MATENB1 Sick test 1

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MATH APM 1A2E

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Dec 22, 2024

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Uploaded by JudgeCobraMaster1211

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICSMODULEMATENB1APPLICATIONS OF CALCULUS FOR ENGINEERSCAMPUSAPKASSESSMENTSEMESTER SICK TEST 1DATE 11/09/2023TIME 08:00ASSESSORSDR 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 SICK TEST 1 – 11 SEPTEMBER 20231/11Question 1[10 marks]For questions 1.1 – 1.10, chooseonecorrect answer, and make a cross (X) in the correct block.Questionabcde1.11.21.31.41.51.61.71.81.91.10For 1.1 and 1.2 below, considerP(x)Q(x)=x3−7x2+ 4xx2−5x+ 1=S(x) +R(x)Q(x).After long division:1.1S(x) =(1)(a)x−2(b)x+ 2(c)x+ 1(d)x−1(e) None of the above.1.2R(x) =(1)(a) 7x+ 2(b) 2x−7(c)−2x+ 7(d)−7x+ 2(e) None of the above.

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20232/111.3 When we use integration by parts to evaluate the integralZx·2xdx=, we chooseuanddvsuch that(1)(a)du=x·2x−1dxandv=x22.(b)du=dxandv=2xln 2.(c)du= 2xln 2dxandx2.(d)du=dxandv= 2x+1.(e) None of the above.1.4 To evaluate the integralZcot4xcsc6x dx=, we use(1)(a) the identity 1 + csc2x= cot2xand the substitutionu= cotx.(b) the identity 1 + cot2x= csc2xand the substitutionu= cscx.(c) the identity 1 + cot2x= csc2xand the substitutionu= cotx.(d) the identity 1 + cot2x=−csc2xand the substitutionu= cscx.(e) None of the above.1.5 Choose the correct trigonometric substitution for the given integralZ√a2−b2x2dx(1)(a)x=abtan(θ),−π2≤θ≤π2(b)x=abcos(θ),−π2≤θ≤π2(c)x=absin(θ),−π2≤θ≤π2(d)x=absec(θ),−π2≤θ≤π2(e) None of the above.1.6 Why does the Mean Value Theorem not apply tof(x) =|x−3|on [1,4]?(1)(a)f(x) is not continuous on [1,4](b)f(x) is not differentiable on (1,4)(c)f(1)̸=f(4)(d)f(1)> f(4)(e) None of the above.

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20233/111.7 Choose the correct partial fraction decomposition for the given rational function:(1)t3−2t2+ 2t−5t4+ 4t2+ 3(a)At+B(t2+ 1)(t2+ 3)(b)At+B(t2+ 1)+C(t2+ 3)(c)At+B(t2+ 1)+Ct+D(t2+ 3)(d)A(t2+ 1)+C(t2+ 3)(e) None of the above.1.8 Consider the following integrals:(1)I1=Z104dxex−10I2=Z8−8dx((x+ 1)−7)((x+ 1) + 7)Select the correct option:(a) Only the integralI1is improper.(b) Only the integralI2is improper.(c) IntegralsI1andI2are improper.(d) NeitherI1norI2are improper.(e) None of the above..1.9 Letfbe a differentiable function and suppose thatf(1) =f(7). Select the statement thatis true:(1)(a) There exists a numberc∈(1,7) such thatf′(c) = 0.(b) By Rolle’s Theorem there exists a numberc∈(1,7) such thatf′(c) = 0.(c) By the Intermediate Value Theorem there exists a numberc∈(1,7) such thatf(c) = 0.(d) By Fermat’s Theoremfhas a local minimum or local maximum on (1,7).(e) None of the above.1.10 Letfbe a twice-differentiable function ofxsuch that, whenx=c,fis decreasing, concaveup, and has anx-intercept. Which of the following is true?(1)(a)f′(c)< f′′(c)< f(c)(b)f(c)< f′(c)< f′′(c)(c)f′′(c)< f(c)< f′(c)(d)f′(c)< f(c)< f′′(c)(e) None of the above.

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20234/11Question 2[23 marks]Evaluate the following integrals:(a)Zπ20sin5xcos3x dx(3)(b)Z∞−∞e−|x|dx(4)

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20235/11(c) Use trigonometric substitution to show thatZdx√x2+b2= lnx+√x2+b2+C(4)

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20236/11(d) EvaluateZ10x2−2x(2x+ 1)(x2+ 1)dx(4)

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20237/11(e)Zx3√1 +x2dx.Compute this integral by integrating by parts. Letu=x2.(4)Question 3[2 marks]Letfandgbe functions that are strictly positive and increasing(i.e. For eachx∈R, f(x)>0, f′(x)>0 andg(x)>0, g′(x)>0).Consider the functionFdefined byF(x) =f(e3x)g(2x+ 1).Prove thatFhas no critical points.

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20238/11Question 4[4 marks]Prove the Mean Value Theorem, that is, prove that:Iffis a function that satisfies the following hypotheses:1.fis continuous on the closed interval [a, b].2.fis differentiable on the open interval (a, b).Then there is a numbercin (a, b) such thatf′(c) =f(b)−f(a)b−a

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 20239/11Question 5[4 marks]Use Rolle’s Theorem to show thatf(x) =x36+x22+x+ 1 has exactlyone real root.

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 202310/11Question 6[11 marks]Letf(x) =x√2−x2.(a) Find the domain off.(1)(b) Find the intercepts off(x).(1)(c) Determine the intervals of increase and decrease and find the local maximum and/or mini-mum values off.(4)

MATENB1 SEMESTER SICK TEST 1 – 11 SEPTEMBER 202311/11(d) Given:f′′(x) =2x(x2−3)(2−x2)3/2.Discuss the concavity of the graph and find the inflection point(s).(3)(e) Sketch the graph off.(2)