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Test1Sol(1804)(08-092nd)

Test1Sol(1804)(08-092nd) - T1Sol/MATH1804/YMC/2009 THE...

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T1Sol/MATH1804/YMC/2009 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1804 University Mathematics A Outline Solution to Test 1 Name : U. No. : Group : Answer ALL 4 Questions Write your answer in the space provided. For full credits, show your steps clearly and give details to your solution. No books and notes are allowed to use. 1. ( 6 points ) (a) Let A = ( - 3 , 2] \ (0 , 1] , B = { x Z : | x +1 | ≤ 2 } and C = { x R : x 10 +6 x 8 + x 2 - 1 > 0 } . Find A B C . (b) Let f ( x ) = - 1 - x 2 for 0 x 1 . Find the inverse function of f ( x ) (with explanation) and specify where this inverse function is defined. Solution. (a) A = ( - 3 , 0] (1 , 2] , B = {- 3 , - 2 , - 1 , 0 , 1 } . Then A B = {- 2 , - 1 , 0 } , and hence A B C = {- 2 , - 1 } (since x = 0 does not satisfy the inequality). (b) The function g ( x ) = 1 - x 2 for - 1 x 0 is the inverse function of f ( x ) because: For 0 x 1 , g f ( x ) = g ( f ( x )) = g ( - 1 - x 2 ) = 1 - (1 - x 2 ) = x 2 = x and For - 1 x 0 , f g ( x ) = f ( g ( x )) = f ( 1 - x 2 ) = - 1 - (1 - x 2 ) = - x 2 = - ( - x ) ( since x is negative ) = x
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Alternative method : Let y = f ( x ) = - 1 - x 2 for 0 x 1 . Then by projecting the graph onto the y -axis, we have - 1 y 0 . Now, y 2 = 1 - x 2 and hence x = 1 - y 2 for - 1 y 0 . Then we replace x by f - 1 ( x ) and y by x , we have f - 1 ( x ) = 1 - x 2 for - 1 x 0 which is the inverse function of f ( x ) . For your reference:
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2. ( 8 points ) (a) Compute lim x 1 2 x - x 4 - 3 x 1 - 3 x 4 . (b) Let f ( x ) = 3 x + k, x < 1 x 2 + x + 1 , x 1 Find the value of k for which f ( x ) is differentiable at x = 1 . (c) If g (1) = 1 and g ( x ) = 1 1 + x 2 , use the Mean Value Theorem to estimate g (2) .
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