设f(x,y)是连续函数,而D:x²+y²≤1且y>0,则∫∫f(√x²+y²)dxdy=？ 函数y=f（x）是由方程x|x|+y|y|=1确定的函数，则...
\u8bbe\u51fd\u6570y=y(x)\u7531\u65b9\u7a0by=f(x^2+y^2)+f(x+y)\u786e\u5b9a,\u4e14y(0)=2,f(x)\u662f\u53ef\u5bfc\u51fd\u6570,f'(2)=1/2,f'(4)=1,\u5219f'(0)\u7684\u503cy=f(x²+y²)+f(x+y)
y'=f'(x²+y²)\u00d7(x²+y²)'+f'(x+y)\u00d7(x+y)'
=(2x+2yy')f'(x²+y²)+(1+y')f'(x+y)
\u5f53x=0\u65f6\uff0cy=2\uff0c\u90a3\u4e48y'=(0+4y')f'(4)+(1+y')f'(2)
\u800cf'(4)=1\uff0cf'(2)=1/2\uff0c\u6240\u4ee5y'=4y'\u00d71+(1+y')\u00d7(1/2)
\u5373\uff1ay'=4y'+1/2+y'/2\uff0c\u6240\u4ee5y'=-1/7\uff0c\u5373f'(0)=-1/7
(1) x \u2265 0, y \u2265 0
x² + y² = 1
\u6b64\u4e3a\u5706\u5fc3\u5728\u539f\u70b9\uff0c\u534a\u5f84\u4e3a1\u7684\u5706\u5728\u7b2c1\u8c61\u9650\u7684\u90e8\u5206, \u51cf\u51fd\u6570
(2) x \u2265 0, y < 0
x² - y² = 1
\u6b64\u4e3aa = b = 1\uff0c\u5b9e\u8f74\u4e3ax\u8f74\u7684\u53cc\u66f2\u7ebf\u5728\u7b2c4\u8c61\u9650\u7684\u90e8\u5206, \u51cf\u51fd\u6570
(3) x < 0, y \u2265 0
y² - x² = 1
\u6b64\u4e3aa = b = 1\uff0c\u5b9e\u8f74\u4e3ay\u8f74\u7684\u53cc\u66f2\u7ebf\u5728\u7b2c2\u8c61\u9650\u7684\u90e8\u5206, \u51cf\u51fd\u6570
(3) x < 0, y < 0
\u4e0d\u5b58\u5728
∫∫f(√x²+y²)dxdy
= ∫[0,pi] dt ∫[0,1] f(r) rdr (注意：是上半圆）
绛旓細璁綟'(x)=f(x)d/dx鈭(涓嬮檺a涓婇檺b)f(x+y)dy =d(F(b+x)-F(a+x))/dx =F'(b+x)-F'(a+x)=f(b+x)-f(a+x)
绛旓細鐢变簬绉垎鍖哄煙D={锛坸锛寉锛墊0鈮鈮2锛0鈮鈮2}={(x锛寉)|0鈮鈮4锛寉鈮鈮2}鈭寸Н鍒嗏埆20dx鈭玿20f锛坸锛寉锛塪y=鈭40dy鈭2yf(x锛寉)dx
绛旓細璁緃(x锛寉) = f(x锛寉)-g(x锛寉)锛屽垯h(x锛寉)鍦―涓婃湁杩炵画鍋忓鏁帮紝涓斿湪∂D涓婃亽绛変簬0锛岀敱h(x,y)杩炵画,D鏄湁鐣岄棴鍖哄煙,h(x锛寉)鍙湪D涓婂彇寰楁渶澶ф渶灏忓硷紝鑻ユ渶澶ф渶灏忓奸兘鏄湪∂D涓婂彇寰楋紝鍗虫湁h(x锛寉)鐨勬渶澶ф渶灏忓奸兘鏄0銆俬(x锛寉)鎭掔瓑浜0锛宖(x锛寉) = g(x锛寉)瀵...
绛旓細Y鍨嬶細鈭(0鈫1) dy 鈭(0鈫掆垰(1 - y)) 3x²y² dxx = 鈭(1 - y) ==> x² = 1 - y ==> y = 1 - x²浜ゆ崲绉垎娆″簭鍚庢槸X鍨嬶細鈭(0鈫1) dx 鈭(0鈫1 - x²) 3x²y² dy
绛旓細璁綟'(x)=f(x)d/dx鈭(涓嬮檺a涓婇檺b)f(x+y)dy =d(F(b+x)-F(a+x))/dx =F'(b+x)-F'(a+x)=f(b+x)-f(a+x)
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绛旓細璁緁(x)鏄姝ｅ杩炵画鍑芥暟,D={(x,y)|a<=x<=b,a<=y<=b},鍒橧=鈭埆D2f(x)/[f(x)+f(y)] dxdy=?1涓洖绛 #鐑# 宸插濂虫у氨搴旇鎵挎媴瀹堕噷澶ч儴鍒嗗鍔″悧?cjl414 2012-07-14 路 TA鑾峰緱瓒呰繃108涓禐 鐭ラ亾绛斾富 鍥炵瓟閲:42 閲囩撼鐜:0% 甯姪鐨勪汉:26.4涓 鎴戜篃鍘荤瓟棰樿闂釜浜洪〉 鍏虫敞 灞曞紑...
绛旓細姹俤z杩囩▼濡備笅锛氬綋鍑芥暟 z=f(x,y) 鍦 (x0,y0)鐨勪袱涓亸瀵兼暟 f'x(x0,y0) 涓 f'y(x0,y0)閮藉瓨鍦ㄦ椂锛屾垜浠О f(x,y) 鍦 (x0,y0)澶勫彲瀵笺傚鏋滃嚱鏁 f(x,y) 鍦ㄥ煙 D 鐨勬瘡涓鐐瑰潎鍙锛岄偅涔堢О鍑芥暟 f(x,y) 鍦ㄥ煙 D 鍙銆傛鏃讹紝瀵瑰簲浜庡煙 D 鐨勬瘡涓鐐 (x,y) 锛屽繀鏈変竴涓 x (...
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绛旓細=鈭<0,t>蟻[鈭<0,2蟺>f(蟻cos胃,蟻sin胃)d胃]d蟻 lim<t鈫0+>F`(t)/t =lim<t鈫0+>鈭<0,2蟺>tf(tcos胃,tsin胃)d胃/t =lim<t鈫0+>鈭<0,2蟺>f(tcos胃,tsin胃)d胃(绉垎涓煎畾鐞)=lim<t鈫0+>f(tcos尉,tsin尉)鈭<0,2蟺>d胃 鐢f(x,y)鍦ㄥ師鐐规煇閭诲煙D杩炵画锛寰...
