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1007_Test3_Sol

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

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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

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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 ( 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 = 0 or x = - 2 or x = +2 f (0) = 16 > 0 CU local minimum f ( - 2) = - 32 < 0 CD local maximum f (+2) = - 32 < 0 CD local maximum 6. [2 marks]
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1007_Test3_Sol - MATH 1007 A Test#3 November 3 2006...

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