University:
University of Central Florida
Course:
MAC 2312 | Calculus with Analytic Geometry II
Academic year:

2023
Views:

332

Pages:

2
Author:

Maria

HOME WORK 3 3 4x Q1: -e dx = lim t-0 S-e dx 1-00 t =[(3)+(1)] lim e e 4 4 4x 13 lim Lex 13-00 4 t Taking the limit, we see that ast-1-∞ =) The integral converges and e 4 18 -ex dx = -e12 100 Q2: ∫ 10xedx Leca) and ga) be functions with continuous derivatives on ja, bj. Then, the theorem for integration by parts for definite integrals involving these two functions is udeu The acronym LIATE stands for Logarithmic Functions, Inverse Trigonometric Functions, Algebraic Functions, Trigonometric Functions, and Exponential Functions. The mnemonic suggests the priority for choosing the sa function. L'Hospital's Rule (oooo) Case. Suppose fand gare differentiable functions over an open interval containing a, except possibly at a. Suppose lim f(z)=oc and limg(x)=200. Then f(x) lim (2)6 (2) () assuming the limit on the right exists or is oo orou. This result also holds if the limit is infinite, if a toc, or if the limit is one-sided We start by rewriting the integral as a limit 10kze dr lini 10re da lim 10ne da This integral contains an exponential function e and an algebraic function 102. Using LIATE we choose 10z and dee da. We can now determine du and o 10r de=de Substitute the expressions above into the formula. 10ze delim ((102) () () (10))+( +lim (10)()()(10) de) Um (10)-(10) dr), lim ((102)-(10) de) lim (10 (-1))+lim (10(-1)) -10-lim 10 (t-1)+lim 10 (t-1)+10 lim (10 ((0)-1)-10 ()-1)) + lim (10 (()-1)-10 ((0)-1)) Taking the limit, we see that as +10 (t-1)+oc. Therefore the integral diverges. du = 10 dz da Q3: §2e3x dx = lim f2edx 18 -3x -2e +7-00 3 = lim -20-3(0) Le 3 -3t + 2 e 3 00-47 It = lim = 17-00 -2 + 2 est 3 e Taking the limit, 1, we see that as t→ -∞, Zest) 00 3 => The integral diverges. t Q4: S-xexdx = lim S- 6 6 4=-x du=-1dx -xexdx dv=edx v=-e-x = um xex / - Sexdx = lim t lim (tet-betétte) 6 6