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chap3_notes - Chapter 3 The derivative 3.1 The Derivative...

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Lirong Yu 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 0 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 0 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
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Lirong Yu 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 Δ x in place of h, and Δ y = f ( x + Δ x ) - f ( x ) for the resulting change in y. The The differential quotient f ( x + h ) - f ( x ) h = Δ y Δ x , where the Δ y means the change in y, and the Δ x means the change in x. So f ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim Δ x 0 Δ y Δ 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. f ( x ) , f ( x ): f is the function name and f denote its derivative. dy dx : Differential Notation. As x changes, y changes. x is independent variable, y is dependent on x . f ( x ) and dy dx are used almost interchangeably in mathematics, so we need to be familiar with both version of notation.
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