# B lim x 0 1 x 1 sin x 4 8 marks find an equation of

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(b) lim x ! 0 + ° 1 x ° 1 sin x ± 4. [8 marks ] Find an equation of the tangent to the curve given below, at the point ° 0 ; 1 2 ± : 6 sin ° 1 ( x + y ) = ° ² x 2 + e xy ³ 2
5. [10 marks ] Consider the function y = f ( x ) = 10 ln x x 2 . (a) Determine if the graph of f has any vertical asymptotes and any horizontal asymptotes, justifying your answers by calculating relevant limits. (b) Determine on which intervals the graph of f is increasing, and on which intervals it is decreasing. You may use the fact that f 0 ( x ) = 10 (1 ° 2 ln x ) x 3 . (c) Determine on which intervals the graph of f is concave up, and on which it is concave down. You may use the fact that f 00 ( x ) = 10 (6 ln x ° 5) x 4 . (d) Use the information found in parts (a) to (c) to sketch a graph of y = f ( x ) . Label any key features of the graph. -2 -1 1 2 3 4 5 6 -2 -1 1 2 3
6. [10 marks ] Use the Intermediate Value Theorem to show that the equation sin ( x ) ° 2 x = 1 ° 1 p 2 has at least one solution in the open interval ´ ° ° 4 ; 0 µ . Then use Rolle²s Theorem to prove that, in fact, the equation has exactly one solution. 4
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