Math 150B S22 Midterm 2 Solutions

California State University Northridge MATH 150B: Calculus II, Spring 2022 Midterm Exam 2 Solutions March 14, 2022. Duration: 65 Minutes. Instructor: Dr. Jing Li Student Name: Student number: Instructions: • Take your time to read the entire paper before you begin to write, and read each question carefully. Remember that certain questions are worth more points than others. Make a note of the questions that you feel confident you can do, and then do those first: you do not have to proceed through the paper in the order given. • This exam contains 4 problems on 7 pages (including this title page) for a total of 110 points. • You have 65 minutes to complete this exam. • This is a closed book exam, and no notes of any kind are allowed. No calculators are allowed. The use of cell phones, pagers or any text storage or communication device is not permitted. • The correct answer requires justification written legibly and logically: you must convince me that you know why your solution is correct. Answer these questions in the space provided. Show all work that is used to derive your answers. Use the backs of pages if necessary. • Where it is possible to check your work, do so. • Good Luck! Page 2 3 4 5 6 7 Total Points 20 20 20 10 20 20 110 Your Grades Math 150B Midterm 2 Student Name: Student ID: Question 1. [20 points] Evaluate the following integrals. Z 16x 2 a. (10 points) dx (x ° 6)(x + 2)2 Solution Usingpartial fraction Ifan fo tf we get AMYIBY.fi t tcn En Now Wehave 16 3 A4 44 Bix 6 x 2 01 6 pick x 6 we get 16.62 A1642 pickx 2 Weget 16 27 01 2 6 A 9 8 c 9.2 1316 12 pick x 0 Weget 0 2 18 1 6 13 7 therefore IEggedx 91 hfixblunt b. (10 points) Z Eg 6 tf t Eye dx 9hlx61 7hlay 8 way c 2 2x + x + 1 dx (x ° 1)(x 2 + 1) Solution Usingpartial fraction IIII YE we get AMIIEI.ge Now we have ANID BR Bx tox C AND Bx c ex1 At B It l B tox CA c we Equatingthe likepowerterms on both sides of the aboveequation get 2 4 1 Bt 0 subA 1 24 4 Atc 3 2 into A 2 we get 13 0 sub A 2 into we get c l therefore SEF.fi dx f1ftf dx zenlx il tank t c t c Math 150B Midterm 2 Student Name: Student ID: 3 Question 2. [20 points] Evaluate the following improper integral or state that they diverge. You must write the integrals as limits. Z1 e 3x a. (10 points) dx 6x °1 1 + e SolutionWefirstwrite III dx IEfxdxtf.iexdx aliI Efx Tofindan anti derivativefor Weuse substitution u Sff dx Stfudu tf du Itai e then du 3e'dx ttantle ut c feIdxtbl5yfb e'ax 3du t c therefore 1 541 41tangy Is54sHaiti b. (10 points) Solution 0 ing sin t tiny'sfan le x sin sin I IF 3 dx bing.S.ba B tan 1 dx p 9 ° x2 Wefirstnote that Standx sin Z3 tantalite t s 153 S.ba y I o sin tannery D then we rewrite xdx lb 153 o sine big sin b dx Math 150B Midterm 2 Student Name: Student ID: Question 3. [30 points] Evaluate each series or state that it diverges. 1 µ 4 ∂°k X a. (10 points) 2 3 k=0 Solution Wenotethat 214 0455 214 then Eng y Thisis a convergent geometric series since its ratio r Notethat the firstterm a Therefore the sum 0445 b. (10 points) k=1 µ 2 when ko and irk l is ÉR 1 X 4 1 1 ° k +6 k +7 Ir Ey 4 8 ∂ Solution Wenotethatthegivenseries is atelescopingseries because thesecondpart each termcancels with the firstpartof the succeeding term n of Wethen check its first terms sum Sn E that E L t t I Therefore Eh ta ness hi t int int f Math 150B Midterm 2 Student Name: c. (10 points) Solution Student ID: 5 √ µ ∂ µ ∂k ! 2 k 5 3 °2 5 7 k=0 1 X Weexaminethe two series É 3 5 and É2157 individually E IDFor 315 thefirsttermoftheseries is a theratio r f and w al therefore É315 77 Et For thefirsttermoftheseries is theratio r 57 and in a l therefore EEE 35 I Er a 215 Eq 3 5 2 7 Thus É 315 E4 8 315 EEE I 5 7 2 Math 150B Midterm 2 Student Name: Student ID: Question 4. [40 points] Determine if the series converge or diverge. a. (10 points) Solution 2k 2 + 1 2 k=1 3k ° 2k + 1 1 X Isak 1 44 5 test BytheDivergence we Solution YEI and We notethat ak b. (10 points) 6 0 conclude that the giver series diverges 1 X 1 2 k=2 k(ln k) By the IntegralTest We let fix Note that fix is continuous Sifu dx SEdy dx To find an antiderivative for xp positive and decreasing for x 2 Because ftp.dx blimp Hp we let u thx andthen du I xpdx fated tut c ut c therefore Sig dx bliss pdx bist tux l hence Weconclude that the given series dx t converges bigtubthe ut Math 150B Midterm 2 Student Name: c. (10 points) Student ID: 7 1 X 1 p k=1 k k + 3 Solution Use the comparison Test note that at for kal Because the É is a p series and p 32 21 it converges we conclude that given series converges d. (10 points) Solution 1 X k=3 5 k ° 3k Usethe Limit comparison fight List Note that and the series Fts 1 E E t Test with the series whos II converges because therefore the given series also be É Is It 1 it is a geometric series with converges