MT1710: Calculus 11 January 2017 Test - you have 45 minutes to complete the test Before you start: Have you writen down your candidate number on the front page of your script book? * Is your name and student number written on the right side, which is folded so that is not visible to the marker? 1. Consider the polynomial p(x)=x4+ 2x3 + 2x + 1. (a) Use Descartes Rule of Signs to determine the possible number of zeros of p(x). (8 Marks) (b) Hence, prove that p(x) has just two real zeros and find them; if there is a non- integer zero, it is sufficient to locate the consecutive integers between which that zero lies. (16 Marks) 2. Are the following statements true or false? In each case justify your answer. Given that f(x) = 23-2x2+2x-1 x4+2x3+2x+1 (a) the only vertical asymptote of the graph of y = f(x) is x = 1; (8 Marks) (8 Marks) (b) the graph of y = f(x) has a horizontal asymptote y = 1; (c) the graph of y = f(x) has a slant asymptote y = x + 2. (10 Marks) 3. Find the real solution(s) of the equation 2 sinh x - cosh x = eâ„¢ - 2. 4. Find the general solution of the differential equation (10 Marks) x dy = 2y + x3e-", x > 0. 5. Give the general solution of the differential equation d2y dx2 dy dx - 5 + 4y = 3eâ„¢ (20 Marks) (20 Marks) F. Mota-Furtado