MATH 102 - Review Sheet for Exam #1 - Fall 2016-17 The exam will be given during the common exam slot (8:00-8:50 AM) in the Bossone Auditorium on Friday 10/21 and it will cover sections 11.1, 11.2, 11.3, 11.4, 12.1, 12.2, and 12.3. It is recommended that you arrive at least 10 minutes before the exam is scheduled to begin to get yourself settled in and it is requested that you sit every other seat. In order to make sure that your exam winds up in the appropriate location upon submission, please make sure that you know your recitation section (this is the number starting with 0, not the letter). IDs may be checked at random, so make sure that you have yours with you. You should have a working non-graphing scientific calculator at the exam and it is recommended that you check on the condition of it (batteries, etc.) as you will NOT be allowed to borrow or to use a non-approved one. If you do not have an approved calculator at the exam, you will need to take the exam without use of one and IF YOU ARE CAUGHT TRYING TO USE AN UNAPPROVED CALCULATOR TYPE FOR THE EXAM, YOUR EXAM WILL BE IMMEDIATELY CONFISCATED AND YOU WILL RECEIVE A 0 FOR THE EXAM! for 0 one problem on the exam and this formula will be given to you in the instructions to that problem (it is STRONGLY recommended that you do not attempt to use the limit definition for any problem besides the one for which it is required. No other formulas will be given to you and you will not be allowed to have any notes or scrap paper out during the exam. You will be specifically asked to use the limit definition of the derivative, This sheet is intended to be used as an aid in studying for the exam and it should not be assumed that all problem types on the exam will be represented here or that the exam will be of the same length and/or have the same number of problems. In preparing for the exam it is a good idea to do the problems on this review sheet and to review the example problems done in the lectures, the assigned problems on the syllabus, and the assigned MML problems. The solutions for the problems on this review sheet are listed at the end. 1) Use the limit definition of the derivative to find the derivative for each of the following: 5 a) 4 3 b) 2) Consider the function a) Find the average rate of change from 4 to 8. b) Find the equation of the secant line passing through the points 4, 3) Consider the function 12 a) Find the instantaneous rate of change at 8. b) Find the equation of the line tangent to the graph at 4 and 8, 8 . 8. 4) Find the following limits: a) lim b) lim c) lim 3 d) lim 100 e) lim 8 f) lim 2 g) lim 0 4 5 18 5) A company training program has determined that, on average, a new employee produces after days of the on-the-job training where . Find and interpret lim Ps . s items per day 6) Given that 2 6 5 1 5 1 5 5 , answer the following: 1 ? a) What is b) What is lim f x ? x 1 5 ? c) What is d) What is lim f x ? x 5 e) For what value(s) is 7) Use the below graph of discontinuous? to find the following: a) 1 b) lim f x c) lim f x d) lim f x e) 3 f) lim f x g) lim f x h) lim f x i) lim f x x x 1 x 3 x 1 x 3 x 1 x 3 8) Identify where each of the following functions is continuous: 18 a) 1.98 18 b) 9) Find the derivative of each of the following (DO NOT SIMPLIFY): 6 a) 4 b) 2 c) d) 8 6 5 10 e) f) 5 10) Assume that the total number of bacteria (in millions) present in a culture at a certain time (in hours) is given by 2 8 32. Find (and interpret) the rate at which the total number of bacteria is changing at the following times: a) 3 hours b) 10 hours 11) Suppose that the profit (in hundreds of dollars) from selling x units of a product is given by: Find and interpret the marginal profit when 2 units are sold. Solutions 10 5 1a) 5 10 1b) 4 5 4 3, lim 0 4, 10 , 10 lim 0 , 2a) 1/5 b) 3a) 1 b) 16 4a) 0 b) -1/9 c) 99 5 4 , 4 , d) 1/20 e) 5/8 f) 0 g) DNE 5a) If a new employee was able to train for an infinite amount of time, the number of items that they could produce per day approaches (but never equals or exceeds) 55. 1 6a) 0 b) DNE c) 0 d) 0 e) 7a) 2 b) 2 c) 2 d) 2 e) 4 8a) All real values of x. 9a) b) 48 4 11 2 f) 4 g) 2 h) DNE b) All real values of x except x = -12 and x = 3. 2 5 2 2 5 3 i) c) d) e) / / 8 / 14 / 5 20 / 6 5 10 / 8 f) / / / 10a) 3 10 million/hr, In the time from 3 hours to 4 hours, the number of bacteria will decrease by approximately 10,000,000. b) 10 88 million/hr, In the time from 10 hours to 11 hours, the number of bacteria will increase by approximately 88,000,000. 11) 2 hundreds of dollars/unit = $48/unit. The profit generated from the sale of the next (specifically the 3rd) item will be approximately $48