数据库文化基础 (24).pdf

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1、Basic Calculus14thweek/Optimization(III)Review and refresh some of basic but important topics in calculus:derivatives,gradient,Hessian,Taylors theoremObjectives of This Session Suppose:The(1storder)derivative of at 0 Instantaneous rate of change of with respect to The 1storder(linear)approximation o

2、f at 0 If 0,is increasing at 0=()=0:=lim0 0+(0)()0+(0)(0)Derivative3where 0 0 0+0+0 Secant line 0=slope oftangent line0Derivative Suppose:The 2ndorder derivative of at 0 The 2ndorder(quadratic)approximation of at 040=22=0()0+0 0+02 02where 0Basic Derivative Rules Constant rule:the derivative of a co

3、nstant=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 composite function Polynomials:=1 Exponentiation:=log =log =Logarithms:log=16

4、 ()=()=()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 partial derivatives of the function at certain point Direction of steepest ascent(i.e.,maximum incre

5、ase of the function)from Thus,is the direction of steepest descent(maximum decrease)Gradient8=1Hessian Suppose:Matrix comprising the second-order partial derivatives of a function Symmetric if is continuous 92=212212122Taylors Theorem Suppose:=0+1 0+2 02+3 03+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