Name______
MAT 131 – Calculus for Non-Science MajorsMarch 29, 2012
Professor PestieauExam 2
Multiple-Choice Questions[5 pts each]
Circle the correct answer for the following questions.
1.Which of the following is the derivative function of ?
a)b)
c)d)
2.Suppose a profit function has the cubic form , where are constants. Then which of the following expressions gives the marginal profit when the production level is set to?
a)b)
c)d)
3.What is the slope of the tangent line to the curve at ?
a)0b)
c)d)12
4.Suppose is a revenue function that gives the total revenue,in dollars, generated from the sale ofluxurywatches. Then what is the marginal revenue (in dollars per item) when 4of these watches are sold?
a)$1,876b)$2,150
c)$2,372 d)$3,272
5.Suppose a cost function is given by , where corresponds to a production level and is in dollars. Then what is the approximate cost for manufacturing the 101th item?
a)$202b)$268
c)$376d)$456
6.At what rate is the quantity changing at , rounded to the nearest hundredth?
a)-0.97b)-1.12
c)-1.17d)-2.03
7.Suppose a cost function is given by , where corresponds to a production level and is in dollars. Then what is the average costwhenthe production level is set to 200?
a)$71.60 per itemb)$73.80 per item
c)$471.60 per itemd)$473.80per item
Show ALL your work for the problems below.
Problem 1[20 pts]
Let be the function defined for all real numbers by
.
a)Find the derivative function of using the definition of the derivative (i.e. by computing the limit of the difference quotient as ).
b)Factor and confirm your previous result by using the Product Rulefor differentiation.
Problem 2[15 pts]
The number of Facebook members from the start of 2006 to mid-2008 can be approximated by
,()
where is in millions and is the year since 2004. Here represents the start of 2004.
a)According to this model, how fast was Facebook membership changing at the start of 2008? Interpret this result.
b)According to this model, did Facebook membership…
(A)…increase at a faster and faster rate
(B)…increase at a slower and slower rate
(C)…decrease at a faster and faster rate
(D)…decrease at a slower and slower rate
during 2007 (i.e. the interval [3, 4])? Circle the correct choice above and justify your answer below.
Problem 3[20 pts]
Suppose you are the President of the Italian club on campus and you are organizing a fundraising “Pasta Night,” where you will charge students $1.10 per serving of pasta. The school cafeteria estimates that the cost of producing servings of pasta at the event is given by
,
where is in dollars.
a)Find the marginal revenue and profit functions.
b)Compute the revenue, profit, marginal revenue, and marginal profit if you have produced and sold 200 servings of pasta. Interpret your results.
c)For which value of is the marginal profit zero? Interpret this result.
Problem 4[5 pts]
Suppose represents the profit on the sale of DVDs. If and , then what can you conclude about future profits?
Problem 5[15 pts]
Russia’s crude oil production during the period 2000 – 2005can be approximated by
,()
where is in millions of barrelsper day and is the year since 2000. Here represents the start of 2000.
In addition, the spot price of crude oil during that same 5-year period can by approximated by
,()
where is in dollars per barrel.
Based on these models, how much money can Russia expect to earn from crude oil revenues in 2003?