Question 1

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

Accepted Answer

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)   $N = 0.2 , A = 2 , b = 11$ 
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 2

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.

Accepted Answer

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

Question 3

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

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

Accepted Answer

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)  $Q _ { 0 } = 23.1$ , $k = 0.006503$ 
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 4

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.
​

Accepted Answer

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

Question 5

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.

Accepted Answer

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

Question 6

Encouraged by the popularity of your Dungeons and Dragons website, www.mudbeast.net, you have decided to charge users who log on to the site. When you charged a $1.50 access fee, your web counter showed a demand of 270 "hits" per month. After you lowered the price to $0.50, activity increased to 350 "hits" per month. Obtain the monthly revenue R as a function of the access fee x.

Accepted Answer

A)  $R = - 80 x ^ { 2 } - 390 x$ 
B)   $R = 80 x ^ { 2 } - 230 x$ 
C)   $R = - 80 x ^ { 2 } + 390 x$ 
D)   $R = - 80 x + 390$ 
E)   $R = 80 x - 230$ 
A)  $R = - 80 x ^ { 2 } - 390 x$ 
B)   $R = 80 x ^ { 2 } - 230 x$ 
C)   $R = - 80 x ^ { 2 } + 390 x$ 
D)   $R = - 80 x + 390$ 
E)   $R = 80 x - 230$

Question 7

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

Accepted Answer

A)    
B)      
C)      
D)      
A)    
B)      
C)      
D)

Question 8

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

Accepted Answer

A)  $f ( x ) = x ^ { - 36 }$ 
B)  $f ( x ) = x ^ { - 216 }$ 
C)  $g ( x ) = 6 ^ { - x }$ 
D)  $f ( x ) = x ^ { - 6 }$ 
E)  $h ( x ) = x ^ { - 100 }$ 
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 9

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)

Accepted Answer

The answer of The table lists interest rates on long-term...

Question 10

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

Accepted Answer

A) True 
 B)False

Question 11

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.

Accepted Answer

A) 58,929 years old 
B)  2,629 years old 
C)  1,985 years old 
D)  0 years old 
E)  23,234 years old 
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 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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)  $A = 12.9802$ , $b = 0.6934$ 
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 14

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.

Accepted Answer

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)    $f ( x ) = \frac { 20 } { 1 + 1.5 ^ { x } }$ 
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 15

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)  $A ( t ) = \frac { 5.1 } { 1 + 4.5 ( 1.2 ) ^ { - t } }$ 
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)    $A ( t ) = \frac { 4.5 } { 1 + 4.2 ( 0.7 ) ^ { - t } }$ 
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)    $A ( t ) = \frac { 4.5 } { 1 + 4.2 ( 0.7 ) ^ { - t } }$

Question 16

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 = __________%

Accepted Answer

The answer of The graph shows the actual percentage of...

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.) $$\begin{array} { | l | l | l | l | l | l | l | } 
\hline x & 0 & 20 & 40 & 60 & 80 & 100 \
\hline y & 2.2 & 3.8 & 5.0 & 6.1 & 6.8 & 6.9 \
\hline
\end{array}$$

Accepted Answer

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)    $y = \frac { 7.2 } { 1 + 2.2 \left( 2.08 ^ { - x } \right) }$ 
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

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.

Accepted 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)  $f ( x ) = 0.5 ( 350 ) ^ { - x }$ 
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 19

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.

Accepted Answer

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)    $P ( x ) = 14.65 ( 2.1 ) ^ { - x }$ 
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 20

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

Accepted Answer

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

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

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.

Convert the exponential function to the form indicated. Round all coefficients to four significant digits. $f ( t ) = 23.2 ( 0.997 ) ^ { t }$ ; $f ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }$

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.

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

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

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

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.

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 f(x) 300 75 18.75

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

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.

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.) x 0 20 40 60 80 100 y 2.2 3.8 5.0 6.1 6.8 6.9

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 ( ) 350 175 87.5 Select the correct answer.

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

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Exam 2: Nonlinear Functions and Models

Find N, A, and b for the function. f(x)=111+2(0.2−x)f ( x ) = \frac { 11 } { 1 + 2 \left( 0.2 ^ { - x } \right) }f(x)=1+2(0.2−x)11​

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.

Convert the exponential function to the form indicated. Round all coefficients to four significant digits. f(t)=23.2(0.997)tf ( t ) = 23.2 ( 0.997 ) ^ { t }f(t)=23.2(0.997)t ; f(t)=Q0e−k′tf ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }f(t)=Q0​e−k′t

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.

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

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

The given table corresponds to the function f(x)=0.042(4.2−51.7)−xf ( x ) = 0.042 \left( 4.2 - \frac { 5 } { 1.7 } \right) ^ { - x }f(x)=0.042(4.2−1.75​)−x . -3 -2 -1 0 1 2 3 ( ) 0.084 0.067 0.067 1 0.033 0.027 0.021

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . 0 1 2 f(x) 300 75 18.75

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

Find the logistic function f with the given properties. f(0)=1f ( 0 ) = 1f(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.

Use technology to find a logistic regression curve y=N1+Ab−xy = \frac { N } { 1 + A b ^ { - x } }y=1+Ab−xN​ approximating the given data. (Round b to three significant digits and A and N to two significant digits.) x 0 20 40 60 80 100 y 2.2 3.8 5.0 6.1 6.8 6.9

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . 0 1 2 ( ) 350 175 87.5 Select the correct answer.

Use logarithms to solve the equation. Round your answer to four decimal places. 6−2x=406 ^ { - 2 x } = 406−2x=40

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

Convert the exponential function to the form indicated. Round all coefficients to four significant digits. $f ( t ) = 23.2 ( 0.997 ) ^ { t }$ ; $f ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }$

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

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

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 f(x) 300 75 18.75

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

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.

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.) x 0 20 40 60 80 100 y 2.2 3.8 5.0 6.1 6.8 6.9

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . 0 1 2 ( ) 350 175 87.5 Select the correct answer.

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