# Final - MAT 21B Final Exam Wednesday Last Name First Name...

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MAT 21B Final Exam, Wednesday, March 20, 2013 Last Name: First Name: Student ID #: Discussion Section Time: (Circle 3pm, 4pm, 5pm, 6pm, or 7pm) Name of Left Neighbor: Name of Right Neighbor: If you are next to the aisle or wall, then please write “aisle” or “wall” appropriately as your left or right neighbor. • Read each problem carefully. Write every step of your reasoning clearly. • Usually, a better strategy is to solve the easiest problem first. • This is a closed-book exam. You may not use the textbook, crib sheets, notes, or any other outside material. Do not bring your own scratch paper. Do not bring blue books. • No calculators/laptop computers/cell phones are allowed for the exam. The exam is to test your basic understanding of the material. • Everyone works on their own exams. Any suspicions of collaboration, copying, or otherwise violating the Student Code of Conduct will be forwarded to the Student Judicial Board. Problem # Score 1 (10 pts) 2 (10 pts) 3 (10 pts) 4 (10 pts) 5 (10 pts) 6 (10 pts) 7 (10 pts) 8 (10 pts) 9 (10 pts) 10 (10 pts) Total (100 pts)
2 Problem 1 (10 pts) Consider the following simple definite integral: Z 1 0 x 2 d x = 1 3 . (a) (5 pts) Find the numerical approximation of the above integral using the trapezoidal rule , T n , for arbitrary n N , and simplify the resulting expression so that you have a formula for T n in terms of n (and no summation symbols or other variables).
(b) (3 pts) Define the absolute error E T n : = fl fl fl fl 1 3 - T n fl fl fl fl . Then, find the upper bound of E T n derived from the error estimate theorem involving the second derivative of the integrand, and show that this upper bound is achievable in this problem using the result of Part (a). Z b a f ( x )d x states that E T n