Solved Assignments and Study Material for Desklib

1. The number of major crimes committed in the city of Bronxville between 1987 and 1994 is
approximated by the function
Where denotes the number of crimes committed in year t (t=0 corresponds to 1987).
Encouraged by the increase in the crime rate, the citizens of Bronxville, with the help of the local
police organized “Neighborhood Crime Watch” groups in 1991 to combat this menace.
a) On the interval in question ( ) did the crime rate ever decrease? [3 Marks]
b) On the interval in question ( ) did the crime rate break (inflection point)? If so,
what year [3
Marks]
c) Is the “Neighborhood Crime Watch” program working? Explain your answer [2
Marks]
Solution 1a: Given function is . Differentiate with respect to t we get
Since, at and at and at
That is, the crime rate decreases from 0 to 8 and then decrease from 8 to 9. Therefore crime rate
decreases in .
Solution 1b: Differentiate with respect to t we get,
Solve .
Since, for , , this implies that function is concave upward
And for , , this implies that function is concave downward.
Hence, is the inflection point.
Solution 1c: Yes, “Neighborhood Crime Watch” program working. Because the crime rate
decreases in this means that this will also decreases for

2. The average speed of a vehicle on a stretch of route 184 between 6 A.M and 10 A.M on a
typical weekday is approximated by the function
Where is measured in miles per hour and t is measured in hours, with corresponding to
6 A.M.
a) At what time in the morning is the traffic moving at the slowest rate? [6
Marks]
b) What is the average speed of a vehicle at that time? [2 Marks]
Solution 2a: Given that the average speed of a vehicle on a typical weekday is approximated by
the function
Differentiate with respect to t we get,
For maximum, minimum rate,
So, the critical points are
Now,
At
At
At
From the above calculation, it is observe that the traffic is slowest at that is at 7 A. M. in the
morning is the traffic moving at the slowest rate.

Solution 2b: Since, the traffic is slowest at , so the average speed at that time is 27 miles per
hour.
3. Based on current production techniques, the rate of oil production from a certain oil well t
years from now is estimated to be thousand barrels/year. Based on a new
production technique, however, it is estimated that the rate of oil production from that oil
well t years from now will be t thousand barrels/year. Determine how
much additional oil will be produced over the next 10 years if the new technique is
adopted. [8 marks]
Solution 3: Given
Additional oil will be produced over the next 10 years if the new technique is adopted is
4. The weekly demand for the Pulsar 25 LED television set is
Where denotes the wholesale unit price in dollars and denotes the quantity demanded.
The weekly total cost function associated with manufacturing the Pulsar 25 is given by
Where denotes the total cost incurred in producing sets
a) Find the revenue function and the profit function . [1 mark]
b) Find the marginal cost function, the marginal Revenue function, and the marginal
Profit function. [2 marks]