微積分 第七版 課件 二、二元函數(shù)的二階偏導(dǎo)數(shù)

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二元函數(shù)的二階偏導(dǎo)數(shù)本節(jié)學(xué)習(xí)目標(biāo)010203掌握二元函數(shù)二階偏導(dǎo)數(shù)的計(jì)算規(guī)則理解二元函數(shù)二階偏導(dǎo)數(shù)定義能熟練計(jì)算二元函數(shù)二階偏導(dǎo)數(shù)一、二元函數(shù)的二階偏導(dǎo)數(shù)二元函數(shù)的一階偏導(dǎo)數(shù)仍為自變量的二元函數(shù),還可以考慮它們對(duì)自變量求一階偏導(dǎo)數(shù)1.定義后.2二元函數(shù)z=f(x,y)的一階偏導(dǎo)數(shù)f'x(x,y),f'y(x,y)再分別對(duì)自變量x,y求一階偏導(dǎo)數(shù),所得到的偏導(dǎo)數(shù)稱為二元函數(shù)z=f(x,y)的二階偏導(dǎo)數(shù),共有四個(gè),分別記作3二元函數(shù)的二階偏導(dǎo)數(shù)

42.二元函數(shù)的二階偏導(dǎo)數(shù)計(jì)算方法已知二元函數(shù),若求其二階偏導(dǎo)數(shù),必須先求出一階偏導(dǎo)數(shù),一階偏導(dǎo)數(shù)表達(dá)式再對(duì)自變量求一階偏導(dǎo)數(shù),就得到二階偏導(dǎo)數(shù).5例1求二元函數(shù)z=x3y2-5xy4的二階偏導(dǎo)數(shù)解:計(jì)算一階偏導(dǎo)數(shù)z'x=3x2y2-5y4z'y=2x3y-20xy3所以二階偏導(dǎo)數(shù)6z″xx=(3x2y2-5y4)'x=6xy2z″xy=(3x2y2-5y4)'y=6x2y-20y3z″yx=(2x3y-20xy3)'x=6x2y-20y3z″yy=(2x3y-20xy3)'y=2x3-60xy2例2

解:計(jì)算一階偏導(dǎo)數(shù)

所以二階偏導(dǎo)數(shù)7

在例1與例2中都有關(guān)系式:z″xy=z″yx,這反映出在某種條件下計(jì)算二階偏導(dǎo)數(shù)的一種規(guī)律經(jīng)過(guò)深入的討論可以得到結(jié)論:如果二階偏導(dǎo)數(shù)z″xy與z″yx都連續(xù),則有關(guān)系式z″xy=z″yx下面所討論的二元函數(shù)都滿足這個(gè)結(jié)論的條件,因此只需計(jì)算三個(gè)二階偏導(dǎo)數(shù).8例3求二元函數(shù)z=ex-y的二階偏導(dǎo)數(shù)解:計(jì)算一階偏導(dǎo)數(shù)z'x=ex-y(x-y)'x=ex-yz'y=ex-y(x-y)'y=-ex-y所以二階偏導(dǎo)數(shù)9

例4求二元函數(shù)z=ysinex的二階偏導(dǎo)數(shù)解:計(jì)算一階偏導(dǎo)數(shù)z'x=ycosex·(ex)'x=yexcosexz'y=sinex所以二階偏導(dǎo)數(shù)10

=yex(cosex-exsinex)z″xy=z″yx=excosexz″yy=0二元函數(shù)在定義域內(nèi)點(diǎn)(x0,y0)處的二階偏導(dǎo)數(shù)值為二階偏導(dǎo)數(shù)的表達(dá)式中自變量x,y分別用數(shù)x0,y0代入所得到的數(shù)值.11例5

(a)-2 (b)2(c)-1 (d)112解