Midterm2 - Math 31B Lecture 4 Fall 2011 Exam#2 9 November...

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Math 31B, Lecture 4, Fall 2011 Exam #2, 9 November 2011 Name: ID Number: Section and TA: You have 50 minutes for the exam. No calculators, phones, notes, or books allowed. You must show your work for credit. Hint: n =0 cr n = c 1 - r if | r | < 1 and c 6 = 0. Question: 1 2 3 4 5 6 7 Total Points: 20 20 10 20 10 10 10 100 Score:
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0 1 2 3 4 0 0.2 0.4 1. Consider the above plot of f ( x ) = x e - x on the interval [0 , 4]. (a) Draw in the trapezoids used to approximate the integral of f ( x ) with the midpoint rule M 4 . You should have four trapezoids, which you can number from the left as 1, 2, 3, and 4. State which trapezoids overestimate the integral and which underestimate the integral. Write down the M 4 approximation to the integral (which is the sum of the area of the four trapezoids), but do not sum the terms. (10 points) The four trapezoids are those tangent to the curve at points 1 2 , 3 2 , 5 2 , 7 2 . 1,2 overestimate, while 3, 4 underestimate the integral. M 4 = 1 2 e - 1 2 + 3 2 e - 3 2 + 5 2 e - 5 2 + 7 2 e - 7 2
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(b) Evaluate the integral R N a x e - x d x. Hint: Integrate by parts. You will need this in question 2. (5 points) Integrate by parts with u = x and v 0 = e - x Z N a x e - x d x = - x e - x | N a + Z N a e - x d x = a e - a - N e - N - e - N + e - a . (0.1) (c) Find a bound for R 4 0 x e - x d x - M 4 . Note that the absolute value of ( x e - x ) 00
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