Semester test 1- 2023

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

Course

MATH APM 1A2E

Subject

Mathematics

Date

Dec 22, 2024

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Uploaded by JudgeCobraMaster1211

FACULTY OF SCIENCEDEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICSMODULEMATENB1APPLICATIONS OF CALCULUS FOR ENGINEERSCAMPUSAPKASSESSMENTSEMESTER TEST 1DATE 26/08/2023TIME 08:30ASSESSOR(S)DR K SEBOGODIDR S SINGHDURATION 90 MINUTESMARKS 50SURNAME AND INITIALSSTUDENT NUMBERCONTACT NUMBERNUMBER OF PAGES: 1 + 11 PAGESINSTRUCTIONS: 1. ANSWER ALL THE QUESTIONS ON THE PAPER IN PEN.2. NO CALCULATORS ARE ALLOWED.3. SHOW ALL CALCULATIONS AND MOTIVATE ALL ANSWERS.4. IF YOU REQUIRE EXTRA SPACE, CONTINUE ON THEADJACENTBLANK PAGE AND INDICATE THIS CLEARLY.

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20231/11Question 1[10 marks]For questions 1.1 – 1.10, chooseonecorrect answer, and make a cross (X) in the correct block.Questionabcde1.11.21.31.41.51.61.71.81.91.101.1 Which statement is true for the Second Derivative Test:“Letfbe continuous nearc . . .(1)(a) Iff′(c) = 0 andf′′(c)>0, thenfhas a local minimum atc.”(b) Iff′(c) = 0 andf′′(c)>0, thenfhas a local maximum atc.”(c) Iff(c) = 0 andf′′(c)<0, thenfhas a local minimum atc.”(d) Iff(c) = 0 andf′′(c)<0, thenfhas a local maximum atc.”(e) None of the above1.2 How many critical numbers doesg(x) =x2e−3xhave?(1)(a) 0(b) 1(c) 2(d) 3(e) None of the above1.3 Consider the integralZx4cos 2x dx. If we use integration by partsto evaluate this integral and chooseu=x4, thenv=(1)(a) cos 2x.(b)−2 sin 2x.(c)12sin 2x.(d)−2 cos 2x.(e) None of the above

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20232/111.4Zsin 3xcos 2x dx=(1)(a) sin 5x+C(b)12[sin 5x+ sinx] +C(c) cos 3xsin 2x+C(d)−110cos 5x−12cosx+C(e) None of the above1.5 Choose the correct partial fraction decomposition for the given rational function:(1)x+ 4x4+ 5x2+ 4(a)Ax+Bx2+ 1+Dx2+ 4(b)Ax+Bx2+ 1+Cx+Dx2+ 4(c)Axx2+ 1+Bxx2+ 4(d)Ax2+ 1+Bx2+ 4(e) None of the above1.6 Choose the correct result for the given integralZdx√25−16x2(1)(a)14sin−14x5+C(b)14tan−14x5+C(c)14csc−14x5+C(d)14cot−14x5+C(e) None of the above1.7 Which of the following is the slant asymptote of the functionf(x) =x2+ 7x.(1)Select the correct answer.(a)y=x3(b)y=x2(c)y= 1(d)y=x(e) None of the above

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20233/111.8 Letfbe a function that is differentiable at everyx∈R. Consider the following:(1)(i) By the extreme value theorem max{f(x)|2< x <3}exists.(ii)By Fermat’s theorem, iffhas a local maximum atx=cthenf′(c) = 0.(iii) Sincefis continuous it follows that max{f(x)|2< x <3}= max{f(2), f(3)}.Select the correct option:(a) Only statement (i) is true(b) Only statement (ii) is true(c) Only statement (iii) is true(d) Only statements (i) and (ii) are true(e) None of the above1.9 Letfandgbe functions satisfying 0≤f(x)≤g(x) for eachx∈R.(1)Select the statement that is always true:(a) IfZ∞1f(x)dxconverges then so doesZ∞1g(x)dx.(b) IfZ∞2g(x)dxdiverges then so doesZ∞2f(x)dx.(c) IfZ∞3g(x)dxconverges then so doesZ∞3f(x)dx.(d) IfZ∞1f(x)dxdiverges thenZ∞1g(x)dxconverges.(e) None of the above1.10 Which of the following is true aboutf(x) =(x+ 2)2x.(1)Select the correct answer.(a) Domain: (−∞; 0)∪[0;∞)(b) There is noy−intercept(c) The function is odd(d) The function is periodic(e) None of the above

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20234/11Question 2[7 marks]Evaluate the following integrals:(a)Z120cos−1x dx(4)(b)Ztan5xsec3x dx(3)

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20235/11Question 3[5 marks](a) Evaluate the following integralZ√1 + cosx dx(2)(b) Letp >1 be a constant. Prove thatZ∞11xpdxconverges.(3)

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20236/11Question 4[4 marks]EvaluateZθ2(θ2+a2)3/2dθ

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20237/11Question 5[3 marks]IntegrateZx+ 1x2−3x+ 2dx

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20238/11Question 6[2 marks]State Rolle’s Theorem.Question 7[4 marks]Find the number(s)cin the interval [0,2π] satisfying the conclusion of the Mean Value Theoremfor the functionf(x) = cosx−sinx.

MATENB1 SEMESTER TEST 1 – 26 AUGUST 20239/11Question 8[4 marks]Letfbe a function that is differentiable atx=c.Iffhas a local maximum atx=cthen prove thatf′(c) = 0.(4)

MATENB1 SEMESTER TEST 1 – 26 AUGUST 202310/11Question 9[11 marks]Letf(x) =x2x−2.(a) Find the domain off.(1)(b) Find thex−andy−intercepts and state them as points.(1)(c) Find the horizontal, slant and vertical asymptotes ofy=f(x) if they exist.(3)

MATENB1 SEMESTER TEST 1 – 26 AUGUST 202311/11(d) Determine the intervals of increase and decrease and find the local maximumand/or minimum values off. Given:f′(x) =x(x−4)(x−2)2.(4)(e) Sketch the graph off.(2)