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Unformatted text preview: December 2009 Mathematics 100/180 Page 2 of 13 pages Marks [42] 1. ShortAnswer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question. (a) Evaluate lim h (3 + h ) 2 9 h or determine that this limit does not exist. Answer (b) Evaluate lim x ( p 4 x 2 + x 2 x ) or determine that this limit does not exist. Answer (c) Find all values of the constant c that make the function f continuous everywhere, or determine that no such value exists: f ( x ) = sin(4 x ) x if x 6 = 0 , c if x = 0 Answer Continued on page 3 December 2009 Mathematics 100/180 Page 3 of 13 pages (d) Find the derivative of ( t 3 + 2 t ) e t . Answer (e) Find the derivative of y = sin x x 4 . Answer (f) Find f ( x ), if f ( x ) = e cos x . Answer (g) Find the slope of the tangent line to the curve x + 3 y = 5 at the point (4 , 1). Answer Continued on page 4 December 2009 Mathematics 100/180 Page 4 of 13 pages (h) Find y if y = sin 1 ( x 3 ). [ Note: Another notation for sin 1 is arcsin.] Answer (i) Find f ( x ) if f ( x ) = x sin x . Answer (j) Use a linear approximation to estimate (1 . 999) 4 ....
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