lec07 - 7. Definition of the Derivative P. K. Lamm 09/22/11...

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Unformatted text preview: 7. Definition of the Derivative P. K. Lamm 09/22/11 (15:28) p. 1 / 11 Lecture Notes: 7. Definition of the Derivative These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1(a). Definition of the Derivative at a Point Definition: The derivative of the function f at x is the real number f ( x ) = lim h f ( x + h )- f ( x ) h , (1) provided this limit exists and is finite. Example 1.1: Given f ( x ) = 2 x 2 + 1, find f (2). We will use the formula (1) with x = 2: f (2) = lim h f (2 + h )- f (2) h = lim h (2(2 + h ) 2 + 1)- (2 2 2 + 1) h = lim h (2(4 + 4 h + h 2 ) + 1)- (2 4 + 1) h = lim h 8 + 8 h + 2 h 2 + 1- 9 h = lim h 8 h + 2 h 2 h = lim h h (8 + 2 h ) h = lim h 8 + 2 h = 8 . Example 1.2: Given f ( x ) = 1 x , find f (1). As before we use the formula (1) only now with x = 1: f (1) = lim h f (1 + h )- f (1) h = lim h 1 1+ h- 1 1 h 1 7. Definition of the Derivative P. K. Lamm 09/22/11 (15:28) p. 2 / 11 = lim h 1 h 1- (1 + h ) 1 + h = lim h 1 h - h 1 + h = lim h - 1 1 + h =- 1 . 1(b). The Tangent Line to the Curve y = f ( x ) at x The derivative of the function f at x also has a meaning with regard to the graph of the curve y = f ( x ). Recall that the ratio f ( x + h )- f ( x ) h = f ( x + h )- f ( x ) ( x + h )- x is the slope of the secant line joining two points ( x ,f ( x )) and ( x + h, f ( x + h )) on the curve y = f ( x ). If we let h approach zero, then were asking that the point ( x + h, f ( x + h )) slide along the curve y = f ( x ) and approach the point ( x ,f ( x )). 2 7. Definition of the Derivative P. K. Lamm 09/22/11 (15:28) p. 3 / 11 For smooth curves, this secant line approaches a line that touches the curve at the point ( x ,f ( x )), a line which we know to be the tangent line to the curve y = f ( x ) at that point. Thus, the slope of the tangent line to the curve y = f ( x ) at the point ( x , f ( x )) is given by lim h [ the slope of the secant line joining ( x , f ( x )) to ( x + h, f ( x + h ) ] lim h f ( x + h )- f ( x ) ( x + h )- x lim h f ( x + h )- f ( x ) h = f ( x ) , so it follows that the slope of the tangent line is given by the derivative f ( x ) of f at the point x . We can use the point-slope formula to write the equation of the tangent line to the curve at the point ( x , f ( x )): y- f ( x ) = f ( x )( x- x ) , or y = f ( x ) + f ( x )( x- x ) . Example 1.3: Find the equation of the tangent line to the curve y = 1 x at the point x = 1....
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lec07 - 7. Definition of the Derivative P. K. Lamm 09/22/11...

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