1. Given the function M(t) = 2t3 - 3t2 - 36t, find the criticalvalues and determine, using both the second derivative test and asign chart, the nature of these values.
2. A projectile is launched with a velocity of 22 m/s at 50° to theground. Determine its horizontal and vertical velocities.
3. Two trains start from the same point at the same time, one goingeast at a rate of 40 km/h and the other going south at 60 km/h, asshown in the diagram at right. Find the rate at which they areseparating after 1 h of travel.
4. A professional basketball team plays in a stadium that holds23,000 spectators. With ticket prices at $60, the averageattendance had been 18,000. When ticket prices were lowered to $55,the average attendance rose to 20,000. Based on this pattern, howshould ticket prices be set to maximize ticket revenue?
5. Corey is asked to find the maximum value of a function. Nothaving a complete understanding of the process, Corey decides tofind the derivative of the function, set it equal to zero, andsolve. The resulting value, Corey reasons, will yield the maximumpoint. Explain fully why Corey's method is flawed.
6. A 5,000 m_ rectangular area of a field is to be enclosed by afence, with a moveable inner fence built across the narrow part ofthe field, as shown.The perimeter fence costs $10/m and the innerfence costs $4/m. Determine the dimensions of the field to minimizethe cost.
7. The following table displays the number of HIV diagnoses peryear in a particular country.
Year 1997 1998 1999 2000 2001 2002 2003 2004 2005
Diagnoses 2512 2343 2230 2113 2178 2495 2496 2538 2518
a. Using Curve Expert or another curve modelling program, determinean equation that can be used to model this data.
b. Using this model, estimate the number of diagnoses in 1996 andin 2006.
c. At what rate would the number of diagnoses be changing in2006?
d. Halfway through 2006, the number of new HIV diagnoses was foundto be 1232. Assuming this rate stays fairly constant for theremainder of the year, does this new information change themodelling equation? If so, how would this change your answer topart (c)? If you were an advocate for furthering HIV and AIDSresearch and treatment programs, would you be encouraged ordiscouraged by these results?
