solsmt1 - SOLUTIONS TO MID TERM #1, MATH 100 1. [6 marks]...

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Unformatted text preview: SOLUTIONS TO MID TERM #1, MATH 100 1. [6 marks] Using only the definition of the derivative, and not the rules, find f ( x ) for the function f ( x ) = √ x 2 + 1 . Solution: f ( x ) = lim h → f ( x + h )- f ( x ) h = lim h → p ( x + h ) 2 + 1- √ x 2 + 1 h = lim h → p ( x + h ) 2 + 1- √ x 2 + 1 h × p ( x + h ) 2 + 1 + √ x 2 + 1 p ( x + h ) 2 + 1 + √ x 2 + 1 ! = lim h → ( x + h ) 2 + 1- ( x 2 + 1) h ( p ( x + h ) 2 + 1 + √ x 2 + 1) = lim h → 2 hx + h 2 h ( p ( x + h ) 2 + 1 + √ x 2 + 1) = lim h → 2 x + h p ( x + h ) 2 + 1 + √ x 2 + 1 = x √ x 2 + 1 2. [12 marks] Find the derivatives of the following functions. (a) f ( x ) = ( sin 3 x + cos 3 x ) 2 . (b) f ( x ) = q 1 + √ x + x 2 . (c) f ( x ) = x 2 + 1 x 2- 1 . (d) f ( x ) = ( x 2 + x + 1)( x 3 + 1) . Solution: (a) f ( x ) = 2 ( sin 3 x + cos 3 x ) (3 sin 2 x × cos x + 3 cos 2 x × (- sin x )) . (b) f ( x ) = 1 2 1 + √ x + x 2- 1 / 2 × 1 2 ( x + x 2 )- 1 / 2 (1 + 2 x ) ....
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This note was uploaded on 01/12/2010 for the course STAT 100 taught by Professor Denissjerve during the Winter '06 term at San Jose State.

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solsmt1 - SOLUTIONS TO MID TERM #1, MATH 100 1. [6 marks]...

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