TEST III Math 232 November 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 get full credit if I cannot see how you arrived at 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 is asked for Justify your answers algebraically whenever possible For most problems work done by calculator will 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 for rambling and for incorrect or irrelevant statements This test has six problems and is worth 100 points plus some extra credit at the end Make sure that you have all of the pages Good luck 1 8 pts Determine whether each of the following statements is true T or false F Z x a 2 pts T F The function A x 3 t dt is increasing on 0 3 0 b 2 pts T F If f 0 x F x then Z a c 2 pts T F b b f x dx F x a Z If f is negative on 3 2 then 2 f x dx is negative 3 Z d 2 pts T 2 F 1 dx 2 x 2 ln 1 x C 1 x 20 pts Fill in the blanks and circle answers as appropriate Z 4 a 3 pts If f 4 6 and f 0 x dx 8 then f 1 1 Z b 3 pts The function A x x2 sin t dt is a composition of functions A x g h x with 3 g x and h x Z c 3 pts To solve the integral 1 you would start by using the trigonometric x 5 2 9 substitution d 3 pts Write down two integrals that will have the form R eu du after a substitution of variables e 7 pts The Riemann sum below approximates the area under the graph of some function f x from x a to x b Determine the type of approximation and identify f x a b N x and xk 4 X 1 k2 2 12 Circle one LHS N x RHS Midpoint xk Trapezoid k 1 f x f 1 pt Write any word in this blank for one point a b 3 24 pts Solve each of the following integrals Show all work clearly and in order Z a 8 pts tan5 x sec3 x dx Z 4 x2 dx x Z x2 b 8 pts c 8 pts 3 1 x2 2 dx 4 5 8 pts Find the exact average value of the function f x Z 16 pts This problem concerns the definite integral 1 on the interval 2 5 x2 3 x ex dx 1 a 8 pts Approximate the definite integral above using the Midpoint Sum approximation with N 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 Theorem of Calculus and integration techniques 6 24 pts Solve each of the following integrals Show all work clearly and in order Z a 8 pts tan3 x dx Z sin 1 x dx Z sin4 x dx b 8 pts c 8 pts Survey 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 WORK