solutions_09_28

solutions_09_28 - The Derivative(Solutions 1 Find the...

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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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solutions_09_28 - The Derivative(Solutions 1 Find the...

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