Question 1

Find the associated exponential decay model.
​ $Q = 6,000$ when $t = 0$ ; Half-life = 5

Accepted Answer

$Q = 6,000 e ^ { \frac { - t \cdot \ln 2 } { 5 } }$

Question 2

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

Accepted Answer

14.56; 1.05

Question 3

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $$\begin{array} { | l | l | l | } 
\hline \boldsymbol { x } & 1 & 2 \
\hline f ( x ) & 170 & 289 \
\hline
\end{array}$$

Accepted Answer

$f ( x ) = 100 \cdot 1.7 ^ { x }$

Question 4

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.

Accepted 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)  $f ( x ) = 40 ( 40 ) ^ { x }$ 
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 5

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.)

Accepted Answer

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)  $f ( t ) = 2 t ^ { 2 } - 4 t + 35$ 
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 6

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

Accepted Answer

The answer of Plutonium-239 is used as a fuel for...

Question 7

The Better Baby Buggy Co. has just come out with a new model, the Turbo. The market research department predicts that the demand equation for Turbos is given by 
$q = - 4 x + 220$ 
where q is the number of buggies it can sell in one month if the price is $x per buggy. At what price should it sell the buggies to get the largest revenue  Round the result to the nearest dollar.

Accepted Answer

A) $14 
B)  $110 
C)  $28 
D)  $128 
E)  $3,025 
A) $14 
B)  $110 
C)  $28 
D)  $128 
E)  $3,025

Question 8

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.

Accepted Answer

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

Question 9

The two fraternities Sigma Alpha Mu and Ep Sig plan to raise money jointly to benefit homeless people on Long Island. They will sell Starship Troopers T-shirts in the Student Center, but they are not sure how much to charge. Sigma Alpha Mu treasurer Solo recalls that they once sold 100 shirts in a week at $4 each, but Ep Sig treasurer Justino claims that, based on past experience, they can sell 400 per week if they charge $2 each. The university administration charges the fraternities $700 per week for use of the Student Center. What is the largest possible weekly profit, rounded to the nearest dollar  ?

Accepted Answer

A) $700 
B)  $117 
C)  $817 
D)  $233 
E)  $1,633 
A) $700 
B)  $117 
C)  $817 
D)  $233 
E)  $1,633

Question 10

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.
?

Accepted Answer

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

Question 11

You have just opened a new nightclub, Russ's Techno Pitstop, but you are unsure how much to charge for the cover charge (entrance fee). One week you charged $9 cover per guest and averaged 372 guests per night. The next week you charged $20 per guest and averaged 240 guests per night. Find the linear demand equation showing the number of guests q per night as a function of the cover charge p.

Accepted Answer

A)  $q = - 12 p + 480$ 
B)   $q = 12 p + 480$ 
C)   $q = - 12 p + 492$ 
D)    $q = - 12 p + 468$ 
E)   $q = 12 p + 492$ 
A)  $q = - 12 p + 480$ 
B)   $q = 12 p + 480$ 
C)   $q = - 12 p + 492$ 
D)    $q = - 12 p + 468$ 
E)   $q = 12 p + 492$

Question 12

Use logarithms to solve the equation. (Round the answer to four decimal places.)
​ $6 \left( 6.5 ^ { 3 x - 2 } \right) = 10$ ​
x = __________

Accepted Answer

The answer of Use logarithms to solve the equation. (Round...

Question 13

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.

Accepted Answer

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)    $f ( x ) = \frac { 500 } { 1 + 2 ^ { - x } }$ 
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 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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 $N = 93 \%$ . 
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 16

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

Accepted Answer

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

Question 17

Pack-Em-In Real Estate is building a new housing development. The more houses it builds, the less people will be willing to pay, due to the crowding and smaller lot sizes. In fact, if the company builds 60 houses in this particular development, it can sell them for $190,000 each, but if it builds 70 houses, it will be able to get only $170,000 each. What is the largest possible revenue the company can get 
 
$ __________

Accepted Answer

The answer of Pack-Em-In Real Estate is building a new...

Question 18

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

Accepted Answer

A) True 
 B)False

Question 19

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.

Accepted Answer

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 
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 20

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

Accepted Answer

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

Find the associated exponential decay model. $Q = 6,000$ when $t = 0$ ; Half-life = 5

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 1 2 f(x) 170 289

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 ( ) 40 80 160 Select the correct answer.

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.)

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

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.

Use logarithms to solve the equation. (Round the answer to four decimal places.) $6 \left( 6.5 ^ { 3 x - 2 } \right) = 10$ x = __________

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.

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

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

The given table corresponds to the function $f ( x ) = 10 ^ { x }$ . -3 -2 -1 0 1 2 3 ( ) 0.001 0.01 0.1 1 10 100 1,000

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

Functions and Applications

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

Functions of Several Variables

Trigonometric Models

Filters

Exam 2: Nonlinear Functions and Models

Find the associated exponential decay model. ​ Q=6,000Q = 6,000Q=6,000 when t=0t = 0t=0 ; Half-life = 5

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . 1 2 f(x) 170 289

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . 0 1 2 ( ) 40 80 160 Select the correct answer.

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

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

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.

Use logarithms to solve the equation. (Round the answer to four decimal places.) ​ 6(6.53x−2)=106 \left( 6.5 ^ { 3 x - 2 } \right) = 106(6.53x−2)=10 ​ x = __________

Find the logistic function f with the given properties. f(0)=20f ( 0 ) = 20f(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.

Find the y-intercept(s) of the graph of the quadratic function. 3x2−30x+753 x ^ { 2 } - 30 x + 753x2−30x+75

Find the vertex of the graph of the quadratic function. −x2+12x−36- x ^ { 2 } + 12 x - 36−x2+12x−36

The given table corresponds to the function f(x)=10xf ( x ) = 10 ^ { x }f(x)=10x . -3 -2 -1 0 1 2 3 ( ) 0.001 0.01 0.1 1 10 100 1,000

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

Functions and Applications

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

Functions of Several Variables

Trigonometric Models

Filters

Find the associated exponential decay model. $Q = 6,000$ when $t = 0$ ; Half-life = 5

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 1 2 f(x) 170 289

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 ( ) 40 80 160 Select the correct answer.

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.)

Use logarithms to solve the equation. (Round the answer to four decimal places.) $6 \left( 6.5 ^ { 3 x - 2 } \right) = 10$ x = __________

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.

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

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

The given table corresponds to the function $f ( x ) = 10 ^ { x }$ . -3 -2 -1 0 1 2 3 ( ) 0.001 0.01 0.1 1 10 100 1,000