samplemt2 - MIDTERM 1 MATH 100 SECTION 109 WEDNESDAY OCT 7...

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MIDTERM 1: MATH 100, SECTION 109, WEDNESDAY OCT. 7 QUESTION 1: [4 marks] Below you are given the graph of y = f 0 ( x ) for some function y = f ( x ) . Graph the function y = f ( x ) assuming that f (0) = 1 . –3 –2 –1 0 1 x Figure 1. The graph of y = f 0 ( x ) Solution to (1): Figure 2. The graph of y = f ( x ) QUESTION 2: [6 marks] Using the rules of differentiation find the derivatives of the following functions. DO NOT SIMPLIFY YOUR ANSWERS. (a) f ( x ) = x 2 + 2 x x 3 - x - 1 . (b) f ( x ) = 1 x + 1 . (c) f ( x ) = ( x - 2 + x 2 ) ( x - x - 1 ) . Date : October 2, 2002. 1
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2 MIDTERM 1: MATH 100, SECTION 109, WEDNESDAY OCT. 7 Solution to (2): (a) f 0 ( x ) = ( x 3 - x - 1)(2 x + 2) - ( x 2 + 2 x )(3 x 2 - 1) ( x 3 - x - 1) 2 . (b) f 0 ( x ) = - ( x + 1) - 2 1 2 x . (c) f 0 ( x ) = ( - 2 x - 3 + 2 x )( x - x - 1 ) + ( x - 2 + x 2 )(1 + x - 2 ) . QUESTION 3: [4 marks] Let g ( x ) = ( f ( x )) 3 . You are given that f (0) = 2 and f 0 (0) = - 1 . Find g 0 (0) . Solution to (3): By the chain rule g 0 ( x ) = 3( f ( x )) 2 f 0 ( x ) . Putting x = 0 , and using f (0) = 2 and f 0 (0) = - 1 , gives g 0 (0) = - 12 . QUESTION 4: [6 marks] Let f ( x ) = x 4 - 2 x 2 , -∞ < x < . (a) Determine where f ( x ) is increasing and where it is decreasing.
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