1、Basic Calculus14th week / Optimization (III)Review and refresh some of basic but important topics in calculus: derivatives, gradient, Hessian, Taylors theoremObjectives of This Session Suppose : The (1st order) derivative of at 0 Instantaneous rate of change of with respect to The 1st order (linear)
2、 approximation of at 0 If 0, is increasing at 0 = () =0:= lim0 0 + (0)() 0 + (0)( 0)Derivative3where 0 0 0 + 0 + 0 Secantline 0 = slope oftangentline0Derivative Suppose : The 2nd order derivative of at 0 The 2nd order (quadratic) approximation of at 04 0 = 22 =0 () 0 + 0 0 + 02 02 where 0Basic Deriv
3、ative Rules Constant rule: the derivative of a constant = 0 Constant multiple rule: () = () where is a constant Multiple rule: () = () where is a constant Sum rule: +() = +() Product rule: () = ()+()() Quotient rule: ()() = () 25Basic Derivative Rules Chain rule: to calculate derivatives for a compo
4、site function Polynomials: = 1 Exponentiation: = log = log = Logarithms: log = 16 () = ( ) = ()Partial Derivative Suppose : The partial derivative of (1,) at The second-order partial derivatives are computed recursively7 1, = lim0 1, + , 1,2() =()2()2 =() Suppose : Gradient ()A vector containing the
5、 partial derivatives of the function at certain point Direction of steepest ascent (i.e., maximum increase of the function ) from Thus, is the direction of steepest descent (maximum decrease)Gradient8= 1 Hessian Suppose : Matrix comprising the second-order partial derivatives of a function Symmetric
6、 if is continuous 92=212 21 21 22Taylors Theorem Suppose : = 0 +1 0 + 2 0 2 +3 0 3 + Suppose : 10 = (0) +(0)1! 0 +(0)2! 02 +(0)3! 03 + = () +() +12 2() +Thomas Calculus (13 ed.), G.B. Thomas Jr., M.D. Weir and J.R. Hass, PearsonReferencesOR/MS (Management Science) and Probability Models, Ho Woo Lee, Sigma Press
