962 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function THEOREM Equation of a Tangent Line If mtan exists, an equation of the tangent line to the graph of a function ( ) = y f x at the point ( ) ( ) = P c f c , is ( ) ( ) − = − y f c m x c tan (2) Finding an Equation of the Tangent Line Find an equation of the tangent line to the graph of ( ) = f x x 4 2 at the point ( ) 1, 1 4 . Graph f and the tangent line. Solution EXAMPLE 1 The tangent line contains the point ( ) 1, 1 4 . The slope of the tangent line to the graph of ( ) = f x x 4 2 at ( ) 1, 1 4 is ( ) ( ) ( ) ( )( ) ( ) = − − = − − = − − = − + − = + = → → → → → m f x f x x x x x x x x x lim 1 1 lim 4 1 4 1 lim 1 4 1 lim 1 1 4 1 lim 1 4 1 2 x x x x x tan 1 1 2 1 2 1 1 An equation of the tangent line is ( ) − = − = − y x y x 1 4 1 2 1 1 2 1 4 ( ) ( ) − = − y f c m x c tan Figure 18 shows the graph of = y x 4 2 and the tangent line at ( ) 1, 1 4 . Now Work PROBLEM 11 2 Find the Derivative of a Function The limit in formula (1) has an important generalization: it is called the derivative of f at c . DEFINITION Derivative of a Function at a Number If ( ) = y f x is a function and c is in the domain of f, then the derivative of f at c , denoted by ( ) ′ f c and read “ f prime of c ,” is the number ( ) ( ) ( ) ′ = − − → f c f x f c x c lim x c (3) provided this limit exists. Figure 18 x y y 5 x 2 1, 1 1 2 1 2 1 4 1 4 1 2 y 5 4 x2 ) (
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