Answer all questions using strategies taught in calculus class

1. Given the function M(t) = 2t - 3t - 36t, find the critical values and determine, using both the second derivative test and a sign chart, the nature of these critical values. 2. A projectile is launched with a velocity of 27 m/s at 55 to the ground. Determine its horizontal and vertical velocities. . . . . . . . ... .................. 550 .. ...... 3. Two trains start from the same point at the same time, one going east at a rate of 40 km/h and the other going south at 60 km/h, as shown in the diagram at right. Find the rate at which they are separating after 1 h of travel Train A 40 km/h Train B 50 km/h Distance Between Trains 4. A professional basketball team plays in a stadium that holds 23,000 spectators. With ticket prices at $60, the average attendance had been 18,000. When ticket prices were lowered to $55, the average attendance rose to 20,000. Based on this pattern, how should ticket prices be set to maximize ticket revenue? 5. Higher-order derivatives are derivatives that are taken found after a previous derivative has been taken. For example, we can take the first derivative (y') and the second derivative (y") and the 3rd derivative (y") if we are given the original function (y). Quite often, multiple higher-order derivatives can be found. Higher-order derivatives are used in Science (physics) and mathematics (Differential Equations). A differential equation is an equation that contains functions and some higher-order derivatives. Using a formal proof structure (LHS = RHS), prove the differential equation xy' - 3y + y" + 4 = 2x2 given the function y = x3 - 2x2 + 3x. 6. A 5,000 m rectangular area of a field is to be enclosed by a fence, with a moveable inner fence built across the narrow part of the field, as shown. The perimeter fence costs $10/m and the inner fence costs $4/m. Determine the dimensions of the field to minimize the cost 7. The following table displays the historical number of HIV diagnoses per year 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 Desmos, Excel or Curve Expert or another curve modelling program, determine an equation that can be used to model this data. b. Using this model, estimate the number of diagnoses in 1996 and in 2006. C. At what rate would the number of diagnoses be changing in 2006? d. Halfway through 2006, the number of new HIV diagnoses was found to be 1232. Assuming this rate stays fairly constant for the remainder of the year, does this new information change the modelling equation? If so, how would this change your answer to part (c)? If you were an advocate for furthering HIV and AIDS research and treatment programs, would you be encouraged or discouraged by these results