# 31B_mid1 - MATH 31B Lecture 4 Fall 2006 Midterm 1 NAME...

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MATH 31B - Lecture 4 - Fall 2006 Midterm 1 - October 27, 2006 NAME: STUDENT ID #: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 6 problems for a total of 100 points. 1

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2 1. (10 points) Let f ( x ) = 4 x + cos x . Find the value of the derivative ( f - 1 ) 0 (2 π ). Solution. Note that f ( π/ 2) = 2 π and f 0 ( x ) = 4 - sin x . Thus ( f - 1 ) 0 (2 π ) = 1 f 0 ( π/ 2) = 1 / 3 . 2. (20 points) Evaluate the integral Z π/ 4 0 sin 3 x cos 2 x dx Solution. Z π/ 4 0 sin 3 x cos 2 x dx = - Z cos x =1 / 2 cos x =1 (1 - cos 2 x ) cos 2 x d (cos x ) = - 1 / 3 cos 3 x + 1 / 5 cos 5 x | cos x =1 / 2 cos x =1 = - 1 3 ( 1 2 2 - 1) + 1 5 ( 1 4 2 - 1) = - 5 15 ( 1 - 2 2 2 2 ) + 3 15 ( 1 - 4 2 4 2 ) = - 10(1 - 2 2) + 3(1 - 4 2) 60 2 = - 7 + 8 2 60 2 = 16 - 7 2 120
3 3. (20 points) Evaluate the limit lim x →∞ x (ln( x + 5) - ln x ) Solution. We observe that lim x →∞ 1 x = 0 and lim x →∞ ln( x + 5 x ) = 0.

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