MATH101 Study Guide - Final Guide: The Sketch, Osculating Plane, Ibm System P

• School: University of Alberta
• Department: Mathematics
• Course: MATH101
• Professor: All

Document Summary

Solutions for math 101 2007 final: part (a). Z x p4x (cid:0) x2 dx =z x p4 (cid:0) 4 + 4x (cid:0) x2 dx =z. Introducing the substitution x q4 (cid:0) (x (cid:0) 2)2 dx: x (cid:0) 2 = 2 sin ((cid:18)) =) dx = 2 cos ((cid:18)) d(cid:18); leads to. We have cos ((cid:18)) d(cid:18) x dx = 2z 1 + sin ((cid:18)) cos ((cid:18)) q4 (cid:0) (x (cid:0) 2)2. = 2z 1 + sin ((cid:18)) d(cid:18) = 2 [(cid:18) (cid:0) cos ((cid:18))] + c. = 2 sin(cid:0)1(cid:18) x (cid:0) 2 dx (cid:20)z 1. 2(cid:12)(cid:12)(cid:12) dx = (cid:0)2 x(cid:0) 1 sin2 (x) px (1 + x) 2 dx and is therefore convergent: part (a). To begin with, let us observe that x2 ln (2 + 3x) = x2 ln(cid:20)2(cid:18)1 + 2 x(cid:19)(cid:21) = ln (2) x2 + x2 ln(cid:18)1 + =) x2 ln (2 + 3x) = ln (2) x2 (cid:0) xn+3 n + 1.