Deck 2: Nonlinear Functions and Models

Question 1

Find the logistic function f with the given properties. f has limiting value 18 and passes through (0, 9) and (1, 16).

A) $f ( x ) = \frac { 1 } { 1 + 2 \left( 8 ^ { - x } \right) }$
B) $f ( x ) = \frac { 18 } { 1 + 2 \left( 16 ^ { - x } \right) }$
C) $f ( x ) = \frac { 18 } { 1 + \left( 8 ^ { - x } \right) }$
D) $f ( x ) = \frac { 9 } { 1 + \left( 8 ^ { - x } \right) }$
E) $f ( x ) = \frac { 18 } { 1 + \left( 16 ^ { - x } \right) }$

Answer 1

$f ( x ) = \frac { 18 } { 1 + \left( 8 ^ { - x } \right) }$

Question 2

The chart shows the number of research articles in the prominent journal Physics Review that were written by researchers in Europe during 1983 - 2003 ( $$t = 0$$ represents 1983).
Which of the following logistic functions best models the data (t is the number of years since 1983.) Try to determine the correct model without actually computing data points.

A) $A ( t ) = \frac { 5.1 } { 1 + 4.5 ( 1.2 ) ^ { - t } }$
B) $A ( t ) = \frac { 4.5 } { 1 + 4.2 ( 1.2 ) ^ { - t } }$
C) $A ( t ) = \frac { 7.0 } { 1 + 5.4 ( 1.2 ) ^ { - t } }$
D) $A ( t ) = \frac { 7.0 } { 1 + 5.4 ( 0.7 ) ^ { - t } }$
E) $A ( t ) = \frac { 4.5 } { 1 + 4.2 ( 0.7 ) ^ { - t } }$

Answer 2

$A ( t ) = \frac { 7.0 } { 1 + 5.4 ( 1.2 ) ^ { - t } }$

Question 3

There are currently 1,000 cases of Venusian flu in a total susceptible population of 10,000 and the number of cases is increasing by 25% each day. Find a logistic model for the number of cases of Venusian flu and use your model to predict the number of flu cases a week from now. Round your answer to the nearest integer.

A) $F ( 7 ) = 1,732$ cases
B) $F ( 7 ) = 5,195$ cases
C) $F ( 7 ) = 3,463$ cases
D) $F ( 7 ) = 5,888$ cases
E) $F ( 7 ) = 1,154$ cases

Answer 3

$F ( 7 ) = 3,463$ cases

Question 4

Find N, A, and b for the function. $$f ( x ) = \frac { 11 } { 1 + 2 \left( 0.2 ^ { - x } \right) }$$

A) $N = 2 , A = 0.2 , b = 11$
B) $N = 2 , A = 11 , b = 0.2$
C) $N = 11 , A = 2 , b = 0.2$
D) $N = 11 , A = 0.2 , b = 2$
E) $N = 0.2 , A = 2 , b = 11$

Question 5

Choose the logistic function that best approximates the curve.

A) $f ( x ) = \frac { 8 } { 1 + 7 ( 0.5 ) ^ { - x } }$
B) $f ( x ) = \frac { 8 } { 1 + 3 ( 0.5 ) ^ { - x } }$
C) $f ( x ) = \frac { 8 } { 1 + 3 ( 3 ) ^ { - x } }$

Question 6

The chart shows the number of research articles in the prominent journal Physics Review that were written by researchers in Europe during 1983 - 2003 ( $t = 0$ represents 1983). $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Year, } \boldsymbol { t } & 0 & 5 & 10 & 15 & 20 \\\hline \text { Research Articles, } \boldsymbol { A } ( \mathbf { 1 } , \mathbf { 0 0 0 } ) & 1.2 & 2.1 & 3.8 & 5.1 & 5.7 \\\hline\end{array}$$ Determine the logistic regression model for the data (Round all coefficients to two significant digits.) According to the model, how many Physics Review articles were published by U.S. researchers in 2001 ( $$t = 18$$ )

A)5,300
B) 5,000
C) 6,100
D) 5,800
E) 5,700

Question 7

The graph shows the actual percentage of U.S. households with a computer as a function of household income (the data points) and a logistic model of these data (the curve). The logistic model is

$P ( x ) = \frac { 90 } { 1 + 5.45 ( 1.05 ) ^ { - x } }$
where x is the household income in thousands of dollars. According to the model, what percentage of extremely wealthy households had computers

P = __________%

Question 8

In Russia the average consumer drank two servings of Coca-Cola in 1993. This amount appeared to be increasing exponentially with a doubling time of 2 years. Given a long-range market saturation estimate of 100 servings per year, find a logistic model for the consumption of Coca-Cola in Russia and use your model to predict when, to the nearest year, the average consumption will be 43 servings per year. ​

A)Sometime in 2005.
B) Sometime in 1993.
C) Sometime in 2003.
D) Sometime in 2004.
E) Sometime in 1994.

Question 9

Find the logistic function f with the given properties. f has limiting value 12 and passes through (0, 3) and (1, 10).

A) $f ( x ) = \frac { 12 } { 1 + 4 \left( 30 ^ { - x } \right) }$
B) $f ( x ) = \frac { 12 } { 1 + 3 \left( 15 ^ { - x } \right) }$
C) $f ( x ) = \frac { 3 } { 1 + 3 \left( 30 ^ { - x } \right) }$
D) $f ( x ) = \frac { 3 } { 1 + 3 \left( 15 ^ { - x } \right) }$
E) $f ( x ) = \frac { 12 } { 1 + 3 \left( 30 ^ { - x } \right) }$

Question 10

Find the logistic function f with the given properties. $f ( 0 ) = 1$ , f has limiting value 20, and for small values of x, f is approximately exponential and grows by 50% with every increase of 1 in x.

A) $f ( x ) = \frac { 1 } { 1 + 1.5 ^ { - x } }$
B) $f ( x ) = \frac { 20 } { 1 + 19 \left( 1.5 ^ { - x } \right) }$
C) $f ( x ) = \frac { 20 } { 1 + 2 ^ { - x } }$
D) $f ( x ) = \frac { 20 } { 1 + 19 \left( 2 ^ { - x } \right) }$
E) $f ( x ) = \frac { 20 } { 1 + 1.5 ^ { x } }$

Question 11

Last year's epidemic of Martian flu began with a single case in a total susceptible population of 10,000. The number of cases was increasing initially by 38% per day. Find a logistic model for the number of cases of Martian flu and use your model to predict the number of flu cases 2 weeks into the epidemic. Round your answer to the nearest integer.

A) $P ( 14 ) = 153$ cases
B) $P ( 14 ) = 45$ cases
C) $P ( 14 ) = 90$ cases
D) $P ( 14 ) = 30$ cases
E) $P ( 14 ) = 135$ cases

Question 12

Find N, A, and b for the function given. $$f ( x ) = \frac { 2 } { 0.5 + 2.5 \left( 1.5 ^ { - x } \right) }$$

A) $N = 5 , A = 1.5 , b = 4$
B) $N = 5 , A = 4 , b = 1.5$
C) $N = 1.5 , A = 5 , b = 4$
D) $N = 4 , A = 5 , b = 1.5$
E) $N = 4 , A = 1.5 , b = 5$

Question 13

The following graph shows the actual percentage of U.S. households with a computer as a function of household income (the data points) and a logistic model of these data (the curve). The logistic model is
$P ( x ) = \frac { 93 } { 1 + 5.35 ( 1.05 ) ^ { - x } }$
Where x is the household income in thousands of dollars. For low incomes, the logistic model is approximately exponential. Which exponential model best approximates P(x) for small x Round the coefficients to the nearest hundredth.

A) $P ( x ) = 17.65 ( 2.1 ) ^ { x }$
B) $P ( x ) = 14.65 ( 1.05 ) ^ { - x }$
C) $P ( x ) = 17.65 ( 1.05 ) ^ { x }$
D) $P ( x ) = 14.65 ( 1.05 ) ^ { x }$
E) $P ( x ) = 14.65 ( 2.1 ) ^ { - x }$

Question 14

Choose the logistic function that best approximates the given curve.

A) $f ( x ) = \frac { 10 } { 1 + 9 \left( 5 ^ { - x } \right) }$
B) $f ( x ) = \frac { 10 } { 1 + 4 \left( 5 ^ { - x } \right) }$
C) $f ( x ) = \frac { 10 } { 1 + 4 \left( 0.5 ^ { - x } \right) }$
D) $f ( x ) = \frac { 10 } { 1 + 6 \left( 0.5 ^ { - x } \right) }$
E) $f ( x ) = \frac { 10 } { 1 + 9 \left( 0.5 ^ { - x } \right) }$

Question 15

Choose the logistic function that best approximates the curve.

A) $f ( x ) = \frac { 10 } { 1 + 1.5 \left( 3 ^ { - x } \right) }$
B) $f ( x ) = \frac { 9 } { 1 + 3.5 \left( 2 ^ { - x } \right) }$
C) $f ( x ) = \frac { 10 } { 1 + 1.5 \left( 1.05 ^ { - x } \right) }$

Question 16

Choose the logistic function that best approximates the curve.

A) $f ( x ) = \frac { 18 } { 1 + 2 ( 1.1 ) ^ { - x } }$
B) $f ( x ) = \frac { 18 } { 2 + 2 ( 4 ) ^ { - x } }$
C) $f ( x ) = \frac { 18 } { 2 + 7 ( 1.1 ) ^ { - x } }$
D) $f ( x ) = \frac { 18 } { 1 + 7 ( 1.1 ) ^ { - x } }$
E) $f ( x ) = \frac { 18 } { 2 + 7 ( 4 ) ^ { - x } }$

Question 17

Use technology to find a logistic regression curve $y = \frac { N } { 1 + A b ^ { - x } }$ approximating the given data. (Round b to three significant digits and A and N to two significant digits.) $$\begin{array} { | l | l | l | l | l | l | l | } \hline x & 0 & 20 & 40 & 60 & 80 & 100 \\\hline y & 2.2 & 3.8 & 5.0 & 6.1 & 6.8 & 6.9 \\\hline\end{array}$$

A) $y = \frac { 7.2 } { 1 + 2.2 \left( 1.04 ^ { - x } \right) }$
B) $y = \frac { 7.2 } { 4.4 \left( 1.04 ^ { - x } \right) }$
C) $y = \frac { 5.2 } { 1 + 2.2 \left( 1.04 ^ { - x } \right) }$
D) $y = \frac { 5.2 } { 1 + 4.4 \left( 2.08 ^ { - x } \right) }$
E) $y = \frac { 7.2 } { 1 + 2.2 \left( 2.08 ^ { - x } \right) }$

Question 18

The graph shows the actual percentage of U.S. households with a computer as a function of household income (the data points) and a logistic model of these data (the curve). The logistic model is $P ( x ) = \frac { 93 } { 1 + 5.35 ( 1.05 ) ^ { - x } }$
Where x is the household income in thousands of dollars. According to the model, what percentage of extremely wealthy households had computers

A) $P ( x )$ is close to $N = 94 \%$ .
B) $P ( x )$ is close to $N = 100 \%$ .
C) $P ( x )$ is close to $N = 92 \%$ .
D) $P ( x )$ is close to $N = 88 \%$ .
E) $P ( x )$ is close to $N = 93 \%$ .

Question 19

Find the logistic function f with the given properties. $f ( 0 ) = 20$ , f has limiting value 500, and for small values of x, f is approximately exponential and doubles with every increase of 1 in x.

A) $f ( x ) = \frac { 500 } { 1 + 24 \left( 2 ^ { - x } \right) }$
B) $f ( x ) = \frac { 500 } { 1 + 24 \left( 1.5 ^ { - x } \right) }$
C) $f ( x ) = \frac { 500 } { 1 + 1.5 ^ { - x } }$
D) $f ( x ) = \frac { 1 } { 1 + 24 \left( 2 ^ { - x } \right) }$
E) $f ( x ) = \frac { 500 } { 1 + 2 ^ { - x } }$

Question 20

Choose the logistic function that best approximates the given curve.

A) $f ( x ) = \frac { 10 } { 1 + 0.5 \left( 1.01 ^ { - x } \right) }$
B) $f ( x ) = \frac { 10 } { 1 + 4 \left( 2 ^ { - x } \right) }$
C) $f ( x ) = \frac { 6 } { 1 + 4 \left( 3 ^ { - x } \right) }$
D) $f ( x ) = \frac { 6 } { 1 + 0.5 \left( 2 ^ { - x } \right) }$
E) $f ( x ) = \frac { 6 } { 1 + 0.5 \left( 3 ^ { - x } \right) }$

Question 21

Find the associated doubling time. $Q = 1,000 e$ ^0.5t

A) $\frac { 2 } { \ln 2 }$
B) $0.5 \ln 2$
C) $\ln 2$
D) $\frac { 0.5 } { \ln 2 }$
E) $2 \ln 2$

Question 22

Choose the logistic function that best approximates the curve.

A) $f ( x ) = \frac { 16 } { 2 + 6 ( 4 ) ^ { - x } }$
B) $f ( x ) = \frac { 16 } { 2 + 5 ( 1.1 ) ^ { - x } }$
C) $f ( x ) = \frac { 16 } { 2 + 6 ( 1.1 ) ^ { - x } }$

Question 23

Soon after taking an aspirin, a patient has absorbed 350 mg of the drug. If the amount of aspirin in the bloodstream decays exponentially, with half being removed every 2 hours, find the time it will take for the amount of aspirin in the bloodstream to decrease to 260 mg. ​
Select the answer rounded to three decimal places.
​

A)1.715 hours
B) 0.858 hours
C) 0.429 hours
D) 2.573 hours
E) 2.144 hours

Question 24

How long, to the nearest year, will it take an investment in U.S. to double its value if the interest is compounded every six months Please round the answer to the nearest year. $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Country } & \text { U.S. } & \text { Japan } & \text { Canada } & \text { Korea } & \text { Australia } \\\hline \text { Yield } & 5.3 \% & 1.5 \% & 5.2 \% & 5.4 \% & 6.0 \% \\\hline\end{array}$$

A)14 years
B) 16 years
C) 13 years
D) 8 years
E) 15 years

Question 25

You are trying to determine the half-life of a new radioactive element you have isolated. You start with 2 grams, and 4 days later you determine that it has decayed down to 0.1 gram. What is the half-life Round your answer to three decimal places. ?
Select the answer rounded to three decimal places.
?

A)1.388 days
B) 1.851 days
C) 2.777 days
D) 0.463 days
E) 0.926 days

Question 26

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.

$f ( t ) = 23.2 ( 0.997 ) ^ { t }$ ; $f ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }$

A) $Q _ { 0 } = 23.2$ , $k = 0.004754$
B) $Q _ { 0 } = 23.2$ , $k = 0.003005$
C) $Q _ { 0 } = 23.2$ , $k = 0.001256$
D) $Q _ { 0 } = 23.1$ , $k = 0.008252$
E) $Q _ { 0 } = 23.1$ , $k = 0.006503$

Question 27

The chart shows the number of research articles in the prominent journal Physics Review that were written by researchers in Europe during 1983 - 2003 ( $$t = 0$$ represents 1983).
Which of the following logistic functions best models the data ( t is the number of years since 1983.) Try to determine the correct model without actually computing data points.

A) $A ( t ) = \frac { 7.0 } { 1 + 5.4 ( 1.2 ) ^ { - t } }$
B) $A ( t ) = \frac { 4.9 } { 1 + 4.3 ( 1.2 ) ^ { - t } }$
C) $A ( t ) = \frac { 4.9 } { 1 + 4.3 ( 0.7 ) ^ { - t } }$
D) $A ( t ) = \frac { 7.0 } { 1 + 5.4 ( 0.7 ) ^ { - t } }$

According to the model you selected, at what percentage was the number of articles growing around 1984

__________%

Question 28

Find the associated exponential decay model. $Q = 7,000$ when $t = 0$ ; Half-life = 9

A) $Q = 7,000 - e ^ { - \frac { t } { 9 } }$
B) $Q = 7,000 e ^ { - \frac { t ( \ln 2 ) } { 9 } }$
C) $Q = 7,000 + e ^ { \frac { t ( \ln 2 ) } { 9 } }$
D) $Q = 7,000 e ^ { - \frac { t } { 9 } }$
E) $Q = 7,000 e ^ { \frac { t } { 9 } }$

Question 29

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.
$f ( x ) = 2.7 e ^ { - 0.7 x }$ ; $f ( x ) = A b ^ { x }$

A) $A = 3.7$ , $b = 0.62$
B) $A = 2.7$ , $b = 0.4966$
C) $A = 3.7$ , $b = 0.4946$
D) $A = 4.7$ , $b = 0.3621$
E) $A = 2.7$ , $b = 0.4969$

Question 30

How long will it take an investment to triple if it is continuously compounded at 15% per year ?
Select the correct answer rounded to the nearest year.

A)15 years
B) 13 years
C) 8 years
D) 4 years
E) 7 years

Question 31

Use logarithms to solve the equation. (Round the answer to four decimal places.)
​ ​
x = __________

Question 32

Graph the function. $f ( x ) = \log _ { \frac { 1 } { 6 } } x$

A)
B)
C)
D)
E)

Question 33

Use logarithms to solve the equation. Round your answer to four decimal places. $8 \left( 2.5 ^ { 2 x - 1 } \right) = 10$

A) $x = 1.2436$
B) $x = 0.6218$
C) $x = 0.25$
D) $x = - 0.6218$
E) $x = - 1.2436$

Question 34

Plutonium-239 is used as a fuel for some nuclear reactors and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long will it take 12 grams of plutonium-239 to decay to 2 grams ?
Round your answer to the nearest hundreds.

A)31,600 years
B) 15,800 years
C) 126,200 years
D) 63,100 years
E) 63,200 years

Question 35

Graph the function. $f ( x ) = \log _ { 5 } x$

A)
B)
C)
D)
E)

Question 36

The chart shows the number of research articles in the prominent journal Physics Review that were written by researchers in Europe during 1983 - 2003 ( t=0 represents 1983)
$$\begin{array} { | l | l | l | l | l | l | } \hline \text { Year, } \boldsymbol { t } & 0 & 5 & 10 & 15 & 20 \\\hline \text { Research Articles, } \boldsymbol { A } ( \mathbf { 1 } , \mathbf { 0 0 0 } ) & 1.2 & 2.1 & 3.8 & 5.1 & 5.7 \\\hline\end{array}$$
a. What is the logistic regression model for the data (Round all coefficients to two significant digits.)
$A(t)=\frac{\_\_\_\_}{1+\_\_\_\_ ^. (\_\_\_\_)^{-t}}$
b. At what value does the model predict that the number of research articles will level off

__________ articles
b.c. According to the model, how many Physics Review articles were published by U.S. researchers in 1990 ( t=7)

Question 37

Use logarithms to solve the equation. Round your answer to four decimal places. $6 ^ { - 2 x } = 40$

A) $x = - 0.8333$
B) $x = - 1.8493$
C) $x = 0.8333$
D) $x = 1.0294$
E) $x = - 1.0294$

Question 38

The following graph shows the actual percentage of U.S. households with a computer as a function of household income (the data points) and a logistic model of these data (the curve). The logistic model is
$P ( x ) = \frac { 91 } { 1 + 5.25 ( 1.05 ) ^ { - x } }$
where x is the household income in thousands of dollars. For low incomes, the logistic model is approximately exponential. Which exponential model best approximates P(x) for small x Round the coefficients to the nearest hundredth.

P(x) = ________ ·( ________)x

Question 39

The amount of carbon-14 remaining in a sample that weighs B is given by $X ( t ) = B ( 0.999879 ) ^ { t }$
where t is time in years. If tests on a fossilized skull reveal that 99.92% of the carbon-14 has decayed, how old is the skull
Select the correct answer rounded to the nearest integer.

A)58,929 years old
B) 2,629 years old
C) 1,985 years old
D) 0 years old
E) 23,234 years old

Question 40

The table lists interest rates on long-term investments (based on 10-year government bonds) in several countries in 2004-2005. Assuming that you invest $13,000 in Japan, how long (to the nearest year) must you wait before your investment is worth $19,000 if the interest is compounded annually Round your answer to the nearest year. $$\begin{array} { | l | l | l | l | } \hline \text { Country } & \text { U.S. } & \text { Japan } & \text { Canada } \\\hline \text { Yield } & 5.3 \% & 1.5 \% & 5.2 \% \\\hline\end{array}$$

A)26 years
B) 19 years
C) 25 years
D) 22 years
E) 28 years

Question 41

The table below is filled correctly. $$\begin{array} { | l | l | } \hline \text { Exponential form } & \text { Logarithmic form } \\\hline 5 ^ { 4 } = 625 & \log _ { 5 } 625 = 4 \\\hline 0.6 ^ { 2 } = 0.36 & \log _ { 0.6 } 0.36 = 2 \\\hline 7 ^ { 0 } = 1 & \log _ { 7 } 1 = 0 \\\hline 8 ^ { - 1 } = 0.125 & \log _ { 8 } 0.125 = - 1 \\\hline\end{array}$$

Question 42

Model the data using an exponential function $$f ( x ) = A b ^ { x }$$ . $$\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 350 & 175 & 87.5 \\\hline\end{array}$$ Select the correct answer.

A) $f ( x ) = 0.5 ( 0.5 ) ^ { x }$
B) $f ( x ) = 350 ( 350 ) ^ { x }$
C) $f ( x ) = 350 ( 0.5 ) ^ { x }$
D) $f ( x ) = 350 ( 0.5 ) ^ { - x }$
E) $f ( x ) = 0.5 ( 350 ) ^ { - x }$

Question 43

Plutonium-239 is used as a fuel for some nuclear reactors and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long will it take 40 grams of plutonium-239 to decay to 3 grams

Round your answer to the nearest hundreds.

__________ years

Question 44

Model the data using an exponential function $$f ( x ) = A b ^ { x }$$ . $$\begin{array} { | l | l | l | } \hline \boldsymbol { x } & 1 & 2 \\\hline f ( x ) & 13 & 16.9 \\\hline\end{array}$$ Select the correct answer.

A) $f ( x ) = 10 ( 10 ) ^ { x }$
B) $f ( x ) = 1.3 ( 1.3 ) ^ { - x }$
C) $f ( x ) = 10 ( 1.3 ) ^ { x }$
D) $f ( x ) = 10 ( 1.3 ) ^ { - x }$
E) $f ( x ) = 1.3 ( 10 ) ^ { x }$

Question 45

Find the associated exponential decay model.
​ when ; Half-life = 5

Question 46

How long, to the nearest year, will it take an investment in Canada to double its value if the interest is compounded every six months Please round the answer to the nearest year. $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Country } & \text { U.S. } & \text { Japan } & \text { Canada } & \text { Korea } & \text { Australia } \\\hline \text { Yield } & 5.3 \% & 1.5 \% & 5.2 \% & 5.4 \% & 6.0 \% \\\hline\end{array}$$
t = __________ year(s)

Question 47

Rock Solid Bank & Trust is offering a CD that pays 5% compounded continuously. How much interest would a $1,000 deposit earn over 12 years (Round your answer to the nearest dollar.)
Select the correct answer.

A)$2,822
B) $822
C) $1,822
D) $1,820
E) $1,796

Question 48

Which of the following five functions will be largest for large values of x
Select the correct answer.

A) $h ( x ) = x ^ { 10 }$
B) $f ( x ) = 64 x ^ { 8 }$
C) $f ( x ) = x ^ { 8 }$
D) $g ( x ) = 8 ^ { x }$
E) $f ( x ) = 8 x ^ { 8 }$

Question 49

Given the graph of the functions $f _ { 1 } ( x ) = 2.5 ^ { x }$ and $f _ { 2 } ( x ) = 2.8 ^ { x }$ . Identify which graph corresponds to $f _ { 2 } ( x ) = 2.8 ^ { x }$ .
Select the correct answer.

A)Blue
B) Red

Question 50

The U.S. population was 170 million in 1950 and 240 million in 1990. Assuming exponential population growth, what will the population be in the year 2020 Round your answer to the nearest million.
Select the correct answer.

A)250 million
B) 243 million
C) 486 million
D) 972 million
E) 311 million

Question 51

The table lists interest rates on long-term investments (based on 10-year government bonds) in several countries in 2004-2005. Assuming that you invest $12,000 in Japan, how long (to the nearest year) must you wait before your investment is worth $18,000 if the interest is compounded annually $$\begin{array} { | l | l | l | l | } \hline \text { Country } & \text { U.S. } & \text { Japan } & \text { Canada } \\\hline \text { Yield } & 5.3 \% & 1.5 \% & 5.2 \% \\\hline\end{array}$$
__________ year(s)

Question 52

Soon after taking an aspirin, a patient has absorbed 310 mg of the drug. If the amount of aspirin in the bloodstream decays exponentially with half being removed every 2 hours, find the amount of aspirin in the bloodstream after 9 hours. ​
Select the correct answer.
​

A)1,370
B) 137
C) 50
D) 13.7
E) 2.21

Question 53

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.
​ ; ​ ​

Question 54

Graph the function. $$f ( x ) = 4 \left( 2 ^ { - x } \right)$$
Select the correct answer.

A)
B)
C)
D)

Question 55

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $$\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 40 & 80 & 160 \\\hline\end{array}$$ Select the correct answer.

A) $f ( x ) = 40 ( 2 ) ^ { x }$
B) $f ( x ) = 40 ( 2 ) ^ { - x }$
C) $f ( x ) = 2 ( 40 ) ^ { - x }$
D) $f ( x ) = 2 ( 2 ) ^ { x }$
E) $f ( x ) = 40 ( 40 ) ^ { x }$

Question 56

How long, to the nearest year, will it take an investment to triple if it is continuously compounded at 15% per year

Round the answer to the nearest year.

__________ years

Question 57

The table below is filled correctly. $$\begin{array} { | l | l | } \hline \text { Exponential form } & \text { Logarithmic form } \\\hline 5 ^ { 1 } = 5 & \log _ { 5 } 5 = 1 \\\hline 7 ^ { 4 } = 2,401 & \log _ { 7 } 2,401 = 4 \\\hline 9 ^ { - 1 } = \frac { 1 } { 9 } & \log _ { 9 } \frac { 1 } { 9 } = - 1 \\\hline 10 ^ { 3 } = 1,000 & \log _ { 10 } 1,000 = 3 \\\hline\end{array}$$

Question 58

Find an equation for an exponential function that passes through the pair of points $( 4,3 )$ and $( 7,1 )$ . $y = A b ^ { x } ( b > 0 )$

A) $A = 12.9805$ , $b = 0.693$
B) $A = 12.9799$ , $b = 0.6935$
C) $A = 12.9792$ , $b = 0.6924$
D) $A = 13.0162$ , $b = 0.5584$
E) $A = 12.9802$ , $b = 0.6934$

Question 59

Given the graph of the functions $f _ { 1 } ( x ) = 2.3 ^ { - x }$ and $f _ { 2 } ( x ) = e ^ { - x }$ . Determine the color of the graph that corresponds to $f _ { 1 } ( x )$ .
Select the correct answer.

A)Blue
B) Red

Question 60

The table is filled correctly. $$\begin{array} { | l | l | } \hline \text { Exponential form } & \text { Logarithmic form } \\\hline 8 ^ { - 3 } = 0.001953 & \log _ { 8 } 0.001953 = 3 \\\hline 2 ^ { 0 } = 1 & \log _ { 2 } 1 = 0 \\\hline 0.4 ^ { 4 } = 0.0256 & \log _ { 0.4 } 0.0256 = 4 \\\hline 0.8 ^ { - 3 } = 1.953125 & \log _ { 0.8 } 1.953125 = 3 \\\hline\end{array}$$

Question 61

Suppose the amount of carbon dioxide (in pounds per 15,000 miles) released by a typical sport utility vehicle (SUV) depends on its fuel efficiency according to the formula
$10 x ^ { 2 } - 580 x + 30,803$
Where x is a fuel efficiency of an SUV in miles per gallon. According to the model, what is the fuel efficiency of an SUV that has the least carbon dioxide pollution

A)14 miles per gallon
B) 19 miles per gallon
C) 18 miles per gallon
D) 32 miles per gallon
E) 29 miles per gallon

Question 62

The following chart shows the value of trade between two countries for the period 1994 - 2004 ( $t = 0$ represents 1994).
Which of the following models best approximates the data given (Try to answer this without actually computing values.)

A) $f ( t ) = 2 t ^ { 2 } - 4 t - 45$
B) $f ( t ) = - 2 t ^ { 2 } - 4 t - 35$
C) $f ( t ) = - 2 t ^ { 2 } + 4 t - 45$
D) $f ( t ) = - 2 t ^ { 2 } - 4 t + 35$
E) $f ( t ) = 2 t ^ { 2 } - 4 t + 35$

Question 63

Find the vertex of the graph of the quadratic function. $- x ^ { 2 } + 12 x - 36$

A) $( 6,0 )$
B) $( - 6,6 )$
C) $( - 6,0 )$
D) $( 0,6 )$
E) $( - 6 , - 6 )$

Question 64

Model the data using an exponential function $$f ( x ) = A b ^ { x }$$ . $$\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline f ( x ) & 300 & 75 & 18.75 \\\hline\end{array}$$

Question 65

For the following demand equation, find the largest possible revenue. $q = - 4 p + 6,400$

A)3,200
B) 5,120,000
C) 7,680,000
D) 800
E) 2,560,000

Question 66

The given table corresponds to the function $$f ( x ) = 0.042 \left( 4.2 - \frac { 5 } { 1.7 } \right) ^ { - x }$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.084 & 0.067 & 0.067 & 1 & 0.033 & 0.027 & 0.021 \\\hline\end{array}$$

Question 67

The U.S. population was 170 million in 1970 and 250 million in 1995. Assuming exponential population growth, what will the population be in the year 2025

Round your answer to the nearest million.

__________ million

Question 68

The fuel efficiency (in miles per gallon) of a sport utility vehicle (SUV) depends on its weight according to the formula
$E = 0.00005 x ^ { 2 } - 0.3 x + 41$
Where x is the weight of an SUV in pounds. According to the model, what is the weight of the least fuel-efficient SUV

A)3,000 pounds
B) 3,100 pounds
C) 2,000 pounds
D) 6,000 pounds
E) 491 pounds

Question 69

Find the y-intercept(s) of the graph of the quadratic function. $3 x ^ { 2 } - 30 x + 75$

A) $( 0 , - 75 )$
B) $( 5,0 ) , ( 5,0 )$
C) $( 0,5 ) , ( 0,5 )$
D) $( 75,0 )$
E) $( 0,75 )$

Question 70

Which of the following five functions will be smallest for large values of x
Select the correct answer.

A) $f ( x ) = x ^ { - 36 }$
B) $f ( x ) = x ^ { - 216 }$
C) $g ( x ) = 6 ^ { - x }$
D) $f ( x ) = x ^ { - 6 }$
E) $h ( x ) = x ^ { - 100 }$

Question 71

The given table corresponds to the function $$f ( x ) = e ^ { \frac { x } { 5 } }$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.549 & 0.67 & 0.819 & 1 & 1.221 & 1.492 & 1.822 \\\hline\end{array}$$

Question 72

The given table corresponds to the function $f ( x ) = 10 ^ { x }$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 & 1,000 \\\hline\end{array}$$

Question 73

Model the data using an exponential function .

Question 74

Find the equation for the exponential function that passes through the pair of points and .
​ ​
A = __________
​
b = __________
​
Round your answer to four decimal places.

Question 75

The given table corresponds to the function $$f ( x ) = 10 ^ { - x }$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 & 1,000 \\\hline\end{array}$$

Question 76

The given table corresponds to the function $$f ( x ) = 5 ^ { x } - 1$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & - 0.992 & - 0.96 & - 0.8 & 0 & 4 & 24 & 124 \\\hline\end{array}$$

Question 77

Model the data using an exponential function .

Question 78

The given table corresponds to the function $f ( x ) = 10 ^ { x - 1 }$ . $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.0001 & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 \\\hline\end{array}$

Question 79

For the following demand equation, find the largest possible revenue.
$q = - 4 p + 3,600$

A)1,620,000
B) 450
C) 900
D) 810,000
E) 648,000

Question 80

For the following demand equation, express the total revenue R as a function of the price p per item. $q = - 2 p + 1600$

A) $R = - 2 p ^ { 2 } + 1600 p$
B) $R = - 2 p ^ { 2 } + 1600$
C) $R = 1598 p$
D) $R = - 2 + \frac { 1600 } { p }$
E) $R = - 2 p ^ { 2 }$