Paper Code: MATH500 Mathematical Concepts Lecturers: Puna Raass, Amjad Ali Assignment 2 Due: Thursday, 25 May 2017 at 4pm Name ........................................ ID number.......................... Question 1 Marks 10 2 10 3 10 4 10 5 15 6 20 7 10 8 15 Total 100 Score Instructions: Please attach this sheet to the front of your assignment. The assignment must be dropped in the Assignment Boxes on WT Level 1 (City students) or the MB foyer (AUT South). Answer all questions and show your working. No working = no marks. This is an individual assignment. The point of this assignment is for you to go through the process of discovery for yourself. Copying someone else's work will not achieve this. Plagiarism has occurred where a person effectively and without acknowledgement presents as their own work the work of others. That may include published material, such as books, newspapers, lecture notes or handouts, material from the internet or other students' written work. It also includes computer output. The Department of Mathematical Sciences regards any act of cheating including plagiarism, unauthorised collaboration and theft of another student's work most seriously. Any such act will result in a mark of zero being given for this part of the assessment and may lead to disciplinary action. Please sign to signify that you understand what this means, and that the assignment is your own work. Signature: ........................................ 1 Question 1. [10 marks] (a) Determine whether the expression can be the partial sum of an arithmetic or a geometric sequence. Then find the sum using the formula for a partial sum. Show your working. 3 + 3.7 + 4.4 + + 10.7 3 4 + 1 2 + 1 3 + + 8 81 (b) Is the infinite series 34 + of the infinite series. 1 2 + 1 3 + . . . a convergent or divergent series? If it is convergent, find the sum Question 2. [10 marks] (a) Find the equation of the line with y-intercept 15 and slope 21 . (b) Find the equation of the line that passes through the point (1, 1) and has slope 34 . (c) Find the equation of the line that passes through the points (5, 7) and (2, 14). (d) Find the equation of the line with x-intercept 2 and y-intercept 4. (e) Find the equation of the vertical line through the point (4, 3). Question 3. [10 marks] At the beginning of 2006 Facebook membership was 5.5 million, and at the beginning of 2008 the membership had increased to 58 million. (a) Assume that the membership increases exponentially, and find an exponential growth model for Facebook membership. Let t be the time in years with t = 0 in 2006 and F (t) the number of Facebook members in millions at time t. (b) Use your model to estimate Facebook membership in 2017. (c) What is Facebook membership in 2017? (Google it.) How well does your estimate in (b) compare with this number? Question 4. [10 marks] Atmospheric pressure P (in kilopascals, kPa) at an altitude h (in kilometers, km) is given by the expression k ln P0 k ln P = h. P0 is the atmospheric pressure at sea level and k is a positive constant. (a) Use the logarithm rules to rewrite this formula in exponential form. Does atmospheric pressure follow an exponential growth rule or an exponential decay rule? (b) The atmospheric pressure at sea level is 100 kPa and at an altitude of 4 km the pressure is 55 kPa. When a large aircraft is flying at cruising height of approximately 10 km, the cabin pressure is kept equal to the atmospheric pressure at an altitude of 2.1 km. What is the atmospheric pressure at an altitude of 2.1 km? (c) At what altitude will the atmospheric pressure be half of the pressure at sea level? Question 5. [15 marks] Much of the fish sold in supermarkets today is raised on commercial fish farms, not caught in the wild. A pond on one such farm is initially stocked with catfish. The logistic function that models this population is P (t) = 8000 2 + 14e0.05t where P (t) is the number of fish in the pond after t weeks. (a) How many fish are initially released into the pond, according to this model? (b) According to this model, what is the long term prediction for the number of fish in the pond? 2 (c) After how many weeks will the fish population reach 2000 fish? (d) What is the average rate (in fish per week) at which the population is increasing in the first 5 weeks? (e) Sketch a graph of the population size P versus t. Which time interval will you use for the graph? Why? Question 6. [20 marks] (a) There are four points, labeled A to D, on the graph for the function y = x3 + 3x2 + 5x. Find the coordinates for these points to 2 decimal digits, by using derivatives and algebra. Show your working. (b) Find the equation of the tangent to the curve at the point where x = 2. (c) At which other point on the graph, will the tangent be parallel to the tangent at x = 2? Question 7. [10 marks] Use the rules for differentiation to find the derivatives of the following functions. Show your working and simplify your answers. Use correct notation for the derivatives. t4 2 t 6 (Do not use the quotient rule.) (a) f (t) = 2t3 (b) y = 3x4 + 4 7 2x (c) v = (t2 2t) ln t (d) w = (4s3 5s + 2)es Question 8. [15 marks] The fuel efficiency M (in km per litre) when a car is driven at a speed of v km/h is given by M= 1750 . v + 900v 1 (a) Sketch a graph of fuel efficiency M versus speed v for 0 < v < 150 km/h. (b) What is the fuel efficiency when the car is driven at 100 km/h? (c) At which speed should you drive the car in order to have a fuel efficiency of 20 km/litre? (d) At which speed should you drive the car in order to have maximum fuel efficiency? 3 \f\f