《高数双语》课件section 9.8.pptx
- 【下载声明】
1. 本站全部试题类文档,若标题没写含答案,则无答案;标题注明含答案的文档,主观题也可能无答案。请谨慎下单,一旦售出,不予退换。
2. 本站全部PPT文档均不含视频和音频,PPT中出现的音频或视频标识(或文字)仅表示流程,实际无音频或视频文件。请谨慎下单,一旦售出,不予退换。
3. 本页资料《《高数双语》课件section 9.8.pptx》由用户(momomo)主动上传,其收益全归该用户。163文库仅提供信息存储空间,仅对该用户上传内容的表现方式做保护处理,对上传内容本身不做任何修改或编辑。 若此文所含内容侵犯了您的版权或隐私,请立即通知163文库(点击联系客服),我们立即给予删除!
4. 请根据预览情况,自愿下载本文。本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
5. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007及以上版本和PDF阅读器,压缩文件请下载最新的WinRAR软件解压。
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 高数双语 高数双语课件section 9.8 双语 课件 section
- 资源描述:
-
1、Section 9.81Gauss,Karl Friedrich Unrestricted Extreme Valuesis defined in00(,).U xyDefinition Suppose the that the function(,)f x ysuch that0,If 000000(,)(,)(,)(,),(,)(,),)f x yf xyf x yf xyx yUxy then we say the function f has an unrestricted local maximum(minimum),or f has a maximum(minimum)00(,)f
2、 xyat the point 00(,)xy00(,)xyis called a maximal(minimal)point or for short,and the point Maximum and minimum values are called by the jointextreme point.name extreme value.attains a minimum at the point22zxyFor example,the function(0,0)attains a maximum at the point(0,0).and 221zxy3The Necessary C
3、ondition for An Extreme ValueTheorem Suppose that both partial derivatives of the function(,)f x yis an extreme point of the function00(,)xyexist at the point,and f.Then0000(,)(,)(,)0.xyxyf xyff00(,)xy00(,)xyis called a stationary point(驻点)A point 00(,)0f xy,which satisfies However,just as for funct
4、ions of one variable,a stationary of the function f.point is not necessarily an extreme point.,the point(0,0)is a stationary22(,)f x yxy For example,for the function point but not a extreme point.4Critical Point1.Interior points where0.xyffdo not exist.yf2.Interior points where one or both of xfandT
5、he theorem of necessary condition for a extreme value says that thecan ever have an extreme value areplaces a function(,)f x y(,)f x ywhere Definition An interior point of the domain of a function both fx and fy are zero or where one or both fx and fy do notexist is a critical point of f.5Saddle Poi
6、nt can assume extreme(,)f x yThus,the only points where a function As with differentiablevalues are critical points and boundary points.functions of a single variable,not every critical point gives rise to a A differentiable function of a single variable mightlocal extremum.A differentiable function
7、 of two variables have a point of inflection.might have a saddle point(鞍点鞍点).has a saddle point at a(,)f x yDefinition A differentiable functionstationary point(a,b)if in every open disk centered at (a,b)there areand domain points(x,y)domain points (x,y)where(,)(,)f x yf a b(,)(,).f x yf a b The cor
8、responding point(a,b,f(a,b)on thewhere is called a saddle point of the surface.(,)zf x y surface 6Sufficient Condition for Extreme Valueshas continuous second(,)zf x y Theorem Suppose that function order partial derivatives in a neighbourhood of the point 000(,),P xyLet0Pand is a stationary point of
9、 the function f.000(),(),().xxxyyyAfPBfPCfPThen 4.Can not be determined if 20.ACBand0()f P1.0A is a minimum value of the function f if 20;ACBand0()f P2.0A is a maximum value of the function f if 20;ACB0()f P3.is not an extreme value of the function f if 20;ACB7Finding Local Extreme ValuesExample Fin
10、d the local extreme values of 22(,)224.f x yxyxyxyThe function is defined and differentiable for all x and y and Solution The function therefore has extremeits domain has no boundary points.These values only at the points where fx and fy are simultaneously zero.leads to 2.xy or220,220,xyfyxfxyTheref
11、ore,the point(-2,-2)is the only point f may take on an extremevalue.8Finding Local Extreme ValuesSolution(continued)To see if it does so,we calculate 2,2,1.xxyyxyfff The discriminant of f at(-2,-2)is22(2)(2)(1)413.xxyyxyfff The combination 20 xxyyxyfffand0 xyf The value of f at this pointtell us tha
12、t f has a local maximum at(-2,-2).is(2,2)8.f Example Find the local extreme values of 22(,)224.f x yxyxyxy9Searching for Local Extreme ValuesExample Find the local extreme values of(,).f x yxy Since f is differentiable everywhere,it can assume extremeSolution values only where0.yfxand0 xfyThus,the o
13、rigin is the only point where f might have an extreme value.To see what happens there,we calculate0,0,1.xxyyxyfffThe discriminant,21,xxyyxyfff is negative.Therefore,the function has a saddle point at(0,0).we has no local extreme values.(,)f x yxy conclude that10Global Maxima and Minima(最值最值)Suppose
14、that the function f is continuous on a closed bounded region Then the function must attain its maximum and minimum in.xyand theythe maximum or minimum must be an extreme valueIf the maximum or minimum is attained in the interior of the region,zmust be stationary point of f if the partial derivatives
15、 of f existat the extreme point.11Finding Global ExtremaExample Find the global maximum and minimum values of 22(,)222f x yxyxyon the triangular plate in the first quadrant bounded by the line 0,x 0,9.yyxthe only places where f can assume these values are pointsand point on the boundary.0 xyffinside
16、 the triangle where Solution Since f is differentiable,xOy0y 0 x (0,9)B(9,0)A9yx12Finding Global ExtremaSolution(continued)xOy0y 0 x (0,9)B(9,0)A9yx(1,1)Boundary PointsFor these we have 220,yfy220,xfxThe valueyielding the single point(1,1).of f there is(1,1)4.f Interior PointsThe function 0.y 1.On t
17、he segment OA,2(,)(,0)22f x yf xxxmay now be regarded as a function of x defined on the closed interval09.x13Finding Global ExtremaxOy0y 0 x (0,9)B(9,0)A9yx9 9,2 2(1,1)2(,)(,0)22f x yf xxxIts extreme values may occur at the endpoints(0,0)2f where 0 x (9,0)61f where 9x and at the interior points wher
18、e(,0)220.fxx where(,0)0fx The only interior point where1,x is(,0)(1,0)3.f xf14Finding Global ExtremaxOy0y 0 x (0,9)B(9,0)A9yx(1,1)2.On the segment OB,The function 0.x 2(,)(0,)22.f x yfyyyWe know from the symmetry of f inin x and y and from the analysis wejust carried out that the candidateson the se
展开阅读全文