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Unformatted text preview: SOLUTIONS TO MID TERM #2, MATH 100 1. [6 marks] (a) Find the derivative of f ( x ) = arcsin( √ x ) . Do not simplify. (b) Find f ( x ) f ( x ) if f ( x ) = (ln x ) x and simplify. (c) Find f ( x ) for f ( x ) = arctan x 1 x + 1 and simplify. Solution: (a) f ( x ) = 1 √ 1 x × 1 2 √ x . (b) f ( x ) f ( x ) = d dx (ln f ( x )) = d dx ( x ln(ln x )) = ln(ln x ) + 1 ln x . (c) f ( x ) = 1 1 + ( x 1 x +1 ) 2 × x + 1 ( x 1) ( x + 1) 2 = 2 ( x + 1) 2 + ( x 1) 2 = 1 1 + x 2 . 2. [8 marks] Let f ( x ) be the function f ( x ) = x (ln x ) 2 , x > . (a) Find all x where f ( x ) = 0 . (b) Find all x where f 00 ( x ) = 0 . (c) Find all intervals where f ( x ) is decreasing. (d) Find all intervals where f 00 ( x ) < . Solution: (a) f ( x ) = (ln x ) 2 + 2 ln x = ln x (ln x + 2) = 0 ⇐⇒ x = 1 , e 2 . (b) f 00 ( x ) = 2 ln x × 1 x + 2 x = 2 x (ln x + 1) = 0 ⇐⇒ x = e 1 . (c) f ( x ) is decreasing for e 2 < x < 1 ....
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 Winter '06
 denissjerve
 Approximation

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