MAT137Y1 Study Guide - Midterm Guide: Infimum And Supremum, Bounded Function, Upper And Lower Bounds

• School: UTSG
• Department: Mathematics
• Course: MAT137Y1
• Professor: All

Document Summary

1: [6 points] (a) calculate (cid:90) e2x dx. + c (b) find a function f satisfying: f (cid:48)(x) = x + 1 for all x, f (1) = 1. 2: [6 points] (a) calculate the area of the region bounded between the curves y = x4 and y = 1. Your answer: the area is 8/5. (cid:90) 1 (cid:18) (cid:0)1 x4(cid:1) dx = (cid:19) (cid:18) Area = (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)x=1 x= 1 (cid:18) (cid:19) x x5. 5 (b) given the function g(x) = (cid:90) 3 sin t. Your answer: g(cid:48)(1) = 2 sin(2) (cid:90) 2x2. [4 points] let r be the region bounded by the x-axis and the curve y = 1 x2. Calculate the volume of the solid of revolution obtained by rotating the region. The region we rotate is as in the picture (cid:90) 1. Using the method of cylindrical shells: (cid:0)1 x2(cid:1)2 dx = (cid:18) x 2x3.