1007_Test3_Sol

1007_Test3_Sol - MATH 1007 A Test #3 November 3, 2006...

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Unformatted text preview: MATH 1007 A Test #3 November 3, 2006 SOLUTIONS 1. [3 marks] Sketch the graphs of the functions f ( x ) = e x and g ( x ) = ln x on the same set of axes. Label at least 1 point on each curve. 2. [2 marks] Simplify each of the following expressions (i) e ln( x 2 ) Solution: e ln( x 2 ) = x 2 (ii) ln ( 1 e ) Solution: ln ( 1 e ) = ln( e- 1 ) =- 1 3. [3 marks] Please label each of the following statements as True or False: (i) If f ( c ) = 0, then f has a local maximum or minimum at c . True False (ii) If f ( x ) < 0 for 1 < x < 6, then f is decreasing on (1 , 6). True False (iii) If f ( x ) = g ( x ) for 0 < x < 1, then f ( x ) = g ( x ) for 0 < x < 1. True False 4. [1 mark] Prove that f ( x ) = x 5 + 3 x 3 is an odd function. Solution: f (- x ) = (- x ) 5 + 3(- x ) 3 =- x 5- 3 x 3 =- ( x 5 + 3 x 3 ) =- f ( x ) Since f (- x ) =- f ( x ), f is odd. 1 5. [2 marks] Let f ( x ) = 8 x 2- x 4 = x 2 (8- x 2 ) f ( x ) = 16 x- 4 x 3 = 4 x (2 + x )(2- x ) f 00 ( x ) = 16- 12 x 2 = 4(4- 3 x 2 ) Use the second derivative test to determine the local maximum(s) and local minimum(s) of f . (No marks for any other method.) Solution: f ( x ) = 0 = x...
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This note was uploaded on 03/22/2010 for the course MATH 1007 taught by Professor Unknown during the Fall '07 term at Carleton.

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1007_Test3_Sol - MATH 1007 A Test #3 November 3, 2006...

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