Question 1

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 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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)  $A = 2.7$ , $b = 0.4969$ 
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 4

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

Accepted Answer

A) True 
 B)False

Question 5

Choose the logistic function that best approximates the curve.

Accepted Answer

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)    $f ( x ) = \frac { 18 } { 2 + 7 ( 4 ) ^ { - x } }$ 
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 6

For the following demand equation, find the largest possible revenue.
​ $q = - 6 p + 3,600$ ​
$ __________

Accepted Answer

The answer of For the following demand equation, find the...

Question 7

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

Accepted Answer

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

Question 8

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)

Accepted Answer

a. 6.3; 4.

Question 9

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.
​ $f ( t ) = 23.7 ( 0.996 ) ^ { t }$ ; $f ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }$ ​ $Q _ { 0 } =\_\_\_\_$ ​ $k =\_\_\_\_$

Accepted Answer

The answer of Convert the exponential function to the form...

Question 10

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.

Accepted Answer

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 
 
__________% 
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 11

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $$\begin{array} { | l | l | l | l | } 
\hline \boldsymbol { x } & 0 & 1 & 2 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 10 & 20 & 40 \
\hline
\end{array}$$

Accepted Answer

The answer of Model the data using an exponential function...

Question 12

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

Accepted Answer

A) True 
 B)False

Question 13

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. ​

Accepted Answer

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

Question 14

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)

Accepted Answer

The answer of How long, to the nearest year, will...

Question 15

Choose the correct letter for each question.
​
-none

Accepted Answer

A) $x ^ { 2 } - 6 x + 8$ 
B) $x ^ { 2 } + 10 x + 25$ 
C) $x ^ { 2 } + 10 x + 30$ 
A) $x ^ { 2 } - 6 x + 8$ 
B) $x ^ { 2 } + 10 x + 25$ 
C) $x ^ { 2 } + 10 x + 30$

Question 16

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 ( $$t = 18$$ )

Accepted Answer

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

Question 17

Choose the logistic function that best approximates the curve.

Accepted Answer

A)  $f ( x ) = \frac { 16 } { 2 + 6 ( 4 ) ^ { - x } }$ 
B)  $f ( x ) = \frac { 16 } { 2 + 5 ( 1.1 ) ^ { - x } }$ 
C)  $f ( x ) = \frac { 16 } { 2 + 6 ( 1.1 ) ^ { - x } }$ 
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 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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)    $N = 4 , A = 1.5 , b = 5$ 
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 20

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.

Accepted Answer

A) Blue 
B)  Red 
A) Blue 
B)  Red

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

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

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

The table below is filled correctly. Exponential form Logarithmic form =625 625=4 0.=0.36 0.36=2 =1 1=0 =0.125 0.125=-1

Choose the logistic function that best approximates the curve.

For the following demand equation, find the largest possible revenue. $q = - 6 p + 3,600$ $ __________

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

Convert the exponential function to the form indicated. Round all coefficients to four significant digits. $f ( t ) = 23.7 ( 0.996 ) ^ { t }$ ; $f ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }$ $Q _ { 0 } =\_\_\_\_$ $k =\_\_\_\_$

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 ( ) 10 20 40

The table is filled correctly. Exponential form Logarithmic form =0.001953 0.001953=3 =1 1=0 0.=0.0256 0.0256=4 0.=1.953125 1.953125=3

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. Country U.S. Japan Canada Korea Australia Yield 5.3\% 1.5\% 5.2\% 5.4\% 6.0\% t = __________ year(s)

Choose the correct letter for each question. -none

Choose the logistic function that best approximates the curve.

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

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

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Exam 2: Nonlinear Functions and Models

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

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

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

The table below is filled correctly. Exponential form Logarithmic form =625 625=4 0.=0.36 0.36=2 =1 1=0 =0.125 0.125=-1

Choose the logistic function that best approximates the curve.

For the following demand equation, find the largest possible revenue. ​ q=−6p+3,600q = - 6 p + 3,600q=−6p+3,600 ​ $ __________

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

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . 0 1 2 ( ) 10 20 40

The table is filled correctly. Exponential form Logarithmic form =0.001953 0.001953=3 =1 1=0 0.=0.0256 0.0256=4 0.=1.953125 1.953125=3

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. Country U.S. Japan Canada Korea Australia Yield 5.3\% 1.5\% 5.2\% 5.4\% 6.0\% t = __________ year(s)

Choose the correct letter for each question. ​ -none

Choose the logistic function that best approximates the curve.

Use logarithms to solve the equation. Round your answer to four decimal places. 8(2.52x−1)=108 \left( 2.5 ^ { 2 x - 1 } \right) = 108(2.52x−1)=10

Find N, A, and b for the function given. f(x)=20.5+2.5(1.5−x)f ( x ) = \frac { 2 } { 0.5 + 2.5 \left( 1.5 ^ { - x } \right) }f(x)=0.5+2.5(1.5−x)2​

Given the graph of the functions f1(x)=2.5xf _ { 1 } ( x ) = 2.5 ^ { x }f1​(x)=2.5x and f2(x)=2.8xf _ { 2 } ( x ) = 2.8 ^ { x }f2​(x)=2.8x . Identify which graph corresponds to f2(x)=2.8xf _ { 2 } ( x ) = 2.8 ^ { x }f2​(x)=2.8x . 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

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

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

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

For the following demand equation, find the largest possible revenue. $q = - 6 p + 3,600$ $ __________

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

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 ( ) 10 20 40

Choose the correct letter for each question. -none

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

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