Math 115 - Team Homework Assignment #8, Winter 2017 Due Date: Apr 17-18. (Your instructor will tell you the exact date and time.) This assignment covers material from Chapters 5 and 6 of the course textbook, which is the 6th edition of Calculus: Single Variable by Hughes-Hallett, Gleason, McCallum, et al.. Make sure to understand the material in this chapter before attempting these problems. It is important that you try these problems before your first meeting with your team. Remember, the first meeting with your team is for the members of the team to discuss the arguments used by each member of the team to find their solutions. It is also a time to address any questions that each member of the team may have about the assignment. Remember to follow the guidelines from the \"Doing Team Homework\" and \"Team HW Tutorial\" links in the sidebar of the course website. Do not forget to rotate roles and include a reporter's page each week. This is an assignment that evaluates mathematical skills and your ability to justify carefully your solutions. 1. The first case of a new strain of feline calcivirus (FC) was reported in August 2014. Fortunately for the feline community, a new vaccine that protects cats from being infected with FC was available in the market in January 2015. Consider the following functions: Let C(m) be the number of cats (in thousands) in the region where the FC outbreak is reported m months after August 20141 . Let I(m) be the rate at which cats are being infected (in thousands of cats per month) with FC m months after August 2014. Let V (x) be the rate (in thousands cats per day) at which cats are being vaccinated x days after January 1, 2015. m C(m) I(m) x V (x) 0 1.5 0 320 2.3 2 327 3.5 4 331 4.2 3 1.2 6 0.95 9 0.85 6 335 5.2 12 0.72 8 337 4.7 10 340 4.2 15 0.65 12 345 3.75 18 0.53 14 349 3.45 21 0.48 16 353 3.1 24 0.43 18 358 2.4 27 0.37 20 360 1.9 30 0.35 33 0.32 In questions a)-c), your answers must be written in terms of some (or all) of the functions defined above. (a) Write a mathematical expression for the exact value of the average number of cats infected with FC per month between February 2015 and February 2016. (b) Write a mathematical expression for the number of cats that have not been vaccinated with the by the end of January 2015. (c) Estimate the values of all the mathematical expressions you found in part a) and b). For your estimate in part a) use m = 2 and for part b) use 5 subintervals in your Riemann sums. 1 In this case m = 0 counts as August 2014, m = 1 as September 2014, etc. 1 Z (d) Write a sentence interpreting the meaning of 27 V (z)dz. Assuming that V (x) is an 3 increasing function, how often do you need to measure the values for the function V (x) in Z 27 V (z)dz within 2000 cats using a left-hand sum with order to obtain an estimate of subintervals of the same size? 3 2. Jasper is chasing a red dot along a wall. The wall is 24 feet long, and when the game starts, Jasper is on the left end of the wall and the red dot is in the center. Let j(t) and r(t) be Jasper's and the red dot's velocity, respectively, in feet per second, t seconds into the game, where a positive velocity means movement to the right, and a negative velocity means movement to the left. Below is a graph showing both j(t) (solid line, in blue) and r(t) (dotted, in red) for the first 24 seconds of the game. y 5 4 y = r(t) 3 2 y = j(t) 1 2 4 6 8 10 12 14 16 18 20 22 t 24 1 2 3 4 5 (a) (b) (c) (d) (e) (f) (g) (h) Over what intervals of time is Jasper moving to the right? At what time(s) is Jasper's velocity greatest? At what time(s) is Jasper furthest to the right? At what time(s) are Jasper and the red dot moving at the same velocity? How many times are Jasper and the red dot in the same location during these 24 seconds? How far, total, has Jasper traveled in these 24 seconds? How far from the left end of the wall is Jasper after 24 seconds? After 22 seconds, Jasper stays where he is while the red dot continues to move at a constant velocity. At what time are Jasper and the red dot in the same location along the wall? 2 3. Let f (x) and h(x) be an antiderivative and the derivative of the continuous function g(x) respectively. The graphs below show pieces of the graphs of each of these functions: (a) The graph of f (x) is only shown on 2 x 8. (b) The graph of g(x) is only shown on 1.5 x 7. (c) The graph of h(x) is only shown on 0 x < 2 and 6 < x 10. 3 y y = f (x) 2 1 x 1 2 3 4 5 6 7 8 9 10 6 7 8 9 10 6 7 8 9 10 1 3 y y = g(x) 2 1 x 1 2 3 4 5 1 5 y 4 y = h(x) 3 2 1 x 1 1 2 3 4 5 2 3 4 Use the information on the partial graphs of f (x), g(x) and h(x) in order to: (a) Sketch the possible graphs of f (x), g(x) and h(x) on 0 x 10. Make sure that your graph clearly shows the important features of each function. (b) Find the exact value of each of the following quantities: Z 5 i. (3g(x) 2)dx. 2 Z 4 ii. h(x + 3)dx. 7 Z 4 iii. Assume that g(x) is an odd function. Find (g(t) + f (6))dt. 2 3