Math 132Midterm Examination 1 – February 8, 20126 multiple choice, 4 long answer. 100 points.General Instructions: Please answer the following, without use of calculators. Youmay refer to a 3x5 card, but no other notes. Part I of the exam is multiple choice,while Part II is long answer.Part I Instructions: If you do not have a pencil to fill out your answer card, please askto borrow one from your proctor. Write your Student ID number on the six blank lineson the top of your answer card, and shade in the corresponding bubbles to the rightof each digit.Fill in the bubble corresponding to each of the following 6 questions. Each is worth 4points. On Part I, no partial credit will be given.1. Let xi=3i2n− 1 for i = 0, 1, . . . , n. These xi’s form a partition of the interval:(a) [−1, 0](b) [−1,12](c) [−1, 2](d) [0,12](e) [0, 1](f) [0, 2](g) [0, 3](h) [1,32](i) [1, 3](j) [1,52](k) None of the above.2. Which of the following is equal to19Xi=11i(a) 0 + 1 +12+13+ ··· +119.(b) 0 + 1 +12+13+ ··· +120.(c) 1 +12+13+ ··· +119.(d) 1 +12+13+ ··· +120.(e)Z1911xdx.(f)Z1901xdx.(g)Z2001xdx.(h)Z2011xdx.(i) None of the above.3. Which of the following is an antiderivative of11+4x2?(a)14tan−1(x)(b)12tan−1(x)(c) tan−1(2x)(d)12tan−1(2x)(e) ln(1 + 4x2)(f)14ln(1 + 4x2)(g)12xln(1 + 4x2)(h) 1 + ln(4x2)(i) None of the above.4.ddxZx−πsin t2dx is equal to(a) sin x2(b) cos x2(c) 2x · sin x2(d) 2x · cos x2(e) 0(f) cos x2+ C(g)12x · cos x2(h)12x · cos x2+ C(i) None of the above.5. If f is a continuous function such thatR120f(t) dt = 3,R122f(t) dt = 4, andR42f(t) dt =1, then findR40f(t) dt.(a) −4(b) −3(c) −2(d) −1(e) 0(f) 1(g) 2(h) 3(i) 4(j) None of the above.6. Which of the following definite integrals havenXi=1(1 +in)4·2nas an associated Riemann sum?I.Z20(1 +x2)4dxII.R31x4dxIII.R102 · (1 + x)4dx(a) None of the above.(b) I only.(c) II only.(d) III only.(e) I and II only.(f) I and III only.(g) II and III only.(h) All of the above.Name: Id #: Math 132Part II Instructions: Answer the following on the exam sheet, showing all your work.Correct answers without correct supporting work may not receive full credit. You mayuse the back of each page for additional answer space (please clearly indicate if youhave done so), or scratch work.Please put your name and student id number on each page of Part II now.1. Integration(a) (5 points) EvaluateZ1−1sin πx dx.(b) (5 points) EvaluateZx sin(x2) dx.(c) (5 points) EvaluateZln 3ln 2e2x√1 + exdx.(d) (7 points) Solve the initial value problem: Ifddtf(t) = 8t · (2t2+ 1)4and f(0) = 1,then find f(t).Name: Id #: Math 1322. Areas and volumes(a) (15 points) Find the volume of the object formed by rotatingy =rx1 + x2about the x-axis for 0 ≤ x ≤ 2.(b) (10 points) Find the area between the curves y = sin x and y = cos x for0 ≤ x ≤π2.Name: Id #: Math 1323. The Fundamental Theorem of Calculus(a) (6 points) Using definite integrals, give an antiderivative of sin x2. For full credit,explain clearly what theorems and/or properties of sin x2that you are using.(b) (8 points) FindZπ/20(ddxex sin x) dx.Name: Id #: Math 1324. Riemann sums and definite integrals(a) (5 points) The points1 = x0< x1< x2< ··· < xn−1< xn= 3form the uniform partition of [1, 3]. Find xi, and give the length of each part.(b) (6 points) Give any Riemann sum forZ311xdx. Be sure to explain the choice ofpartition and points that you make.(c) (4 points) In 1-3 sentences, explain why the functionf(x) =(1 if x is rational0 otherwisefrom Worksheet 1 is not integrable on [0,
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