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a-cl-linc.2011t - Linear Algebra MAS4105 6137 Prereq-A Prof...

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Linear Algebra MAS4105 6137 Prereq-A Prof. JLF King 24Aug2011 A1: On your own sheets of paper, please write ( double-spaced ) a proof of the following, in complete English sentences. Do not restate the problem. Let L ( n ) := [5 [2 n ] ] - 1. By induction on k , prove that k N : L ( k ) is a multiple of 3. A2: Show no work. NOTE : The inverse-fnc of g , often written as g 1 , is different from the reciprocal fnc 1 /g . E.g, suppose g is invertible with g ( 2) = 3 and g (3) = 8: Then g 1 (3) = 2, yet [1 /g ](3) def === 1 /g (3) = 1 / 8. Please write DNE in a blank if the described object does not exist or if the indicated operation cannot be performed. a 2 27 3 = . . . . . . . . . . . . log 8 (4)= . . . . . . . . . . . . b Line y = [ M · x ] + B owns points (3 , 1) and ( 3 , 17). Hence M = . . . . . . . . . . . . . . . and B = . . . . . . . . . . . . . . . . c Quadratic 15 x 2 + 23 x + 6 = [ Ax - α ] · [ Bx - β ], for numbers A = . . . . . , α = . . . . . ; B = . . . . . , β = . . . . . . d Below, f and g are differentiable fncs with f (2) = 3 , f 0 (2) = 19 , f (3) = 5 , f 0 (3) = 17 , g (2) = 11 , g 0 (2) = 1 2 , g (3) = 13 , g 0 (3) = 7 , g (5) = 23 , g 0 (5) = 29 . Define the composition C := g f . Then C (2) = . . . . . . . ; C 0 (2) = . . . . . . . . . . . . . . . . . . . . . . . . . . . . Please write each answer as a product of numbers; do not multiply out. [ Hint: The Chain rule. ] e Let y = f ( x ) := 7 + 3 2 x 5, its inverse-function is f 1 ( y )= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f Let g ( x ) := x 3 + x . Then g 1 (10)= . . . . . . . . . . . . . . .
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