求曲面x2+2y2+3z2=21在点(1,-2,2)处的切平面和法线方程 求曲面3x2+y2+z2=16在点（-1，-2，3）处的切平...

\u6c42\u66f2\u9762x²+2y²+3z²=21\u5728\u70b9(-1,-2,2)\u5904\u7684\u5207\u5e73\u9762\u4e0e\u6cd5\u7ebf\u65b9\u7a0b

\u8bbef\uff08x,y,z\uff09=x²+2y²+3z²=21.\u5207\u5411\u91cf\uff1an\u5411\u91cf=\uff082x,4y,6z\uff09|\uff08-1\uff0c-2\uff0c2\uff09=\uff08-2\uff0c-8\uff0c12\uff09
\u2234\u5207\u5e73\u9762\u65b9\u7a0b\uff1a\uff08-2\uff09\uff08x+1\uff09+\uff08-8\uff09(y+2)+12(z-2)=0
\u6cd5\u7ebf\u65b9\u7a0b\uff1ax+1/-2=y+2/-8=z-2/12

\u5199\u6210F(x.y.z)=0\u7684\u5f62\u5f0f\uff0c\u7136\u540eF(x.y.z)\u5206\u522b\u5bf9x\uff0cy\uff0cz\u6c42\u504f\u5bfc\uff0c\u4ee3\u5165\uff08-1\uff0c-2\uff0c3\uff09\u4fbf\u662f\u5f97\u5230\u7684\u4e09\u4e2a\u6570\u4fbf\u662f\u66f2\u9762\u6cd5\u5411\u91cf\uff08\u6cd5\u7ebf\uff09\u7684\u4e09\u4e2a\u6570\uff0c\u5207\u5e73\u9762\u53ef\u4ee5\u7528a\uff08x+1\uff09+b\uff08y+2\uff09+c\uff08z-3\uff09=0\u7684\u516c\u5f0f\uff0c1\uff0c2\uff0c-3\u662f\u51cf\u53bb\uff08-1\uff0c-2\uff0c3\uff09\u5bf9\u5e94\u7684\u6570\u3002a\uff0cb\uff0cc\u662f\u6cd5\u5411\u91cf\u5bf9\u5e94\u7684\u4e09\u4e2a\u6570

F(x、y、z)=x²+2y²+3z²-21
n=（Fx、Fy、Fz）=（2x、4y、6z）
n|(1、-2、2)=（2、-8、12）
切平面方程：

2（x-1）-8（y+2）+12（z-2）=0

2x-8y+12z-42=0
法线方程：
（x-1）/2=（y+2）/（-8）=（z-2）/12

法线方程：
（x-1）/1=（y+2）/（-4）=（z-1）/3

姹傛洸闈^2+2y^2+3z^2=12鍦(1,-2,1)澶勭殑鍒囧钩闈㈠強娉曠嚎鏂圭▼.
绛旓細姹傛洸闈^2+2y^2+3z^2=12鍦(1,-2,1)澶勭殑鍒囧钩闈㈠強娉曠嚎鏂圭▼. 鎴戞潵绛 棣栭〉 鐢ㄦ埛 璁よ瘉鐢ㄦ埛 瑙嗛浣滆 甯府鍥 璁よ瘉鍥㈤槦 鍚堜紮浜 浼佷笟 濯掍綋 鏀垮簻 鍏朵粬缁勭粐 鍟嗗煄 娉曞緥 鎵嬫満绛旈 鎴戠殑 姹傛洸闈^2+2y^2+3z^2=12鍦(1,-2,1)澶勭殑鍒囧钩闈㈠強娉曠嚎鏂圭▼.  鎴戞潵绛 1涓洖绛 #鍥藉簡蹇呯湅...

姹傛洸闈2+2y2+3z2=21鐨勫垏骞抽潰,浣垮畠骞宠浜庡钩闈+4y+6z=0銆
绛旓細銆愮瓟妗堛戯細

姹傚嚱鏁癴(x,y,z)=x2 2y2 3z2鍦ㄥ钩闈 y z=11鐨勬渶灏忓
绛旓細鈮(x+y+z)²/(1+1/2+1/3)=11²/(11/6)=66.鈭磝=2y=3z涓攛+y+z=11锛屽嵆x=6锛寉=3锛寊=2鏃讹紝鎵姹傛渶灏忓间负: f(x,y,z)|min=66銆

姹傛洸闈2+2y2+3z2=21骞宠浜庡钩闈+4y+6z=1鐨勫垏骞抽潰鏂圭▼
绛旓細鏇查潰x2+2y2+3z2=21浠绘剰澶勭偣锛坸0,y0,z0锛夌殑娉曞悜閲忎负{2x0,4y0,6} 璁惧垏鐐逛负锛坸,y,z锛夋墍浠1,4,6}={2x,4y,6} 瑙ｅ緱 x=0.5 y=1 甯﹀叆鏇查潰鏂圭▼寰梲=姝ｈ礋5/2 灏嗭紙0.5,1,2.5锛夊拰锛0.5,1,-2.5锛夊垎鍒甫鍏ュ垏骞抽潰鏂圭▼ 瑙ｅ緱涓 x+4y+6z=19.5 鍜 x+4y+6z=-10.5 ...

x2+ 2y2+3z2=9鎵绀虹殑鏇查潰
绛旓細鏄竴涓き鐞冿紝濡備笅涓嬪浘鎵绀猴細

鏂圭▼x2+2y2+3z2=1琛ㄧず鐨勪簩娆鏇查潰鏄( )
绛旓細銆愮瓟妗堛戯細D鏈鑰冩煡浜浜娆鏇查潰鐨勭煡璇嗙偣锛

姹傛洸闈(x^2+y^2+z^2)^2=a^3z(a>0)鎵鍥存垚鐨勭珛浣撲綋绉 濡傞,鍒╃敤鐞冮潰鍧愭爣...
绛旓細鐞冮潰鍧愭爣,(x^2+y^2+z^2)^2=a^3z鍙互鍐欎綔,r^4=a^3rcos蠁寰楀埌r=a(cos蠁)^(1/3)鍥犱负r>0, 鎵浠ハ嗏垐[0,蟺/2]V=鈭埆鈭玶^2sin蠁drd胃d蠁=[鈭(0->2蟺)d胃]* [鈭(0->蟺/2)d蠁]* [鈭(0->a(cos蠁)^(1/3)) r^2sin蠁dr]=蟺a^3/3...

楂樻暟棰樼瓟妗 鏇查潰x2+2y2+3z2=12涓婄偣(1,-2,1)澶勫垏骞抽潰鏂圭▼
绛旓細瑙 :

u=x2+2y2+3z2,鑻2+y2+z2鈮100 姹傚叾鏈澶у间互鍙婃渶灏忓
绛旓細x=y=z鏃讹紝 x2= 100/3 u= 6*100/3= 200 鏈灏 x=y=0鏃讹紝z2=100 u=300 鏈澶

姹傛き鐞冮潰x^2+2y^2+3z^2=21涓婃煇鐐瑰鐨勫垏骞抽潰鐨勬柟绋,璇ュ垏骞抽潰杩囧凡鐭ョ洿绾...
绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝绛旀濡傚浘鎵绀

扩展阅读：1x2+2x3+3x4公式推导 ... x2+y2 ... x2+y2+xy1的图形 ... z x2 ... x1+x2 ... x y ... 2+2 ... x1x2+y1y2=0 ... x230 ips ...