MATH 2B Study Guide - Midterm Guide: Riemann Sum, Antiderivative, Solution Set

• School: UC-Irvine
• Department: Mathematics
• Course: MATH 2B
• Professor: All

Document Summary

Math 2b, winter 2015, lecture j: midterm 1. We know f(cid:48)(x) has to be an antiderivative of f(cid:48)(cid:48)(x), so f(cid:48)(x) = tan x + c1 for some constant c1. The initial condition f(cid:48)(0) = 1 then gives us the equation tan 0 + c1 = 1, which implies c1 = 1 (since tan 0 = 0), and so f(cid:48)(x) = tan x + 1. (b) find f (x). Since f (x) is an antiderivative of f(cid:48)(x), we want to calculate the inde nite integral of our answer to (a). To integrate the rst term, we will rewrite it as sin x cos x and make the substitution u = cos x so that du = sin x dx. We nd (cid:90) f (x) = (tan x + 1) dx (cid:90) (cid:90) (cid:90) sin x (cid:90) 1 cos x tan x dx + 1 dx u dx + x + c du + x + c.