Question 1

If $g ( h , k ) = \frac { h } { k } + \frac { k } { h }$ , calculate the value of $g ( 5.03,1.41 )$ .
A) 0.56
B) 7.13
C) 2
D) 3.85

Accepted Answer

Substituting the given values, we get: 
$g(5.03,1.41) = \frac{5.03}{1.41}+\frac{1.41}{5.03} =3.85$ 
Therefore, the answer is D.

Question 2

Below is a graph of a function $f ( x )$ . What is the value of $f ( 4 )$ ? 
A) 8
B) -8
C) 9
D) None of the above

Accepted Answer

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

Question 3

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 20 to 30 square miles.
A) 0.76 species per square mile
B) 36 species per square mile
C) 0.5 species per square mile
D) None of the above

Accepted Answer

The average rate of change in the number of species per square mile is given by:

$$\frac{\text{change in number of species}}{\text{change in area}} = \frac{W_{30}-W_{20}}{30-20} = \frac{36-31}{10} = \frac{5}{10} = 0.5$$

Therefore, the answer is C) 0.5 species per square mile.

Question 4

The maximum length of a certain type of fish is 63 centimeters. The difference between the maximum length and the length at age  years is given by $D = 59.85 \times 0.83 ^ { t }$ centimeters. Write a for-
Mula that gives the length L , in centimeters, of the fish at age  years. 
A) $L = 63 + 59.85 \times 0.83 ^ { t }$
B) $L = 63 - 59.85 \times 0.83 ^ { t}$
C) $L = 63 \times 59.85 \times 0.83 ^ { t }$
D) $L = 59.85 \times 0.83 ^ { t } - 63$

Accepted Answer

Since the difference between the maximum length and the length at age 11 years is given by $D = 59.85 \times 0.83 ^ {t}$, we need to subtract this difference from the maximum length to get the length at age 11 years. Therefore, the formula that gives the length $L$ of the fish at age 11 years is: 
$$L = 63 - 59.85 \times 0.83 ^ {t}$$ 
This matches answer choice B.

Question 5

If $f ( x , y , z ) = \frac { x y + z } { x + y z }$,
calculate the value of $f ( 16.92,1.56 , - 3.76 )$
A) 2.05
B) 20.31
C) -4.53
D) 26.06

Accepted Answer

Plugging the values into the function $f(x, y, z) = \frac{xy + z}{x + yz}$, we get $f(16.92, 1.56, -3.76) = \frac{16.92 \times 1.56 - 3.76}{16.92 + 1.56 \times -3.76}$. Calculating the numerator and denominator separately, we get approximately $26.3952 - 3.76 = 22.6352$ for the numerator and $16.92 - 5.9056 = 11.0144$ for the denominator. Dividing these, $22.6352 / 11.0144 ≈ 2.05$.

Question 6

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 51 times per minute.
A) 52.75 degrees
B) 78 degrees
C) 40.25 degrees
D) none of the above

Accepted Answer

Using the given rate of change of 0.25 degree per chirp per minute, we can estimate that for every additional 11 chirps (51 minus 40), the temperature increases by 0.25 x 11 = 2.75 degrees Fahrenheit. Starting from the initial temperature of 50 degrees Fahrenheit, the estimated temperature when a cricket chirps 51 times per minute is 50 + 2.75 = 52.75 degrees Fahrenheit. Therefore, the best choice is A.

Question 7

Sugar costs $1.69 per pound, and lemons cost $0.98 each. You sell lemonade for $1.35 per glass. Write S for the pounds of sugar used, L for the number of lemons used, G for the number of glasses
Of lemonade you sell, and P for the profit, in dollars, you make. Express using a formula your profit in
Terms of pounds of sugar used, number of lemons used, and number of glasses of lemonade sold. 
A) $P = 1.35 G - 1.69 S - 0.98 L$
B) $P = 1.35 G + 1.69 S + 0.98 L$
C) $P = 1.69 S + 0.98 L - 1.35 G$
D) $P + 1.69 G + 1.35 S - 0.98 L$

Accepted Answer

The formula for profit in terms of pounds of sugar used (S), number of lemons used (L), and number of glasses of lemonade sold (G) is given by:
P = Revenue - Cost
Revenue = 1.35G (selling price per glass times number of glasses sold)
Cost = cost of sugar + cost of lemons
Cost of sugar = 1.69S (cost per pound of sugar times pounds of sugar used)
Cost of lemons = 0.98L (cost per lemon times number of lemons used)
Therefore,
P = 1.35G - 1.69S - 0.98L

Question 8

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 1994.
A) 261.79 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 is calculated as $\frac{281.42 - 248.71}{2000 - 1990} = \frac{32.71}{10} = 3.271$ million per year. To estimate the population in 1994, we add 4 years' worth of growth to the 1990 population: $248.71 + 4(3.271) = 248.71 + 13.084 = 261.794$ million, which rounds to 261.79 million.

Question 9

$$\text { If } g ( x , y ) = \frac { e ^ { x } + e ^ { - y } } { x y ^ { - e } } , \text { calculate the value of } g ( 9.3,3.85 )$$
A) 10938.02
B) 45909.64
C) 45712.3
D) -45712.3

Accepted Answer

Substitute the given values for x and y in the expression for g:
$$g(9.3,3.85)=\frac{e^{9.3}+e^{-3.85}}{9.3\cdot(3.85)^{-e}}=\frac{10,121.5702+0.0209}{9.3\cdot(3.85)^{-2.7183}}=45909.64$$
Therefore the answer is (B).

Question 10

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 7 hours after I heard the whistle.
A) 5 miles
B) 293.47 miles
C) 86125 miles
D) 208.55 miles

Accepted Answer

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

Question 11

$$\text { If } k ( x , y ) = \frac { \pi ^ { x + y } } { x + y } , \text { calculate the value of } k ( 4.56,3.33 ) \text {. }$$
A) 185.35
B) 1837.96
C) 188.99
D) 1060.32

Accepted Answer

The answer of \[\text { If } k ( x...

Question 12

A) 3.47
B) 37.75
C) 5.71
D) None of the above

Accepted Answer

The answer of

A) 3.47
B) 37.75
C) 5.71
D) None of the...

Question 13

The following table shows your monthly payment P, in dollars, if you pay off a loan in  months.

Which formula below fits these data?
A) $P = \frac { 5193 } { t }$
B) $P = \frac { 51.93 \times 1.01 ^ { t } } { 1.01 ^ { t} - 1 }$
C) $P = \frac { 5193 \times ( 1.01 ) ^ { t } } { t - 1.01 }$

D) None of the above

Accepted Answer

The formula for monthly payment on a loan with a fixed interest rate and fixed term is given by:
P = (rPv)/(1 - (1+r)^-n)
where r is the monthly interest rate, Pv is the present value (amount borrowed), and n is the number of monthly payments. 
In this case, we don't have enough information to directly calculate r or Pv, but we can use the formula provided as an approximation. 
Option B matches this formula with a monthly interest rate of 1%, a present value of $5193, and n = 11. Therefore, it is the best choice for this scenario.

Question 14

The temperature of an oven remains constant at 371 degrees Fahrenheit. A potato is placed in the oven to bake. The difference between the temperature of the oven and that of the potato after  mi-
Nutes in the oven is given by $D = 296 \times 0.92 ^ { t}$ degrees Fahrenheit. Write a formula that gives the tempera-
Ture $T$ , in degrees Fahrenheit, of the potato after  minutes in the oven. 
A) $T = 371 - 296 \times 0.92 ^ { t }$
B) $T = 371 + 296 \times 0.92 ^ { t }$
C) $T = 371 \times 296 \times 0.92 ^ { t }$
D) $T = 296 \times 0.92 ^ { t } - 371$

Accepted Answer

The answer of The temperature of an oven remains constant...

Question 15

The following table shows some function values.

Which formula below fits these data?
A) $f ( x ) = 7 x + 5$
B) $f ( x ) = 5 x + 7$
C) $f ( x ) = \frac { x + 7 } { 5 }$
D) $f ( x ) = \frac { 7 } { x + 5 }$

Accepted Answer

The answer of The following table shows some function values....

Question 16

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 of a function is concave up when its second derivative is positive. From the graph, we can see that the second derivative changes from negative to positive at x = 3, and remains positive for x values greater than 3 up until around x = 6. This means that the graph is concave up over the region from 3 to 6. Therefore, the correct choice is C.

Question 17

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 1989.
A) 3.32 tons per hectare
B) 3.55 tons per hectare
C) 3.46 tons per hectare
D) None of the above

Accepted Answer

To estimate the yield in 1989, we need to find the average rate of change between t = 5 and t = 10.

Average rate of change = (change in Y) / (change in t) 
= (3.61 - 3.32) / 5 
= 0.058

Using this average rate of change, we can estimate the yield in 1989 as follows:

Y = 3.32 + (0.058)(9) 
= 3.55

Therefore, the best choice is B, 3.55 tons per hectare.

Question 18

If you work 43 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) $\rho \text { oc WV }$
B) $\rho = 43 \mathrm {~W}$
C) $W = 430$
D) None of the above.

Accepted Answer

Since the weekly pay is proportional to the hourly wage, we can write:

P ∝ W

Now, we need to introduce the constant of proportionality, which we'll call k. So:

P = kW

We know that if we work 43 hours per week, we'll get paid P dollars. So:

P = 43W

Putting this together with the previous equation, we get:

43W = kW

Solving for k:

k = 43/W

Now we can use this value of k to write the formula for P in terms of W:

P = (43/W)W = 43

So the formula is:

P = 43W

Therefore, the correct choice is B.

Question 19

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

Based on the graph, the temperature of the drink started at room temperature (around 70 degrees) and decreased over time as it was in the freezer. Once it was taken out of the freezer, the temperature dropped quickly to about 20 degrees, but then began to increase slowly as it sat on the counter. After 11 minutes, the temperature was around 60 degrees, which is the closest option to the temperature shown on the graph. Therefore, the best choice is C (70 degrees). None of the other options are close to the temperature shown on the graph.

Question 20

The following table shows average rice yield, in tons per hectare, in Asia  years since 1980. $$\begin{array} { | l | l | l | l | l | l | } 
\hline \begin{array} { l } 
\text { t=years since } \
1980
\end{array} & 5 & 10 & 15 & 20 & 25 \
\hline Y = \text { yield } & 3.32 & 3.61 & 3.73 & 3.95 & 4.11 \
\hline
\end{array}$$ Use the average rate of change to estimate the yield in 1997.
A) 3.73 tons per hectare
B) 3.82 tons per hectare
C) 3.87 tons per hectare
D) None of the above

Accepted Answer

The average rate of change for the yield from t=5 to t=10 is:
$$\frac{3.61-3.32}{10-5}=0.058$$
The estimated yield in 1990 (10 years since 1980) is:
$$3.32+0.058(10-5)=3.61$$
Similarly, the estimated yield in 1995 (15 years since 1980) is:
$$3.61+0.058(15-10)=3.90$$
Using the same average rate of change, the estimated yield in 1997 (17 years since 1980) is:
$$3.61+0.058(17-10)=3.82$$
Therefore, the answer is B, 3.82 tons per hectare.

Quiz 1: Functions

If $g ( h , k ) = \frac { h } { k } + \frac { k } { h }$ , calculate the value of $g ( 5.03,1.41 )$ .

Below is a graph of a function $f ( x )$ . What is the value of $f ( 4 )$ ?

If $f ( x , y , z ) = \frac { x y + z } { x + y z }$ , calculate the value of $f ( 16.92,1.56 , - 3.76 )$

$\text { If } g ( x , y ) = \frac { e ^ { x } + e ^ { - y } } { x y ^ { - e } } , \text { calculate the value of } g ( 9.3,3.85 )$

$\text { If } k ( x , y ) = \frac { \pi ^ { x + y } } { x + y } , \text { calculate the value of } k ( 4.56,3.33 ) \text {. }$

The following table shows some function values. $The following table shows some function values. Which formula below fits these data? A) f ( x ) = 7 x + 5 B) f ( x ) = 5 x + 7 C) f ( x ) = \frac { x + 7 } { 5 } D) f ( x ) = \frac { 7 } { x + 5 }$ Which formula below fits these data?

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

If you work 43 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.

Quiz 1: Functions

If g(h,k) =hk+khg ( h , k ) = \frac { h } { k } + \frac { k } { h }g(h,k) =kh​+hk​ , calculate the value of g(5.03,1.41) g ( 5.03,1.41 ) g(5.03,1.41) .

Below is a graph of a function f(x) f ( x ) f(x) . What is the value of f(4) f ( 4 ) f(4) ?

If f(x,y,z) =xy+zx+yzf ( x , y , z ) = \frac { x y + z } { x + y z }f(x,y,z) =x+yzxy+z​ , calculate the value of f(16.92,1.56,−3.76) f ( 16.92,1.56 , - 3.76 ) f(16.92,1.56,−3.76)

If g(x,y) =ex+e−yxy−e, calculate the value of g(9.3,3.85) \text { If } g ( x , y ) = \frac { e ^ { x } + e ^ { - y } } { x y ^ { - e } } , \text { calculate the value of } g ( 9.3,3.85 ) If g(x,y) =xy−eex+e−y​, calculate the value of g(9.3,3.85)

If k(x,y) =πx+yx+y, calculate the value of k(4.56,3.33) . \text { If } k ( x , y ) = \frac { \pi ^ { x + y } } { x + y } , \text { calculate the value of } k ( 4.56,3.33 ) \text {. } If k(x,y) =x+yπx+y​, calculate the value of k(4.56,3.33) .

The following table shows your monthly payment P, in dollars, if you pay off a loan in months. Which formula below fits these data?

The following table shows some function values. Which formula below fits these data?

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

If you work 43 hours per week, then your weekly pay P , in dollars, is proportional to your hourly wage WWW , in dollars per hour. Express this proportionality relationship as a formula.