MATH237 Study Guide - Midterm Guide: Squeeze Theorem, Opata Language

• School: University of Waterloo
• Department: Mathematics
• Course: MATH237
• Professor: Saeed Parto

Document Summary

Here are the answers (and some solutions) to the sample midterms. Of course, on the test you are required to always give full solutions. Sample midterm 1 answers: see course notes, we have t = t (x(t), y(t), z(t)) and when t = 0, we get (x(t), y(t), z(t)) = (0, 0, 0). 3 (1 + 3z)2 x(cid:48)(t) = 2 y(cid:48)(t) = cos t z(cid:48)(t) = et. Since t , x(t), y(t), and z(t) are all di erentiable the chain rule gives dt dt (0) = tx(0, 0, 0)x(cid:48)(0) + ty(0, 0, 0)y(cid:48)(0) + tz(0, 0, 0)z(cid:48)(0) = ( 2)(2) + (2)(1) + ( 3)(1) = 5: (a) domain d = {(x, y)|x2 + y2 1}, range is {z|0 z 1}. Sample midterm 2 answers: short answer problems a) lim (x,y) (a,b) f (x, y) = l means for every > 0 there exists a > 0 such that. 0 < (cid:107)(x, y) (a, b)(cid:107) < .