This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: L i r o n g Y u Chapter 3 The derivative 3.1 The Derivative and Rates of Change Definition : The derivative of a function f is the function f defined by f ( x ) = lim h f ( x + h )- f ( x ) h for all x where the limit exists. (insert graph of the above idea) Remark: Geometric Interpretation of the derivative Note that let x = a, in the above definition, we then have f ( a ) = lim h f ( a + h )- f ( a ) h i.e. f ( a ) is the slope of the tangent line to the graph of f at the point ( a,f ( a )) , So y- f ( a ) = f ( a ) * ( x- a ) is the equation of the tangent line. Here a is x coordinate of the tangent point. Differentiating a given function f by direct evaluation of the limit in the definition involves carrying out four steps: 1). Write the definition equation of the derivative. 2).Substitute the expression f ( x + h ) and f ( x ) as determined by the perticular function f. 3). Simplify the result by algebraic methods until it is possible to .. 4). Apply appropriate limit laws to finally evaluate the limit. Example 1 Let f ( x ) = x x + 3 , (a) Compute f ( x ) (b) Find the equation of the tangent line at (0,f(0)). Remark Even when the function f is rather simple, this four-step process for computing f directly from the definition of the derivative can be time consuming. Also, Step 3 may require considerable ingenuity. Moreover, it would be very repetitious to continue to rely on this process. To avoid tedium, we want a fast, easy, short method for computing f . This is the focus of this chapter: the development of systematic methods (rules) for differentiating those functions that occur most frequently. Such functions include polynomials, rational functions, the trigonometric functions sin x and cos x , and combi- nations of such functions. Rule If f ( x ) = ax 2 + bx + c, Then f ( x ) = 2 ax + b 1 L i r o n g Y u Proof. (given in section 2.1) Example 2 a) f ( x ) = 3 x 2- 4 x + 5 ,f ( x )=? b) g ( t ) = 2 t- 5 t 2 ,g ( t )=? (relation between the graph of f and f ) see homework P117 30-35. (see also Example 8 in textbook) Differential Notation An important alternative notation is to write x in place of h, and y = f ( x + x )- f ( x ) for the resulting change in y. The The differential quotient f ( x + h )- f ( x ) h = y x , where the y means the change in y, and the x means the change in x. So f ( x ) = lim h f ( x + h )- f ( x ) h = lim x y x = dy dx and f ( a ) = dy dx | x = a Example 2 (cont.) try these notation a) dy dx = 6 x- 4 , so dy dx | x =1 = 2 b) denote z = 2 t- 5 t 2 , dz dt = 2- 10 t, dz dt | t =1 =- 8 Remark The benefit of using the Differential Notation is that they mark out which the independent variable, which is the dependent variable....
View Full Document
This document was uploaded on 04/12/2010.
- Fall '09