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Math_135_Test_2_Key - Math 135 Test#2 Fall 2007 Name KEY-1...

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Math 135 Test#2 Fall 2007 Name KEY ------------------------------------------------------------------------------------------------------------------------------------- 1- Find the relative maximum, relative minimum, and points of inflection of 4 3 ( ) 3 4 1 f x x x = - + and then graph it. (Hint: Use Dr. G table for the y x and also for y x ). (12 points) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 4 3 ' 3 2 '' 2 ' 3 2 2 3 4 1 12 12 36 24 0 12 0 or 1 12 0 12 1 0 1 3 1 0 0 4 1 f x x x f x x x f x x x f x x x x x f f x x = - + = - = - = - = - = = = - + = = = + ' - + 0 1 0 0 1 0 ] ] Z x f f - - ( 29 f x has a local min at ( 29 1,0 ( 29 ( 29 '' 2 0 36 2 2 0 or 4 2 3 12 3 0 f x x x x x x x = D - - = = = ( 29 2 2 2 36 24 2 11 0 2 3 3 1 3 7 f f � � � � � � = = - = � � � � � � � � � � � � '' 2 0 3 0 0 11 1 27 x f f - + + + - ( 29 f x has inflection points at ( 29 2 11 0,1 , 2 27 ( 29 ( 29 2 is concave up on ,0 , 3 f x - �ȥ ( 29 2 is concave down on the interval 0, 3 f x 1
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2- Find the absolute maximum and minimum of ( 29 2 4 x f x x + = on [1, 4]. (8 points) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] [ ] ( 29 ( 29 ( 29 ( 29 ( 29 2 ' ' 2 2 2 ' 2 2 2 2 2 2 ' 2 2 2 2 2 4 4 4 2 4 1 2 4 4 0 0 4 0 4 But since 1,4 , 2 only. Critical values Step 1: 4 2 Step on the interval 1,4 1 4 1 5 2: Step 3: 1 4 2 2 4 5 x x x x f x x x x x x x x x f x x x x x x x f x x x x x f f f x + = + - + - + = = - - = = - = = - = = = + = = = = - = [ ] ( 29 ( 29 ( 29 ( 29 has abs maximum On the i at 1,5 & 4 Conclusion it ha nte s ab rval 1 smimim , u 4 ,5 , ,4 , mat 2 f x 3A- A soup company is constructing an open-top, square-based, rectangular metal tank that will have a 32 ft 3 volume. What dimensions will minimize surface area? What is the minimum surface area? (6 points) Only Set up. Let x =# of feet for minimized length and width ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 The box: size of square bottom size of 4 sides, equal dimensions Surface (bottom) 4 sides 4 Given: 32 3 32 4 4 4 2 32 128 xx xy S V V lwh
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