Math_137_Winter_2010_Midterm_2_Solutions

Math_137_Winter_2010_Midterm_2_Solutions - Name (Print): UW...

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Unformatted text preview: Name (Print): UW Student ID Number: University of Waterloo Term Test 2 Solutions Math 137 Calculus 1 for Honours Mathematics Date: March 24, 2010 Time: 4:30 p.m. - 6:20 p.m. Number of pages: 7 Test type: Closed Book (including cover page) Circle your section number Instructor Section Lecture Time Shengda Hu 001 (8:30 a.m. - 9:20 a.m.) Paula Smith 002 (11:30 a.m. - 12:20 p.m.) Paula Smith 003 (10:30 a.m. - 11:20 a.m.) Instructions 1. Write your name and ID number at the top of this page. Please circle your section num- ber up above. 2. Answer the questions in the spaces provided, using the backs of pages for overow or rough work. 3. Show all your work required to obtain your answers. FOR INSTRUCTORS USE ONLY Question Mark 1 /8 2 /10 3 /10 4 /12 5 /10 6 /10 7 /8 8 /8 9 /12 10 /12 Total /100 1. Find the following derivatives: (a) d dx (arcsin( x )) d dx (arcsin( x )) = 1 p 1 ( x ) 2 1 2 x = 1 2 x 1 x = 1 2 x x 2 . (b) d dx ((tan x ) sin x ) Let y = (tan x ) sin x . Then ln y = ln(tan x ) sin x = sin x ln(tan x ), and d dx (ln y ) = 1 y dy dx = cos x ln(tan x ) + sin x sec 2 x tan x = (cos x ln(tan x ) + sec x ). Hence dy dx = y (cos x ln(tan x ) + sec x ) = e sin x ln(tan x ) (cos x ln(tan x ) + sec x ) . 2. Find the following limits: (a) lim x + x (ln x ) 2 lim x + x (ln x ) 2 = lim x + (ln x ) 2 1 x = lim x + 2 ln x 1 x 1 x 2 = lim x + 2 ln x 1 x = lim x + 2 1 x 1 x 2 = lim x + 2 x = 0 (b) lim x (1 2 x ) 1 /x lim x (1 2 x ) 1 /x = lim x e ln(1 2 x ) 1 /x = e lim x ln(1 2 x ) 1 /x by continuity of e x ....
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This note was uploaded on 04/13/2010 for the course MATH 137 taught by Professor Speziale during the Spring '08 term at Waterloo.

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Math_137_Winter_2010_Midterm_2_Solutions - Name (Print): UW...

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