TEST IIIMath 232November 21, 2002 Name:By writing my name I swear by the honor code.Read all of the following information before starting the exam:• Show all work, clearly and in order. You will not ge t full credit if I cannot see how you arrivedat your answer (even if your final answer is correct).• Make sure that you follow the directions in each problem and that your answer matches what isasked for.• Justify your answers algebraically whenever possible. For most problems, work done by calculatorwill not receive any points (although you may use your calculator to check your answers).• Please keep your written answers brief; be clear and to the point. I will take points off forrambling and for incorrect or irrelevant statements.• This test has six problems and is worth 100 points, plus some extra credit at the end. Make surethat you have all of the pages!• Good luck!1. (8 pts) Determine whether each of the following statements is true (T) or false (F).a. (2 pts) T F The function A(x) =Zx0(3 − t) dt is increasing on [0, 3].b. (2 pts) T F If f0(x) = F (x) thenZbaf(x) dx =F (x) ba.c. (2 pts) T F If f is negative on [−3, 2], then −Z2−3f(x) dx is negative.d. (2 pts) T FZ11 +√xdx = 2√x − 2 ln |1 +√x| + C.2. (20 pts) Fill in the blanks and circle answers, as appropriate.a. (3 pts) If f(4) = 6 andZ4−1f0(x) dx = 8, then f(−1) = .b. (3 pts) The function A(x) =Zx23sin t dt is a composition of functions A(x) = g(h(x)) with:g(x) = and h(x) = .c. (3 pts) To solve the integralZ1(x − 5)2+ 9you would start by using the trigonometricsubstitution:= .d. (3 pts) Write down tw o integrals that will have the formReudu after a substitution of variables.e. (7 pts) The Riemann s um below approximates the area under the graph of some functionf(x) from x = a to x = b. Determine the type of approximation and identify f(x),a, b, N, ∆x, and xk.4Xk=1(1 +k2)2(12)Circle one: (LHS) (RHS) (Midpoint) (Trapezoid)N = , ∆x = , xk= .f(x) = , a = , b = .f. (1 pt ) Write any word in this blank for one point: .3. (24 pts) Solve each of the following integrals. Show all work clearly and in order.a. (8 pts)Ztan5x sec3x dxb. (8 pts)Z4 + x2xdxc. (8 pts)Zx2(1 − x2)32dx4. (8 pts) Find the exact average value of the function f (x) =1x2on the interval [2, 5].5. (16 pts) This problem concerns the definite integralZ31x exdx.a. (8 pts) Approximate the definite inte gral above using the Midpoint Sum approximation withN = 4 rectangles. (Note: You do not have to use any sigma notation to do this problem!)b. (8 pts) Find the exact value of the definite integral above (by using the Fundamental Theoremof Calculus and integration techniques).6. (24 pts) Solve each of the following integrals. Show all work clearly and in order.a. (8 pts)Ztan3x dxb. (8 pts)Zsin−1x dxc. (8 pts)Zsin4x dxSurvey Questions: (2 extra credit points)Name a question or topic that could have been on this test, but wasn’t.How do you think you did?SPACE FOR SCRAP