The University of the South Pacific Serving the Cook Islands, Fiji, Kiribati, Marshall Islands, Nauru, Niue, Samoa, Solomons Islands, Tokelau, Tonga, and Vanuatu Faculty of Science, Technology & Environment School of Computing, Information and Mathematical Sciences MA111: Calculus 1 & Algebra 1 ASSIGNMENT 2 - SEMESTER 2, 2016 DUE DATE: 12/10/2016 (Week 13 Wednesday) at 5:00pm INSTRUCTIONS: 1. All questions are compulsory. Show all relevant working 2. Please note that this assignment must be done individually 3. You can type your answers on MS word or scan the hand-written answers and upload on MOODLE as a pdf file 4. Your name and ID number must be written on the cover sheet and upload the soft copy of the solution on MOODLE before the due date MA111 Assignment 2 Semester 2, 2016 Question 1 (4 + (3 + 3 + 3 + 4) + (2 + 2 + 3) + 4 + (2 + 2 + 2) = 34 marks) (a) Calculate the domain of the function f ( x ) = x+3 . x2 9 (b) Evaluate the following lomits. x4 (i) lim x 4 x2 2t + 2t5 t+ 3t 3t5 \u0011 \u0010p 3 6 3 x + 5x x (iv) lim (iii) lim x2 4x + 3 x 3 x2 9 (ii) lim x + (c) Consider the piecewise function x, f ( x ) = 3 x, ( x 3)2 , if x < 0 if 0 x < 3 if x 3 (i) Find lim f ( x ) if it exists. x 0 (ii) Show that f ( x ) is continuous at x = 3. (iii) Sketch the graph of f ( x ). (d) Use the definition of derivative to differentiate f ( x ) = 1 x + x2 . (e) Use the technique of differentiation to find the derivatives of the following functions. (i) y = (1 + x ) x. \u0010 \u00114 p (ii) y = 2 + x3 2 . 2 1 x + ex . (iii) y = 10x 1 Question 2 ((4 + 4 + 4 + 3 + (2 + 2 + 2 + 4) + 6 = 31 marks) dy , if 3ey = xy + x2 + y2 . dx x3 4 8x 12 dy , if y = . (b) Use logarithmic differentiation to solve dx (1 + x 2 )2 (a) Use implicit differentiation to find Page 2 of 4 MA111 Assignment 2 Semester 2, 2016 (c) Find the equation of the tangent line to the graph of the function y = x at the point x = 1. x+1 (d) Find the first derivative and second derivative of f ( x ) = x e x . (e) Consider the function f ( x ) = 3x4 4x3 . (i) Find the intervals over which the function f ( x ) is increasing and the intervals on which it is decreasing. (ii) Find the intervals over which the function f ( x ) is concave upwards and the intervals on which it is concave downwards. (iii) Determine the points of inflection of f ( x ). (iv) Sketch f ( x ) showing all intercept, turning points and points of inflection. (f) For the function f ( x ) = 5x3 + 3x5 , find all critical values and determine whether each represents a local maximum, local minimum or neither. Then find the absolute extrema on the interval [2, 2]. Page 3 of 4 MA111 Assignment 2 Semester 2, 2016 Albert Einstein: Math Genius \"'Do not worry about your difficulties in Mathematics, I assure you that mine are greater\"' Page 4 of 4