practice math exam version 1

practice math exam version 1 - Name: _ Section Number: _ TA...

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Name: _____________________________ Section Number: _____ TA Name: ___________________________ Section Time: _____ Math 10A. Final Examination December 8, 2006 Turn off and put away your cell phone . You may use any type of calculator , but no other electronic devices during this exam . You may use one page of notes , but no books or other assistance during this exam . Read each question carefully , and answer each question completely . Show all of your work ; no credit will be given for unsupported answers . Write your solutions clearly and legibly ; no credit will be given for illegible solutions . If any question is not clear , ask for clarification . Please note : there are multiple versions of this examination . Answers appropriate for a version different from your own will be considered as evidence of academic dishonesty . # Points Score 16 24 34 46 56 66 74 81 0 46 Σ
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1. Find dy dx using implicit differentiation: (a) (3 points) x 4 y 5 – 5 xy = 100. Notation : We shall use the abbreviation y for dy dx . To proceed, we shall use the product rule. 4 x 3 y 5 + 5 x 4 y 4 y – 5( y + xy ) = 0 y (5 x 4 y 4 – 5 x ) = 5 y – 4 x 3 y 5 . Solving for y , we have: 35 44 54 55 yx y y x ′ = . (b) (3 points) cos( y ) = xy . -sin( y ) y = y + xy y (-sin( y ) – x ) = y . Solving for y , we have: sin( ) sin( ) yy y y xy x == −+ .
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2. (4 points) A crate open at the top has vertical sides, a square bottom, and a volume of 108 cubic feet. If the crate has the least possible surface area, find its dimensions. What does this crate look like? We draw a picture: h b b We want to minimize surface area. The equation for surface area of a crate without a top is given by A = 4 bh + b 2 . (Otherwise, it would have been A = 4 bh + 2 b 2 .) We are given a constraint, namely V = hb 2 = 108. Using the constraint equation, we can solve for h in terms of b . Thus, h = 108/ b 2 . Substituting our expression for h into the formula for area, we get: 22 2 108 432 () 4 Ab b b b bb ⎛⎞ =+ = + ⎜⎟ ⎝⎠ . To find where this is minimized, we look at the derivative, set it equal to 0 and look for local minimum(s).
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This note was uploaded on 10/20/2008 for the course MATH 10A taught by Professor Arnold during the Spring '07 term at UCSD.

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practice math exam version 1 - Name: _ Section Number: _ TA...

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