CALCULATOR ALLOWED A. I have the unfortunate delight of hosting a calculus course during the first period of the school day. For many reasons (among them part-time jobs, a major term paper deadline, and a Taylor Swift concert the night before) most of my students have a hard time staying awake in class. (I'm sure it has nothing to do with the way I teach ....) Out of curiosity, one day I pay close attention to the number of my 30 students who are actually paying attention during the 45-minute class period, and I compiled my results in the below chart: Elapsed class time (in minutes) 5 10 15 15 20 25 30 35 40 45 55 Number of students 29 26 22 17 14 11 awake 19 24 28 a) The number of students awake is a function of t, which we'll call it A(t). Using the data in the table, approximate A'(15) and show your computation with the correct units. What does this value represent in the context of the problem? b) Use your answer from part (a) to write the equation of the tangent line at t=15. c) Use your answer from part (b) to estimate the number of students awake at time t =18 minutes. Answer appropriately. d) Calculate the slope of the line connecting (5,29) and (45,28) and explain what its value represents in the context of the problem. e) Use the trapezoidal method (with trapezoids of width 5) to approximate the average number of students paying attention on the time interval [5,45].