极坐标下的坐标方程r(θ)的导数代表什么含义？（如直角坐标下的导数表示切线的斜率一样） 极坐标下的坐标方程r（θ）的导数代表什么含义
\u6570\u5b66 \u76f4\u89d2\u5750\u6807\u7cfb\u65b9\u7a0b\u5bfc\u6570\u8868\u793a\u5207\u7ebf\u7684\u659c\u7387\uff0c\u90a3\u4e48\u6781\u5750\u6807\u7cfb\u65b9\u7a0b\u7684\u5bfc\u6570\u8868\u60a8\u597d\uff0c\u6b65\u9aa4\u5982\u56fe\u6240\u793a\uff1a
\u7528\u6781\u5750\u6807\u89e3\u51b3\u51e0\u4f55\u95ee\u9898\u7684\u65b9\u6cd5\u3002\u5728\u76f4\u89d2\u5750\u6807\u7cfb\u4e2d\uff08x,y\uff09
x\u88ab\u03c1cos\u03b8\u4ee3\u66ff\uff0cy\u88ab\u03c1sin\u03b8\u4ee3\u66ff\uff0c\u03c1=\u221a(x^2+y^2)
\u4ece\u800c\u5f97\u5230\u65b0\u7684\u65b9\u7a0b\u3002\u8fd9\u6837\u7684\u65b9\u7a0b\u5e38\u5e38\u7528\u6765\u89e3\u51b3\u66f2\u7ebf\u95ee\u9898\uff0c\u5982\u692d\u5706\u66f2\u7ebf\u3001\u7ebd\u7ebf\u3001\u87ba\u7ebf\u7b49\u7b49\uff0c\u53ef\u4ee5\u4f7f\u89e3\u9898\u66f4\u52a0\u6e05\u6670\u7b80\u4fbf\u3002\u8bbe\u66f2\u7ebfC\u7684\u6781\u5750\u6807\u65b9\u7a0b\u4e3ar=r\uff08\u03b8\uff09\u3002\u5219C\u7684\u53c2\u6570\u65b9\u7a0b\u4e3a{ x=r\uff08\u03b8\uff09cos\u03b8y=r\uff08\u03b8\uff09sin\u03b8\u5176\u4e2d\u03b8\u4e3a\u6781\u89d2\u3002\u7531\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc\u6cd5\uff0c\u5f97\u66f2\u7ebfC\u7684\u5207\u7ebf\u5bf9x\u8f74\u7684\u659c\u7387\u4e3a y\u02ca=r\u02ca(\u03b8)sin\u03b8+r(\u03b8)cos\u03b8\u2215r\u02ca(\u03b8)cos\u03b8-r(\u03b8)sin\u03b8=r\u02catan\u03b8+r\u2215r\u02ca-rtan\u03b8\u8bbe\u66f2\u7ebfC\u5728\u70b9M\uff08r\uff0c\u03b8\uff09\u5904\u7684\u6781\u534a\u5f84OM\u4e0e\u5207\u7ebfMT\u95f4\u7684\u5939\u89d2\u4e3a\u03a8\uff0c\u5219\u03a8=\u03b1-\u03b8\uff08\u5982\u56fe\uff09\u6545\u6709tan\u03a8=tan\uff08\u03b1-\u03b8\uff09=y\u02ca-tan\u03b8\u22151+y\u02catan\u03b8\u5c06y\u02ca\u4ee3\u5165\uff0c\u5316\u7b80\u5f97tan\u03a8=r\uff08\u03b8\uff09\u2215r\u02ca\uff08\u03b8\uff09\u8fd9\u4e00\u91cd\u8981\u516c\u5f0f\u8868\u660e\uff1a\u5728\u6781\u5750\u6807\u7cfb\u4e0b\uff0c\u66f2\u7ebf\u7684\u6781\u534a\u5f84r\uff08\u03b8\uff09\u4e0e\u5176\u5bfc\u6570r\u02ca\uff08\u03b8\uff09\u4e4b\u6bd4\u7b49\u4e8e\u6781\u534a\u5f84\u4e0e\u66f2\u7ebf\u5207\u7ebf\u4e4b\u5939\u89d2\u7684\u6b63\u5207\u3002
\u5f88\u9ad8\u5174\u80fd\u56de\u7b54\u60a8\u7684\u63d0\u95ee\uff0c\u60a8\u4e0d\u7528\u6dfb\u52a0\u4efb\u4f55\u8d22\u5bcc\uff0c\u53ea\u8981\u53ca\u65f6\u91c7\u7eb3\u5c31\u662f\u5bf9\u6211\u4eec\u6700\u597d\u7684\u56de\u62a5\u3002\u82e5\u63d0\u95ee\u4eba\u8fd8\u6709\u4efb\u4f55\u4e0d\u61c2\u7684\u5730\u65b9\u53ef\u968f\u65f6\u8ffd\u95ee\uff0c\u6211\u4f1a\u5c3d\u91cf\u89e3\u7b54\uff0c\u795d\u60a8\u5b66\u4e1a\u8fdb\u6b65\uff0c\u8c22\u8c22\u3002\u2606\u2312_\u2312\u2606 \u5982\u679c\u95ee\u9898\u89e3\u51b3\u540e\uff0c\u8bf7\u70b9\u51fb\u4e0b\u9762\u7684\u201c\u9009\u4e3a\u6ee1\u610f\u7b54\u6848\u201d
\u6781\u5750\u6807r(\u03b8)\u51fd\u6570\u8868\u793a\u7684\u5c31\u662f
\u534a\u5f84r\u4e0e\u89d2\u5ea6\u03b8\u4e4b\u95f4\u7684\u5173\u7cfb
\u90a3\u4e48\u73b0\u5728r\u5bf9\u03b8\u6c42\u5bfc
\u5f97\u5230\u7684\u5bfc\u6570\u5f53\u7136\u5c31\u662f\u534a\u5f84r \u5728\u67d0\u03b8\u503c\u65f6\u7684\u53d8\u5316\u7387
在极坐标系下，曲线的极半径r(θ)与其导数r‘(θ)之比等于极半径与曲线切线之夹角的正切。
绛旓細dy = sin胃* dr + r*cos胃* d胃銆俤x ^ dy = r*cos胃锛奵os胃锛奷r ^ d胃- r*sin胃锛妔in胃d胃^ dr銆= r * (cos胃锛奵os胃锛媠in胃锛妔in胃锛* dr ^ d胃銆= r dr ^ d胃銆傜浉鍏充俊鎭細鐢ㄦ瀬鍧愭爣绯绘弿杩扮殑鏇茬嚎鏂圭▼绉颁綔鏋佸潗鏍囨柟绋锛岄氬父鐢ㄦ潵琛ㄧず蟻涓鸿嚜鍙橀噺胃鐨勫嚱鏁般傛瀬鍧愭爣鏂圭▼缁忓父浼...
绛旓細鍙傛暟鏂圭▼涓鑸槸涓轰簡鏂逛究璁ㄨ鎴栬绠楄岄夊彇鐨勫弬鏁般傝鏋佸潗鏍閫氬父閮芥槸鍦ㄧ洿瑙掑潗鏍囪璁烘病閭ｄ箞绠渚跨殑鏃跺欒岄夊彇鐨勩傛湰韬篃鍙湅浣滃涓嬬殑鍙傛暟鏂圭▼锛毼=t r=r(t)杩欓噷鐨勫弬鏁皌鍗充负瑙掑害銆傚叾鍖栨垚鐩磋鍧愭爣鏂圭▼涔熷彲鐪嬫垚鏄哥殑鍙傛暟鏂圭▼锛歺=r(胃)cos胃 y=r(胃)sin胃 鍏蜂綋鐨勮浆鍖栬繕闇鏍规嵁瀹為檯鐨勬柟绋嬫潵閫夋嫨鍚堥傜殑...
绛旓細瑙ｏ細鍧囧彲浠ョ洿瑙掑潗鏍囩郴鐨勫師鐐逛负鏋佺偣銆亁杞存鍚戜负鏋佽酱鏂瑰悜锛屽缓绔鏋佸潗鏍绯伙紝璁緓=rcos胃锛寉=rsin胃鍙樻崲姹傝В銆傘愯鍦嗙殑鍗婂緞涓篴銆戜粠宸﹀埌鍙筹紝绗1鍥撅紝绉垎鍖哄煙D={(r,胃)涓0鈮鈮2asin胃锛0鈮の糕墹蟺}銆傜2鍥撅紝绉垎鍖哄煙D={(r,胃)涓0鈮鈮2acos胃锛-蟺/2鈮の糕墹蟺/2}銆傜3鍥撅紝鏋佽酱鍜屾瀬瑙掑彇鍐...
绛旓細楂樹腑鏁板鏋佸潗鏍囧叕寮忓涓 鏋佸潗鏍囩郴涓殑涓や釜鍧愭爣蟻鍜胃鍙互鐢眡=蟻cos胃锛寉=蟻sin胃杞崲涓虹洿瑙掑潗鏍囩郴涓嬬殑鍧愭爣鍊笺備粠鐩磋鍧愭爣绯讳腑x鍜寉涓ゅ潗鏍囪绠楀嚭鏋佸潗鏍囦笅鐨勫潗鏍锛毼=arctan锛坹/x锛(x鈮0)銆傛瀬鍧愭爣鏂圭▼蹇呰儗鍏紡 x=r/cos/theta锛寉=r/sin/theta锛屾瀬鍧愭爣绯讳腑鐨勪袱涓鍧愭爣r鍜屛稿彲浠ョ敱涓婇潰鐨勫叕寮忚浆鎹...
绛旓細涓嶈兘銆傛瀬鍧愭爣绯讳腑鐨勪袱涓鍧愭爣r鍜屛稿彲浠ョ敱涓嬮潰鐨勫叕寮忚浆鎹负鐩磋鍧愭爣绯讳笅鐨勫潗鏍囧笺倄 = rcos锛埼革級锛寉 = rsin锛埼革級锛涚敱涓婅堪浜屽叕寮忥紝鍙緱鍒颁粠鐩磋鍧愭爣绯讳腑x鍜寉涓ゅ潗鏍囧浣曡绠楀嚭鏋佸潗鏍囦笅鐨勫潗鏍锛毼 = arctan(y/x)锛涘湪x = 0鐨勬儏鍐典笅锛氳嫢y涓烘鏁拔 = 90掳锛涜嫢y涓鸿礋鏁帮紝鍒櫸 = 270掳 銆...
绛旓細鎶婄洿瑙掑潗鏍囩郴涓紙x,y锛夛紝x鐢ㄏ乧os胃浠ｆ浛锛寉鐢ㄏ乻in胃浠ｆ浛锛岀洿鎺ュ甫鍏ュ嵆鍙傝鏇茬嚎C鐨鏋佸潗鏍囨柟绋涓簉=r锛埼革級锛屽垯C鐨勫弬鏁版柟绋嬩负x=r锛埼革級cos胃锛寉=r锛埼革級sin胃锛屽叾涓镐负鏋佽銆傜敱鍙傛暟鏂圭▼姹傚娉曪紝寰楁洸绾緾鐨勫垏绾垮x杞寸殑鏂滅巼涓簓藠=r藠(胃)sin胃+r(胃)cos胃鈭時藠(胃)cos胃-r(胃)sin胃...
绛旓細胃=0锛r=2a锛屛=蟺锛宺=0锛屽叧浜庢瀬杞村绉般1銆佹瀬杞村乏杈癸細V=鈭紙0,2a锛壪y²dxx =rcos胃=a(1+cos胃)cos胃 =a(cos胃+cos²胃)dx =a(-sin胃-2sin胃cos胃)d胃y =rsin胃=a(1+cos胃)sin胃 =a(sin胃+sin胃cos胃)锛屼唬鍏ワ細V=鈭紙0,2a锛壪y²dx =蟺鈭紙蟺/2...
绛旓細濡傚浘
绛旓細绠鍗曞垎鏋愪竴涓嬶紝绛旀濡傚浘鎵绀
绛旓細ds=鈭歔(dx)²+(dy)²]=鈭歔(dx/d胃)²+(dy/d胃)²]d胃锛濃垰((r'(胃))^2锛(r(胃))^2)d胃 搴旂敤 寮鏅嫆绗竴瀹氬緥锛氬お闃崇郴涓殑鎵鏈夎鏄熷洿缁曞お闃宠繍鍔ㄧ殑杞ㄩ亾閮芥槸妞渾锛屽お闃冲鍦ㄦ墍鏈夋き鍦嗙殑涓涓劍鐐逛笂銆傚紑鏅嫆绗簩瀹氬緥锛鏋佸潗鏍鎻愪緵浜嗕竴涓〃杈惧紑鏅嫆琛屾槦杩愯瀹氬緥鐨勮嚜鐒舵暟...
