MATH 110

EXAM 3 Review

Jeopardy

Oh RatsShow me the

MoneyThe 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:

• Answer: x ≠ 7, -2

145

22

xx

xxf

Oh Rats 200

• Find all asymptotes and intercepts to the function:

• 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)

2157

452

2

xx

xxf

Oh Rats 300

• Find the equation of the slant asymptote for the function

• Answer: y = 7x + 29

2

2157 2

x

xxxf

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.

• Answer: 32

4

xx

xxf

Oh Rats 500• Find the equation of the rational function shown:

• Answer:

x

y

31

321

xx

xxy

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:

• t is in years

• What is the half-life of plutonium-240?

• Answer: 6301 years

teQQ 00011.00

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 modeled by the exponential function , what is the rate of decay?

• Answer: r =-0.09163 or the substance decays at a continuous rate of 9.163% annually.

rteQQ 0

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 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

ktePP 0

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 , how much of the substance would remain after 1000 years?

• Answer: 652 grams

rteQQ 0

Who are those guys? 100

• Which of the following represents a one-to-one function?

(1)

(2) The function f which assigns each U of A student his/her student I.D. number

(3)

23 2 xxf

x

y

Who are those guys? 200

• Find the inverse for the function below:

• Answer:

54 3 xy

35log4 x

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.

• Answer: The inverse tells you the year you were born given your life expectancy.

101.0

94.66

x

xxL

101.0

94.661

x

xxL

Who are those guys? 400• The graph below shows the equation of a

logarithmic function. Find its equation.

• Answer: 4log3 xy

x

y

Who are those guys? 500

• Find the inverse of the function

• Answer:

32ln2 xxf

2

3

2

1 21 xexf

Famous Log Cabins 100

• Simplify each expression:

x

BA

A

ee

e

e

4log

ln

2

1ln

21

2

2

ln3ln2

5

x

BAee

Ae

e

x

BA

A

4

12

32ln3ln2

55

4log

2ln

21ln

21

2

21

Famous Log Cabins 200

• Expand the following logarithm completely:

25

10log

ab

baab 55525 log2log10log10

log

Famous Log Cabins 300

• Write the following as a single logarithm:

8lnlnln23

1 yx

2ln8lnlnln2

3

1 31

32

yxyx

Famous Log Cabins 400

• Which of the following is/are true?

cMc

M

ee

eeeee

e

1

lnln

12lnln2ln2

ln

2lnlnln2lnln2lnln

2ln

Famous Log Cabins 500

• Solve for x:

• Answer: x = -1

2log3log 22 xx

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?

x

x

x

y

y

y

8.11000

1.11000

5.11000

Top

Middle

Bottom1st Quadrant

Bottom

Top

Middle2nd 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

to show how much pollutant would be left from an initial quantity of 100 grams.

• Answer:

xaQQ 0

xQ 80.0100

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 carbon-14. If the half-life of carbon 14 is 5730 years, determine whether or not this picture is fake.

• Answer: It’s a fake!