SAMPLE MULTIPLE-CHOICE QUESTIONS
1.What is?
Answer: C
a)0b)3
c)d)The limit doesnot exist.
2.Atoy company makes a profit of dollars whenit manufactures and sells toys. If 5toys have been sold already, what profit, rounded to the nearest cent, can the company expect to earn from producing and selling a 6th toy?
Answer: B
a)¢7b)¢8
c)¢9d)¢10
3.If ,then is equal to
Answer: A
a)b)
c)d)
4.The graph of the curve changes concavity at
Answer: D
a)b)
c)d)
SAMPLE PROBLEMS
1.[Temperature Conversion]
There are three widely used scales for measuring temperatures. The vast majority of the world uses the Celsius scale, while the U.S. and other small nations like Belize and Burma still use the antiquated Fahrenheit scale. Scientists, on the other hand, prefer to use the Kelvin scale.
All three of these scales are linearly related.
We know that water freezes at 0° Celsius, or 32° Fahrenheit and boils at 100° Celsius, or 212° Fahrenheit. Moreover, the Celsius and Kelvin scales are related by the formula .
a)Come up with a formula that converts degrees Fahrenheit () to degrees Kelvin ().
b)What temperature is the same in both the Fahrenheit and the Kelvin scale?
2.[Monorail Service in Las Vegas]
In 2005 the Las Vegas monorail charged $3 per ride and had an average ridership of 28,000 people per day. At the end of that year, the Las Vegas Monorail Company (LVMC) decided to raise the fare to $5 per ride. As a result, average ridership plunged to 19,000 people a dayin 2006.
a)Find a linear model for, the monoraildemand in terms of , the price of the monorail fare.
b)Find the price the LVMC should have charged in 2006 to maximize its revenues from ridership.
c)The Las Vegas Monorail Company would have needed $49.9 million in revenues from ridership to break even in 2006. Could that have been a possibility?
3.[Blood Alcohol Level]
You are hosting a party at your house. After several drinks, your friend has reached a blood alcohol level of 440 mg/dL (milligrams per deciliter). If the amount of alcohol in the blood decays exponentially, with one fourth being removed every hour, how long will it take for your friend’s blood alcohol level to decrease to 80 mg/dL and, hence, allow him to legally drive back home?
Here the exponential decay is given by , where is the blood alcohol level in mg/dL, is the time elapsed in hours and are nonzero positive constants.
4.[Average Cost of Soccer Jerseys]
Let the function given below represent the total cost, in dollars, of manufacturing official Team USA jerseys for the 2010 world cup in South Africa.
Show that the minimalaveragecost of a jerseyis less than $200 if the production level is above 4 jerseys. After producing how many jerseys is this minimal average cost reached?
Apply the first-order derivative test to justify your answer.
5.[Inflection Point]
Show that the curve
Has an inflection point at . Is the concavity of the curve changing upwards or downwards at that point? Justify your answer.
6.[Maximizing Profit - Cell Phones]
Worldwide quarterly sales of Nokia cell phones, in millions, were approximately given by
,
where the wholesale price of each phone is $.
Assuming that it costs Nokia $40 to manufacture each cell phone, find the wholesale price that maximizes Nokia’s quarterly profits. What is Nokia’s maximum quarterly profit?
7.[Elasticity of Comic Book Demand]
The demand function for the original Iguanawoman comics is given by
,()
where is the number of copies the publisher can sell per week if it sets the price at $.
a)Is the demand for Iguanawoman comics elasticor inelastic when the price is set at $100 per copy? What if the price is set at $150? Conclude on what the publisher should do as a result in each case.
b)Find the price at which the publisher should sell the comics in order to maximize weekly revenue.
c)What, to the nearest dollar, is the maximum weekly revenue the publisher can realize from sales of Iguanawoman comics?
8.[Mathematical Research]
The number of research articles in the journal Applied Calculus Reviewwritten by mathematicians in Europe can be approximated by
thousand articles per year, ()
where is the numberof years that have elapsed since the start of 1983.
Find , the function that gives the total of research articles that were written by mathematicians in Europe years after 1983, if 2,310articles were already written in the journal Applied Calculus Review at the start of 1983.