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Unformatted text preview: The Derivative (Solutions) September 28, 2009 1. Find the equation of the tangent line to the following functions at the given point. 1. f ( x ) = x 3 + 4 x at x = 1. Solution. By the definition of the derivative, we have f (1) = lim h f (1 + h ) f (1) h = lim h [(1 + h ) 3 + 4(1 + h )] [1 3 + 4 1] h = lim h ( h 3 + 3 h 2 + 3 h + 1) + (4 + 4 h ) 5 h = lim h h 3 + 3 h 2 + 7 h h = lim h ( h 2 + 3 h + 7) = 7 . Therefore the slope of the tangent at the point (1 ,f (1)) is 7. That is, the equation of the tangent has the form y = 7 x + c. To compute c , we use that the line passes through (1 ,f (1)) = (1 , 5), and therefore 5 = 7+ c , or c = 2. So the equation of the tangent line is y = 7 x 2 . 2. f ( x ) = 1 x + 4 at x = 0. Solution. Again, we compute the slope of the tangent line first, which, by definition, is the derivative at 0. We have f (0) = lim x f ( x ) f (0) x = lim x 1 x +4 1 4 x = lim x 4 ( x +4) 4( x +4) x = lim x  x 4 x ( x + 4) = lim x  1 4( x + 4) = 1 16 . 1 Hence the tangent line has the form y = x/ 16 + c....
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 Fall '09
 BenedictGross
 Derivative

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