MAT102H5 Study Guide - Final Guide: Jyj, Joule

• School: UTM
• Department: Mathematics
• Course: MAT102H5
• Professor: Shay Fuchs

Document Summary

Mat102s - introduction to mathematical proofs - utm - spring 2011. Solutions to selected problems from problem set b. To prove that(cid:12)(cid:12)|x| |y|(cid:12)(cid:12) |x y| we use the triangle inequality |a + b| |a| + |b| twice as follows: Now since(cid:12)(cid:12)|x| |y|(cid:12)(cid:12) must be equal to either |x| |y| or |y| |x| we conclude that(cid:12)(cid:12)|x| |y|(cid:12)(cid:12) |x y| . If we set a = x y and b = y we get |x y + y| |x y| + |y| which becomes |x| |y| |x y| . If we set a = y x and b = x we get |y x + x| |y x| + |x| which becomes |y| |x| |x y| . If r, s are two real distinct solutions of ax2 + bx + c = 0, then from the quadratic formula we have. A direct computation shows that r + s = = b a and rs = b2 (b2 4ac)