Name______
MAT 131– Calculus for Non-Science MajorsMarch 1, 2012
Professor PestieauExam 1 – Models & Limits
Multiple-Choice Questions[5 pts each]
Circle the correct answer for the following questions.
For questions 1, 2 and 3 below, let the functions and be defined as
and.
Compute the following:
1.
a.-5b.0
c.5d.The limit does not exist.
2.
a.3b.
c.d.
3.
a.-3b.0
c.d.
4.
a.0b.2.1
c.4.2d.
For questions 4 and 5 below, assume that you are in charge of your college newspaper, The East Enders, which sellsfor 50¢ a copy, has fixed production costs of $70 per edition and has variable printing & distribution costs of 40¢ per copy.
5.What profit/loss results from the sale of 500 copies of The East Enders?
a.$50 profitb.$180 profit
c.$20 lossd.$70 loss
6.After selling how many copies of The East Enders will you break-even?
a.400 copiesb.500 copies
c.600 copiesd.700 copies
Show all your work on the following problems to receive full credit.
Problem 1[15 pts]
a)Ifyou invest $380,000 in a mutual fund with an annual yield of 8.9% and the earnings are compounded monthly, how long will it take for you to become a millionaire? Round your answer to the nearest month.
The formula for discrete compounding is given by.
b)Your good friend who is a savvy investor suggests that you switch to a different mutual fund with a lower annual yield of 8.7%. He argues that because this mutual fund compoundsyour earnings continuously, it will actually speed up your process of becoming a millionaire. Is your friend correct?
The formula for continuous compounding is given by.
Problem 2[20 pts]
In 2005, the Las Vegas Monorail Company (LVMC) charged $3 per ride and had an average ridership of about 28,000 people per day. In December of 2005, the LVMC raised the fare to $5 per ride and, as a result, ridership in 2006 plunged to around 19,000 people per day.
a)Using the given information, find a linear demand equation ofthe average ridership of the monorail per day() as a function of the fare price().
b)Based on this linear demand, find the price the LVMC should have charged to maximize revenue from ridership. What would have been the corresponding daily revenue with this optimal price?
c)The LVMC would have needed $44.9 million in revenues from ridership to break even in 2006. Would it have been possible for the company to break even in 2006 by charging this optimal price?
Problem 3[10 pts]
The following table shows approximate daily oil production by Pemex, Mexico’s national oil company, for the years 2001 – 2009 ( represents the start of 2001):
(year since 2000) / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9
(in millions of barrels) / 3.1 / 3.3 / 3.4 / 3.4 / 3.4 / 3.3 / 3.2 / 3.1 / 3.0
a)Compute the average rate of change of , the approximate daily oil production by Pemex (in millions of barrels), over the period 2002 – 2007. Interpret your result.
b)Which of the following is true? From 2001 to 2008, the one-year average rates of change of daily oil production by Pemex…
(A)…increased in value.
(B)…decreased in value.
(C)…never increased in value.
(D)…never decreased in value.
Problem 4[15 pts]
There are currently 10,000 cases of a nasty flu going around in a total susceptible population of 120,000. The number of cases, , is increasing by 20% each day at the early stages of this flu epidemic.
a)Find a logistic model of the form , where is the limiting value and are nonzero positive constants.
b)Use your model to predict the number of flu cases 2 weeks from now.
c)According to your model, how long will it take for 40% of the susceptible population to become infested with this flu? [Bonus – 5pts]
Problem 5[15 pts]
The Richter scale is used to measure the intensity of earthquakes. This scale rating of an earthquake is given by the formula
,
where is the energy released by the earthquake (measured in ergs).
a)The San Francisco earthquake of 1989 registered on the Richter scale. How many ergs of energy were released?
b)The recent Chilean earthquake of 2010 registered on the Richter scale. How much more intense was this earthquake in comparison with the San Francisco earthquake of 1989?