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

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SOLUTIONS TO MID TERM #1, MATH 100 1. [6 marks] Using only the definition of the derivative, and not the rules, find f 0 ( x ) for the function f ( x ) = x 2 + 1 . Solution: f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 p ( x + h ) 2 + 1 - x 2 + 1 h = lim h 0 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 0 ( x + h ) 2 + 1 - ( x 2 + 1) h ( p ( x + h ) 2 + 1 + x 2 + 1) = lim h 0 2 hx + h 2 h ( p ( x + h ) 2 + 1 + x 2 + 1) = lim h 0 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 0 ( x ) = 2 ( sin 3 x + cos 3 x ) (3 sin 2 x × cos x + 3 cos 2 x × ( - sin x )) . (b) f 0 ( x ) = 1 2 1 + x + x 2 - 1 / 2 × 1 2 ( x + x 2 ) - 1 / 2 (1 + 2 x ) . (c) f 0 ( x ) = ( x 2 - 1)2 x - ( x 2 + 1)2 x ( x 2 - 1) 2 . (d) f 0 ( x ) = (2 x + 1)( x 3 + 1) + ( x 2 + x + 1)3 x 2 . 3. [8 marks] (a) Determine lim θ 0 tan 2 θ θ . (b) Find the absolute maximum and minimum of the function f ( x ) = x 2 + 1 x 2 on the interval 1 2 x 3 . 1
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2 (c) Find all x where the derivative of y = sin x + cos x is 0 .
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