MATECB2MAINEXAM2021

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School

University of Johannesburg**We aren't endorsed by this school

Course

FEBE RKE3B

Subject

Mathematics

Date

Dec 30, 2024

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Uploaded by MajorFlowerVulture41

ENGINEERING CALCULUS 2B1DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICSUNIVERSITY OF JOHANNESBURGASSESSMENT:FINAL EXAMINATION - 10/2021MODULE CODE:MAT0CB2/MATECB2TOTAL POINTS:40ASSESSORS:DR. M. SIASMODERATOR:PROF. R. PANTDR. A. GOSWAMII•Before writing the test, make sure you have read the following honesty declarationand agree with it.•Answer all questions. Show all the steps of your work.Honesty DeclarationBy writing this test, you confirm that you have not•committed academic misconduct in any form;•committed plagiarism;•helped or attempted to help another student in preparing theirsubmission for the assessment;•misrepresented someone else’s work as your own;•obtained help or attempted to obtain help from another person;•obtained help or attempted to obtain help from any source ofinformation, except for explicitly provided lecture notes by themodule assessor;•made use of solutions or answers produced on websites.

Q1.[2×5=10]Check whether the following statements are true or false. WriteTif it is true or writeFif it is false. Do not include explanations.(i) Iff(x,y)=p4-x2-4y2, thenfx(1,0)=1√2[F;-1√3](ii) The gradient of the functionf(x,y)=x2lnyat(3,1)is9j.[T](iii) IfR=[1,2]×[0,π],thenRRRysin(xy)dA=1.[F; 0](iv)1R0s2R0cos(s3)dtds=12sin3.[F;13sin1.](v)F(x,y)=(xy+y2)i+(x2+2xy)jis a conservative vector field.[F].Q2.[5]Find the extreme values offsubject to both constraints.f(x,y,z)=x+y+z,x2+z2=2,x+y=1.Q3.[4]Use polar coordinates to combine the following sum of integrals into a single doubleintegral and then evaluate the integral.Z10Z√4-x2√1-x2sin(x2+y2)dydx+Z√30Z√4-y21sin(x2+y2)dxdy.Q4.[5]Consider the volume represented by the following triple integral.V=Z2-2Z√4-x2-√4-x2Z√4-x2-y2-√4-x2-y2dzdydx-Z1-1Z√1-x2-√1-x2Z√1-x2-y2-√1-x2-y2dzdydx.1

(i) Explain, in words, the represented volume.(1)(ii) Rewrite thefirst term onlyin the orderdxdzdy.(2)(iii) RewriteVinspherical coordinatesusing only one triple integral.(2)Q5.[5]Evaluate the integral by making an appropriate change of variables:ZZRx-yx+y+22dA,whereRis the square enclosed by the linesx-y=-1,x+y=-1,x-y=1andx+y=1.Q6.[6]Show thatF(x,y)=4x3y2-2xy3,2x4y-3x2y2+4y3is conservative and find the work done byFin moving a particle along the pathCgiven byr(t)=ht+sinπt,2t+cosπti; 06t61.Q7.[5]Letr=hx,y,ziand letr=|r|.Determine the value(s) ofpsuch that the vector fieldF=rrphas divergence zero.2