SINGLE VARIABLE EXAM QUESTIONS - Loyola University Chicago

PART I

SINGLE VARIABLE EXAM QUESTIONS

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Chapter 1 Exam Questions

Questions and Solutions for Section 1.1

1. The empirical function ?

? ?? , given in the graph to the right, comes from the Wall Street Journal, September 4,

1992. From the graph, describe the domain of this function and the range of this function. In a sentence, apply the general

definition of the word "function" to explain why you think that the given curve is in fact a function.

??? ??? ? ?? ? ?? ? ?? ???? ???? ? ??

? ?? ?? ??

ANSWER:

Domain: September 1989 to August 1992. Range: 2810 (approx.) ? domain, there is a unique value of ? .

? ? (approx.). For every date in the

2. Consider a ten-story building with a single elevator. From the point of view of a person on the sixth floor, sketch a graph indicating the height of the elevator as a function of time as it travels. Remember to indicate when it stops. Try to take into account all types of cases that can happen, but do not worry about every possible situation. (There are many different possible graphs that could be drawn for this.) ANSWER: A possible diagram: An elevator first goes from the ground floor to the third floor, then to the eighth floor, and finally back to the ground floor.

?? ?

?

? ? ?

(Ground floor)

? 3.

Draw room

a is

gartaph?

which accurately represents the temperature of the contents and the cup is originally filled with water slightly above the

of a cup freezing

left overnight point.

in

a

room.

Assume

the

ANSWER:

? (temp)

??

???

?

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4. Suppose the Long Island Railroad train from Easthampton to Manhattan leaves at 4:30 pm and takes two hours to reach Manhattan, waits two hours at the station and then returns, arriving back in Easthampton at 10:30 pm. Draw a graph representing the distance of the train from the Farmingdale station in Easthampton as a function of time from 4:30 pm to 10:30 pm. The distance from Easthampton to Manhattan is 150 miles. ANSWER: distance(miles)

150

?

4:30 pm

6:30 pm

8:30 pm

10:30 pm

? 5. Suppose we buy quantities ? and ?, respectively, of two goods. The following graph shows the budget constraint

?? ?? , where ? and ? are the prices of the two goods and is the available budget. On the graph, draw the

lines that correspond to the following situations, and for each line, give the equation and the coordinates of both intercepts. Label each line clearly.

(a) The budget is doubled, but prices remain the same.

?

?

?? ? ??

(b) The price of the first good is doubled, but everything else remains the same (the available budget is still ). ANSWER:

?

?

?

? ? budget doubled

a) ?? ??

b) d?o?u?b?lin?g????

?

? ?? ? ??

?? ?

? ?

?

? (a)

?

push

? ?

?

the line up,

? . If the prices remain the

keeping the slope the same,

same, the slope but doubling the

of the line

? and ?

remains the intercepts.

same.

If

the

budget

is

doubled,

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? (b) i?s?double??;????-i?nterc?e?p.tWrehmeaninthsethperiscaemoef.the first good is doubled, we get ?

??

?

?

?

?

.

The

slope

of

the

line

? ? ? ? ? ??? ? 6. A function is linear for ? and also linear for ? . This function has the following values: ? ? . Find formula(s) (or equation(s)) which describe this function.

;

;

ANSWER:

? ? ? ? ? ? For ? , the slope is

and the ?-intercept is 1; thus ?

? . For ?

??? ? ? ?? ? ? find the ?-intercept we substitute:

,

, so that

. Hence, ? ?

? ? ? ?? ? ? ?? This is an example of a piece-wise function: ?

?

when ?

?

when ?

? ? , the slope is .

?. To

?

?

?

?

? ? 7. The empirical function ?

? graphed below represents the population ? of a city (in thousands of people) at time ?.

Describe the domain and range of this function.

?? ? ? ? ?? ? ? ? ??

Figure 1.1.1

? ??? ANSWER:

Domain: 1900 to 1980. Range:

?

??? (approximately).

?? 8. A pond has a population of 500 frogs. Over a ten-year period of time the number of frogs drops quickly by , then

increases slowly for 5 years before dropping to almost zero. Sketch a graph to represent the number of frogs in the pond

over the ten-year period of time.

ANSWER:

??frogs

??

?? time (years)

Figure 1.1.2

9. Below is the graph of a function ?

the graph.

5

? ?? that represents the height of water in a reservoir. Write a short story to match

height (ft)

? ? ?

Sun Tue Thu Sat Figure 1.1.3

ANSWER: The height of the water on Sunday is 90 feet. Over the course of the next two days the water level drops slowly to approximately 85 feet. The water level drops more quickly for the next day and a half and reaches its lowest level of 40 feet during Wednesday. The water level then rises steadily to a height of 75 feet on Saturday.

? 10. Suppose a container of water is placed in the freezer overnight. The next morning, it is put on the counter in a ? room

and then at the end of the day heated to the boiling point.

(a) Sketch a graph representing the temperature of the water during the day. (b) Describe the domain and range of your graph from part (a).

ANSWER: (a)

Temp (?F)

???

??? time Ice Boiling Water melts starts boils off

Figure 1.1.4

?? ??? (b)

wThaetedr o(ma?in)

is to

the the

time the container is boiling temperature

put (

o?n).the

counter

to

the

end

of

the

day.

The

range

is

the

temperature

of

frozen

11. A school library opened in 1980. In January, 2000 they had 30,000 books. One year later, they had 30,480 books. Assuming they acquire the same number of books at the start of each month:

(a) How many books did they have in January, 2003?

(b) How many books did they have in July of 1980?

(c) Find a linear formula for the number of books, ? , in the library as a function of the number of years ? the library has

been open.

(d) If you graph the function with domain 1980-2010, describe in words what the ?-intercept of the graph means.

ANSWER:

?? (a) They acquire ? ?? ???

? ? ? ? books per year. In January, 2003 the library will have

more books

than they did in January, 2000 for a total of 31, 440 books.

(b) From part (a), the number of books the library acquires each year is 480. January, 1980 was 20 years before January,

?? ??? ??? ? ?? ?? ?? 2000, therefore the number of books the library had in January, 1980 was

From

? ?? ? ? ? ? ? part (a), the number of books acquired each month is

By July, 1980 the library acquired

?? ?? ? ? ? ?? ? books therefore the total number of books in the library will be ? (c) We find the slope ? and the intercept in the linear equation ?

??. From part (a), we use ? = 480. We

substitute to find :

?? ??? ? ? ??????

The linear formula is ? ?? ?? ? ??.

?? ??

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