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1024worksheet7

# 1024worksheet7 - (a f x =(1-2 x-1(b f x = x-x 3-x 5 4(c f x...

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Polytechnic Institute of NYU MA 1024/1324 Worksheet 7 Print Name: Signature: ID #: Instructor/Section: Directions: Show all your work for every problem. Problem Possible Points 1 10 2 10 3 10 4 10 5 10 6 15 7 10 8 15 9 10 Total 100 1

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2 (1) Find the derivative of the given function. (a) G ( x ) = 7 x 4 + 11 x + 1 (b) F ( x ) = ( x 2 - 1)( x - 3) (c) H ( x ) = (9 x 9 - 8 x 8 ) x + 1 x (d) J ( x ) = x 2 - 1 2 x + 3
3 (2) Find f 0 (0) given that h (0) = 3 and h 0 (0) = 2. (a) f ( x ) = xh ( x ) (b) f ( x ) = 3 x 2 h ( x ) - 5 x (c) f ( x ) = h ( x ) - 1 h ( x ) (d) f ( x ) = h ( x ) + x h ( x )

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4 (3) Find an equation for the tangent line to the graph of f ( x ) at the point ( a, f ( a )) (a) f ( x ) = x x + 2 ; a = - 4 (b) f ( x ) = ( x 2 - 3)(5 x - x 3 ); a = 1
5 (4) Find the points where the tangent line to the graph of f ( x ) is horizontal. (a) f ( x ) = ( x - 2)( x 2 - x - 11) (b) f ( x ) = 5 x x 2 + 1

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6 (5) (a) Verify that, if, f , g , h are differentiable, then ( fgh ) 0 ( x ) = f 0 ( x ) g ( x ) h ( x ) + f ( x ) g 0 ( x ) h ( x ) + f ( x ) g ( x ) h 0 ( x ) ( Hint: Apply the product rule to [ f ( x ) g ( x )] h ( x ) ) (b) Use the product rule to show that, if f is differentiable, then g ( x ) = [ f ( x )] 2 has derivative g 0 ( x ) = 2 f ( x ) f 0 ( x )
7 (6) Differentiate the given function:

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Unformatted text preview: (a) f ( x ) = (1-2 x )-1 (b) f ( x ) = ( x-x 3-x 5 ) 4 (c) f ( x ) = [(2 x + 1) 2 + ( x + 1) 2 ] 3 (d) f ( x ) = ± 4 x + 3 5 x-2 ² 3 8 (7) Given that f (0) = 1 , f (0) = 2 , f (1) = 0 , f (1) = 1 , f (2) = 1 , f (2) = 1 and g (0) = 2 , g (0) = 1 , g (1) = 1 , g (1) = 0 , g (2) = 2 , f (2) = 1 evaluate the following. (a) ( f ◦ g ) (0) = (b) ( g ◦ f ) (0) = (c) ( f ◦ g ) (1) = (d) ( g ◦ f ) (2) = 9 (8) Find the indicated derivative in terms of f ( x ) (a) d dx [ f ( x 2 + 1)] = (b) d dx ± f ( x )-1 f ( x ) + 1 ² = 10 (9) Determine the values of x for which f ( x ) = 0 and f ( x ) > 0. (a) f ( x ) = (1 + x 2 )-2 (b) f ( x ) = x (1-x 2 ) 3...
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