Question 1

Below is a graph of a function $f ( x )$ . Find the average rate of change in $f$ from 2 to 4 . 
A) 60
B) -5
C) -10
D) None of the above

Accepted Answer

B

Question 2

Choose the answer that best completes the following sentence. A graph that is decreasing and concave up represents a function that is ...
A) increasing at an increasing rate
B) increasing at a decreasing rate
C) decreasing at a decreasing rate
D) decreasing at an increasing rate

Accepted Answer

A graph that is decreasing and concave up indicates that the function's values are getting smaller, but the rate at which they decrease is slowing down, hence it is decreasing at a decreasing rate.

Question 3

$$\text { If } k ( x , y ) = \frac { \pi ^ { x + y } } { x + y } , \text { calculate the value of } k ( 3.62,1.09 )$$
A) 63.28
B) 61.75
C) 64.44
D) 46.62

Accepted Answer

D

Question 4

$$\text { If } g ( h , k ) = \frac { h } { k } + \frac { k } { h } , \text { calculate the value of } g ( 5.03,2.05 )$$
A) 0.82
B) 4.91
C) 2
D) 2.86

Accepted Answer

Substituting $h=5.03$ and $k=2.05$ into the given equation, we get
$$g(5.03,2.05) = \frac{5.03}{2.05}+\frac{2.05}{5.03}=\frac{5.03^2+2.05^2}{2.05\cdot 5.03}=\frac{27.0206}{10.3165}=2.86$$
Therefore, the correct choice is $\boxed{	ext{(D)}}$.

Question 5

Below is a graph of a function $f ( x )$ . What is the value of $f ( 2 )$ ? 
A) 8
B) 9
C) 5
D) None of the above

Accepted Answer

The answer of Below is a graph of a function...

Question 6

Choose the answer that best completes the following sentence. A point of inflection occurs where...
A) concavity changes
B) concavity is at a maximum
C) the graph is increasing
D) the graph is bent

Accepted Answer

A point of inflection occurs where the concavity changes. This means that the graph changes from being curved upwards to being curved downwards or vice versa. It is the point where the second derivative changes sign.

Question 7

The following table shows average rice yield, in tons per hectare, in Asia  years since 1980. $$\begin{array} { | l | l | l | l | l | l | } 
\hline \begin{array} { l } 
\text { t=years since } \
1980
\end{array} & 5 & 10 & 15 & 20 & 25 \
\hline Y = \text { yield } & 3.32 & 3.61 & 3.73 & 3.95 & 4.11 \
\hline
\end{array}$$ Use the average rate of change to estimate the yield in 1993.
A) 3.61 tons per hectare
B) 3.68 tons per hectare
C) 3.75 tons per hectare
D) None of the above

Accepted Answer

To estimate the yield in 1993, we need to find the average rate of change over the interval from 1980 to 1993 (which is 13 years). We can use the formula:

average rate of change = (final value - initial value) / (final time - initial time)

Plugging in the values from the table, we get:

average rate of change = (Y[25] - Y[5]) / (25 - 5) 
                       = (4.11 - 3.32) / 20 
                       = 0.0395

This means that on average, the yield increased by 0.0395 tons per hectare, per year, between 1980 and 1993. To estimate the yield in 1993, we can use the initial value of 3.32, and add the average rate of change for 13 years:

estimated yield in 1993 = 3.32 + 0.0395 * 13 
                       = 3.68

Therefore, the best choice is B, 3.68 tons per hectare.

Question 8

The temperature in degrees Fahrenheit can be estimated from the number of cricket chirps per minute. If a cricket chirps 40 times per minute, the temperature is approximately 50 degrees Fahrenheit.
The average rate of change in temperature is 0.25 degree per chirp per minute. Use these facts to estimate
The temperature when a cricket chirps 67 times per minute.
A) 56.75 degrees
B) 51 degrees
C) 40.25 degrees
D) none of the above

Accepted Answer

We can use the average rate of change in temperature to estimate the temperature when a cricket chirps 67 times per minute.
- From 40 to 67 chirps per minute, there is an increase of 27 chirps per minute.
- Using the average rate of change, we can estimate that for every chirp per minute, the temperature increases by 0.25 degrees.
- Therefore, for 27 chirps per minute, the temperature increases by 6.75 degrees (27 chirps/min x 0.25 degrees/chirp/min = 6.75 degrees).
- Since the temperature is approximately 50 degrees Fahrenheit at 40 chirps per minute, we can estimate that the temperature is 56.75 degrees Fahrenheit at 67 chirps per minute (50 degrees + 6.75 degrees = 56.75 degrees).
Therefore, choice A (56.75 degrees) is the best option.

Question 9

Below is a graph of a function $f ( x )$ . What is the limiting value for $f$ ? 
A) 60
B) 20
C) 50
D) None of the above

Accepted Answer

As the value of x goes to infinity, the graph of the function converges to a horizontal line at y = 50. Therefore, the limiting value of f is 50.

Question 10

$$\text { If } f ( x , y ) = \frac { x ^ { \wedge } 3 + y ^ { \wedge } 3 } { x ^ { \wedge } 2 + y ^ { \wedge } 2 } , \text { calculate the value of } f ( 7.44,3.48 ) \text {. }$$
A) 6.73
B) 424.7
C) 10.92
D) None of the above

Accepted Answer

Explanation:Given, $$f ( x , y ) = \frac { x ^ { \wedge } 3 + y ^ { \wedge } 3 } { x ^ { \wedge } 2 + y ^ { \wedge } 2 }$$Substituting the values of x and y, we get$$f ( 7.44,3.48 ) = \frac { ( 7.44 ) ^ { \wedge } 3 + ( 3.48 ) ^ { \wedge } 3 } { ( 7.44 ) ^ { \wedge } 2 + ( 3.48 ) ^ { \wedge } 2 }$$$$= \frac { 417.723392 + 39.58319488 } { 55.4496 + 12.1584 }$$$$= \frac { 457.30658688 } { 67.608 }$$$$= 6.73$$Therefore, the value of f ( 7.44,3.48 ) is 6.73.

Question 11

A balloon that originally holds 13.6 cubic inches of air springs a leak. Let $V ( t )$ represent the volume, in cubic inches, of air in the balloon  minutes after the balloon starts to leak air. The average rate
Of change of $V$ is $- 1.02$ cubic inches per minute. Use these facts to estimate $V ( 6 )$ .
A) 13.6 cubic inches
B) 19.72 cubic inches
C) 12.93 cubic inches
D) none of the above

Accepted Answer

The answer of A balloon that originally holds 13.6 cubic...

Question 12

$$\text { If } f ( x , y , z ) = \frac { x y + z } { x + y z } , \text { calculate the value of } f ( 9.52,3.04 , - 2.76 )$$
A) 23.18
B) 20.26
C) -5.64
D) 26.5

Accepted Answer

Explanation:$$f ( x , y , z ) = \frac { x y + z } { x + y z }$$$$f ( 9.52,3.04 , - 2.76 ) = \frac { ( 9.52 ) ( 3.04 ) + ( - 2.76 ) } { ( 9.52 ) + ( 3.04 ) ( - 2.76 ) }$$$$= \frac { 29.0368 + ( - 2.76 ) } { 9.52 + ( - 8.3744 ) }$$$$= \frac { 26.2768 } { 1.1456 }$$$$= 23.18$$

Question 13

$$\text { If } g ( x , y ) = \frac { e ^ { x } + e ^ { - y } } { x y ^ { - 2 } } , \text { calculate the value of } g ( 4.65,4.22 )$$
A) 104.59
B) 1126.82
C) 393.76
D) -393.76

Accepted Answer

To calculate $g(4.65, 4.22)$, we substitute $x = 4.65$ and $y = 4.22$ into the given function:$$g(x, y) = \frac{e^x + e^{-y}}{xy^{-2}}$$$$g(4.65, 4.22) = \frac{e^{4.65} + e^{-4.22}}{4.65 \cdot 4.22^{-2}}$$Using a calculator to compute the values:$$e^{4.65} \approx 104.677$$$$e^{-4.22} \approx 0.0147$$$$4.22^{-2} \approx 0.0561$$Substituting these values back into the equation:$$g(4.65, 4.22) \approx \frac{104.677 + 0.0147}{4.65 \cdot 0.0561}$$$$g(4.65, 4.22) \approx \frac{104.6917}{0.260865}$$$$g(4.65, 4.22) \approx 401.3$$It seems there was a mistake in my calculation. Let's correct the calculation error:$$g(4.65, 4.22) = \frac{e^{4.65} + e^{-4.22}}{4.65 \cdot (4.22)^{-2}}$$Correctly calculating the denominator and the numerator with accurate values for $e^{4.65}$ and $e^{-4.22}$, and considering the correct calculation for $4.22^{-2}$, the correct answer should closely match one of the provided options. Given the discrepancy in my initial calculation, let's correct the approach:The correct approach involves accurately calculating the exponential functions and the division as per the formula without rounding errors and ensuring the operations are correctly applied, especially the exponentiation and division in the denominator. Given the options and the nature of the function, which involves exponential growth and decay terms divided by a product involving an inverse square, the result should indeed be a positive value, reflecting the exponential growth's dominance.Upon reevaluation with precise calculations:$$g(4.65, 4.22) = \frac{e^{4.65} + e^{-4.22}}{4.65 \cdot 4.22^{-2}}$$Given the mistake in the manual calculation, it's important to use a precise calculator or computational tool to accurately compute the value, which should align with one of the provided options, considering the exponential terms and the specific values of $x$ and $y$. The correct calculation will accurately reflect the behavior of the function $g(x, y)$ at the given point (4.65, 4.22), taking into account the exponential increase with $e^{4.65}$ and the exponential decay with $e^{-4.22}$, all scaled by the specific values of $x$ and $y$ as per the function's definition.

Question 14

The following table shows the number of bird species found on an island of area $A$ square miles that is part of a particular Pacific island chain. $$\begin{array} { | l | l | l | l | l | l | } 
\hline A = \text { area } & 10 & 20 & 30 & 40 & 50 \
\hline \begin{array} { l } 
W = \text { number } \
\text { of species }
\end{array} & 25 & 31 & 36 & 39 & 42 \
\hline
\end{array}$$ Calculate the average rate of change in the number of species as the area changes from 30 to 40 square miles.
A) 1.52 species per square mile
B) 39 species per square mile
C) 0.3 species per square mile
D) None of the above

Accepted Answer

We need to find the average rate of change in the number of species per square mile from an area of 30 to 40 square miles. The change in the number of species is 39-36 = 3, and the change in area is 10. Therefore, the average rate of change in the number of species per square mile is:

$$\frac{3}{10}=0.3 \text{ species per square mile}$$

Hence, the answer is (C).

Question 15

You put a drink in the freezer to cool. You take it out of the freezer when it is cold. But you forget about the drink and leave it sitting on the kitchen counter. The graph below shows the temperature, in
Degrees, of the drink  minutes after the drink is placed in the freezer. What is the temperature in the kitch-
En? 
A) 20 degrees
B) 60 degrees
C) 70 degrees
D) None of the above

Accepted Answer

The graph shows that the temperature of the drink starts at room temperature (70 degrees), decreases rapidly in the first few minutes in the freezer, and then slowly decreases until it reaches a plateau around 20 degrees. However, since the drink is left on the counter after being in the freezer, the temperature of the drink will increase back towards room temperature. Therefore, the temperature in the kitchen will be around 70 degrees.

Question 16

The distance, in miles, from me to a moving train is given by $D ( t ) = \sqrt { ( 30 t ) ^ { 2 } + ( 5 - 30 t ) ^ { 2 } }$ Here  represents hours since I heard the train whistle. Calculate the distance to the train 5 hours after I heard the whistle.
A) 5 miles
B) 208.63 miles
C) 43525 miles
D) 147.56 miles

Accepted Answer

The answer of The distance, in miles, from me to...

Question 17

If you work 42 hours per week, then your weekly pay $P$ , in dollars, is proportional to your hourly wage $W$ , in dollars per hour. Express this proportionality relationship as a formula. 
A) $P \propto W$
B) $P = 42 W$
C) $W = 42 P$
D) None of the above.

Accepted Answer

The formula for a proportionality relationship is y = kx, where y and x are the variables and k is the constant of proportionality. In this case, P is the dependent variable and W is the independent variable, so the formula is P = kW. Since it is given that the workweek is 42 hours, this means that k = 42, so the formula becomes P = 42W.

Question 18

The following table shows the U.S. population, in millions, in the given year. $$\begin{array} { | l | l | l | l | l | l | } 
\hline d = \text { year } & 1960 & 1970 & 1980 & 1990 & 2000 \
\hline \begin{array} { l } 
N = \text { popula- } \
\text { tion in mil- } \
\text { lions }
\end{array} & 179.32 & 203.3 & 226.54 & 248.71 & 281.42 \
\hline
\end{array}$$ Use the average rate of change to estimate the U.S. population in 1996.
A) 268.34 million
B) 251.98 million
C) 265.07 million
D) None of the above

Accepted Answer

The average rate of change from 1990 to 2000 can be calculated as $\frac{281.42 - 248.71}{2000 - 1990} = \frac{32.71}{10} = 3.271$ million per year. To estimate the population in 1996, we add 6 years' worth of growth to the 1990 population: $248.71 + 6(3.271) = 248.71 + 19.626 = 268.336$, which rounds to 268.34 million.

Question 19

Below is a graph of a function $f ( x )$ . Over what region(s) is the graph concave up? 
A) at $x = 3$
B) from 0 to 3
C) from 3 to 6
D) None of the above

Accepted Answer

The graph is concave up in a region where the second derivative (the rate of change of the slope of the graph) is positive. Looking at the graph, we can see that the second derivative changes from negative to positive at $x = 3$, indicating a point of inflection. Therefore, the graph is concave up in the region from $x = 3$ to $x = 6$.

Question 20

The following table shows average rice yield, in tons per hectare, in Asia  years since 1980. $$\begin{array} { | l | l | l | l | l | l | } 
\hline \begin{array} { l } 
\mathrm { t } = \text { years since } \
1980
\end{array} & 5 & 10 & 15 & 20 & 25 \
\hline Y = \text { yield } & 3.32 & 3.61 & 3.73 & 3.95 & 4.11 \
\hline
\end{array}$$ Use the average rate of change to estimate the yield in 1996.
A) 3.73 tons per hectare
B) 3.77 tons per hectare
C) 3.87 tons per hectare
D) None of the above

Accepted Answer

To estimate the yield in 1996, we need to find the average rate of change of yield over the period from 1980 to 1996, which is 16 years.

The average rate of change between t = 0 and t = 25 (i.e., the entire period from 1980 to 2005) is:

average rate of change = (final value - initial value) / (final time - initial time)
                        = (4.11 - 3.32) / (25 - 0)
                        = 0.03 tons per hectare per year
                        
Since the rate of change is constant over this period, we can use it to estimate the yield in 1996 as follows:

yield in 1996 = yield in 1980 + (16 years) x (average rate of change)
              = 3.32 + (16 x 0.03)
              = 3.80 tons per hectare
              
The closest choice is B) 3.77 tons per hectare.

Quiz 1: Functions

Below is a graph of a function f(x) f ( x ) f(x) . Find the average rate of change in fff from 2 to 4 .

Choose the answer that best completes the following sentence. A graph that is decreasing and concave up represents a function that is ...

If k(x,y) =πx+yx+y, calculate the value of k(3.62,1.09) \text { If } k ( x , y ) = \frac { \pi ^ { x + y } } { x + y } , \text { calculate the value of } k ( 3.62,1.09 ) If k(x,y) =x+yπx+y​, calculate the value of k(3.62,1.09)

If g(h,k) =hk+kh, calculate the value of g(5.03,2.05) \text { If } g ( h , k ) = \frac { h } { k } + \frac { k } { h } , \text { calculate the value of } g ( 5.03,2.05 ) If g(h,k) =kh​+hk​, calculate the value of g(5.03,2.05)

Below is a graph of a function f(x) f ( x ) f(x) . What is the value of f(2) f ( 2 ) f(2) ?

Choose the answer that best completes the following sentence. A point of inflection occurs where...

Below is a graph of a function f(x) f ( x ) f(x) . What is the limiting value for fff ?

If f(x,y,z) =xy+zx+yz, calculate the value of f(9.52,3.04,−2.76) \text { If } f ( x , y , z ) = \frac { x y + z } { x + y z } , \text { calculate the value of } f ( 9.52,3.04 , - 2.76 ) If f(x,y,z) =x+yzxy+z​, calculate the value of f(9.52,3.04,−2.76)

If g(x,y) =ex+e−yxy−2, calculate the value of g(4.65,4.22) \text { If } g ( x , y ) = \frac { e ^ { x } + e ^ { - y } } { x y ^ { - 2 } } , \text { calculate the value of } g ( 4.65,4.22 ) If g(x,y) =xy−2ex+e−y​, calculate the value of g(4.65,4.22)

If you work 42 hours per week, then your weekly pay PPP , in dollars, is proportional to your hourly wage WWW , in dollars per hour. Express this proportionality relationship as a formula.

Below is a graph of a function f(x) f ( x ) f(x) . Over what region(s) is the graph concave up?

Below is a graph of a function $f ( x )$ . Find the average rate of change in $f$ from 2 to 4 .

$\text { If } k ( x , y ) = \frac { \pi ^ { x + y } } { x + y } , \text { calculate the value of } k ( 3.62,1.09 )$

$\text { If } g ( h , k ) = \frac { h } { k } + \frac { k } { h } , \text { calculate the value of } g ( 5.03,2.05 )$

Below is a graph of a function $f ( x )$ . What is the value of $f ( 2 )$ ?

Below is a graph of a function $f ( x )$ . What is the limiting value for $f$ ?

$\text { If } f ( x , y , z ) = \frac { x y + z } { x + y z } , \text { calculate the value of } f ( 9.52,3.04 , - 2.76 )$

$\text { If } g ( x , y ) = \frac { e ^ { x } + e ^ { - y } } { x y ^ { - 2 } } , \text { calculate the value of } g ( 4.65,4.22 )$

If you work 42 hours per week, then your weekly pay $P$ , in dollars, is proportional to your hourly wage $W$ , in dollars per hour. Express this proportionality relationship as a formula.

Below is a graph of a function $f ( x )$ . Over what region(s) is the graph concave up?