M AT H 1 2 0 Differential and Integral Calculus Fa l l 2 0 1 6 - P r o b l e m S e t # 2 Due Wednesday, September 28 Write your name and student number very clearly in the upper right hand corner of your front sheet and staple your sheets if necessary. You are encouraged to collaborate with others when working on your homework assignments, but you must write up solutions independently, on your own. Do not copy the work of others. In accordance with academic integrity regulations, you must acknowledge in writing the assistance of any students, professors, books, calculators, or software. Be sure that your final write-up is clean and clear and effectively communicates your reasoning to the grader. 1) If X and Y are sets, then the union of X and Y is the set X Y consisting of all elements that are in either X or Y (or possibly in both X and Y). The intersection of X and Y is the set X Y consisting of all elements that are in both X and Y. The empty set, denoted by , is the set that contains no element. Two sets X and Y are disjoint if X Y = . Describe each of the following subsets of R as a union of disjoint intervals. \b \b (a) A = x R | x2 + 4x + 13 < 0 x R | 10x2 + 7 > 0 \u001a \u001b 3x + 1 (b) B = { x R | ( x + 2)( x 1)( x 5) < 0} x R | 0 x2 (c) C = { x R | 8| x 1| > x | x + 2|} 2) Suppose that a and b are any real numbers. The goal of this problem is to prove the lower bound part of the triangle inequality: || a| |b|| | a + b|. Throughout this question you are allowed to assume the part of the triangle inequality that we did prove in class. (a) For any two numbers c and d, explain why the inequality |c| d is equivalent to the the two inequalities c d and c d. (This means that you ought to show that if |c| d, then c d and c d and, on the other hand, if both c d and c d holde, then you should show that |c| d.) (b) Use the equation a = ( a + b) b and the upper bound triangle inequality (the one we proved in class) to prove that | a| |b| | a + b|. (c) Similarly show that |b| | a| | a + b|. (d) Finally, prove that || a| |b|| | a + b|. 3) Find the natural domains of the functions: r q p 1 2 f (x) = 1 4 x , g( x ) x , x s h( x ) = \u0012 sin( x ) \u0013 1 sin( x ) , 2 r(x) = x4 1 . x1 \f