world’s population?
• Answer: 42 years
Potpourri 500
• A picture supposedly painted by Vermeer
(1632-1675) contains 99.5% of its carbon14. If the half-life of carbon 14 is 5730
years, determine whether or not this
picture is fake.
• Answer: It’s a fake!