ENR 114 Mathematical Methods for Engineers 1 Assignment 1, Study Period 2, 2016 This assignment is worth 10% of marks for the course. • Show your working. Marks are generally allocated to the various steps through a solution to reach an answer. Show all necessary steps so that the reader can follow your solution procedure. Sometimes a phrase in English is useful to explain your steps. • Write your own solutions. You may discuss your work with other students, but you must write up your solutions yourself. You are not allowed to use anyone else's written work when you are writing up your assignment. • Acknowledge help and joint work. If you receive help from another source (e.g. students, tutors, friends, internet sources) you must make a note of it on your assignment. • Use MATLAB as permitted. You may use MATLAB to check answers but do not use MATLAB to justify your answers (unless asked to do so). See the first dot point on showing working. • Late penalties: Assignments submitted late, without an extension being granted, attract a penalty of 10% of the maximum marks for each day or part thereof beyond the due date and time. Submissions will not be accepted, without prior negotiation, more than one week beyond the due date. 1 Question 1. 5+7=12 marks Solve the following inequality and equation: a) −4 ≤ 3 − 1 2x < 0 b) ||x − 1| − |x + 1|| = x2 Question 2. 2+3+3+4=12 marks Calculate the following limits or show that they do not exist. Do not use l'Hˆopital's rule. a) lim x→3 |1 − x| b) lim x→1 x − 1 √ 3 x − 1 c) lim x→0 x sin 2 cosx x d) lim x→π/2 cos 1 x − tan x Question 3. 5 marks Let f be a function which i is continuous everywhere except at x = −2 and x = 3 ii is differentiable everywhere except at x = −2, x = 0 and x = 3 iii satisfies f 0(x) =          > = 0 for < > 0 for 0 for 0 0 for x < x > −< x < 2 − 3 < x < .2 3 0 iv satisfies f(−2) = f(3) = 2 v has a graph with x-intercept at x = 2 and y-intercept at y = 2. Draw a possible sketch of the graph of f . Question 4. 7 marks Suppose f(t) = √t and g(t) = t2 + 4. Find f(g(t)) and g(f(t)), and give their domains and ranges. 2 Question 5. 6 marks The graph of the equation y = x3 − 9x2 − 16x + 1 has a slope equal to 5 at exactly two points. Find the coordinates of those points. Question 6. 6+4=10 marks Function f(x) is defined as f(x) =        |1 (x x− − −sin 1 1) | 2πx 2 if if 0 if x < x > ≤ x 0 1.≤ 1 a) Does the limit of f(x) at x = 0 or at x = 1 exist? Justify your answer. b) Obtain a plot of f(x) over the interval x ∈ [−2, 2] using MATLAB. Include the code that produced the plot in your submission. Question 7. 8 marks Solve for x ∈ [0, π]: 6 sin x − 4 cos 2x = 1. Question 8. 10 marks Find the equation of the normal to the graph of f(x) = tan √1 + x2 at x = √π2 − 1. total marks: 70 3