final-1004-2005 - MATH 1004 Final Examination December 2005...

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MATH 1004 Final Examination December 2005 1 Multiple-Choice Questions Please choose only one answer and insert in PENCIL in your Scantron sheet. 1. [3 marks] Evaluate lim x 1 x 3 1 x 2 1 . (a) 1 (b) 1 / 2 (c) 3 / 2 (d) 0 2. [4 marks] Let f ( x ) = (ln x ) x . Evaluate f ( e ), where e = 2 . 718 . . . is Euler’s number. In other words, find the derivative of f at x = e . (a) f ( e ) = 0 (b) f ( e ) = 1 (c) f ( e ) = 1 (d) f ( e ) = 2 3. [3 marks] Let f ( x ) = 3 | x 1 | . Calculate L = lim h 0 f (1 + h ) f (1) h . 4. [4 marks] Find the derivative of the function f defined by f ( x ) = 1 + (ln x ) 2 . (a) ln x x 1 + (ln x ) 2 (b) 1 2 1 + (ln x ) 2 (c) 1 2 x 1 + (ln x ) 2 (d) ln x 2 x 2 1 + (ln x ) 2
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2 MATH 1004 Final Examination December 2005 5. [3 marks] Let f ( x ) = 3 x x 3 . Evaluate f (1). In other words, find the derivative of f at x = 1. 6. [3 marks] A differentiable function f with a differentiable inverse, F , has the property that f (1) = 2 and F (0) = 1. What is the value of the derivative of the inverse of f at x = 0? That is, calculate F (0).
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