在几何空间[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]中，取右手直角坐标系[tex=2.357x1.286]KdGqUTe/gacWvoJo2jKuKQ==[/tex]。用[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]表示绕[tex=0.571x1.286]XubEW9+1+hkJqH7jXe5MrA==[/tex]轴按右手螺旋方向旋转[tex=1.429x1.071]HrADEgZoqo90D/eowIUddQ==[/tex]的变换，用[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]表示绕[tex=0.571x1.286]Hz6y44ELFVLLNrLVhO3CQA==[/tex]轴右旋[tex=1.429x1.071]HrADEgZoqo90D/eowIUddQ==[/tex]的变换，用[tex=0.857x1.0]oXAqKViyEOXeAjRP4JQG3g==[/tex]表示绕[tex=0.5x0.786]gdMkE6SnyZedYLxpUxdkaQ==[/tex]轴右旋[tex=1.429x1.071]HrADEgZoqo90D/eowIUddQ==[/tex]的变换。证明：[tex=6.286x1.214]nNW4SjHKg9l4+lvt7yxOV8ohSYFxCozdR9tGz9/odtY=[/tex]，[tex=4.357x1.286]7zPmGc5p06lu73qQqBk42KQ4K3CRxaGNqouTIk1qDg0=[/tex]，[tex=6.071x1.286]8YDjIMsJw+Fl1yEpDeuNKIfze3JUgeXp6SjNr7JK09Y=[/tex]，并检验[tex=6.357x1.286]0OnkK91gsieklRPIkF9T5msFfzRg+66UXNiBLxjYk8c=[/tex]是否成立。

2022-06-27

在几何空间[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]中，取右手直角坐标系[tex=2.357x1.286]KdGqUTe/gacWvoJo2jKuKQ==[/tex]。用[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]表示绕[tex=0.571x1.286]XubEW9+1+hkJqH7jXe5MrA==[/tex]轴按右手螺旋方向旋转[tex=1.429x1.071]HrADEgZoqo90D/eowIUddQ==[/tex]的变换，用[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]表示绕[tex=0.571x1.286]Hz6y44ELFVLLNrLVhO3CQA==[/tex]轴右旋[tex=1.429x1.071]HrADEgZoqo90D/eowIUddQ==[/tex]的变换，用[tex=0.857x1.0]oXAqKViyEOXeAjRP4JQG3g==[/tex]表示绕[tex=0.5x0.786]gdMkE6SnyZedYLxpUxdkaQ==[/tex]轴右旋[tex=1.429x1.071]HrADEgZoqo90D/eowIUddQ==[/tex]的变换。证明：[tex=6.286x1.214]nNW4SjHKg9l4+lvt7yxOV8ohSYFxCozdR9tGz9/odtY=[/tex]，[tex=4.357x1.286]7zPmGc5p06lu73qQqBk42KQ4K3CRxaGNqouTIk1qDg0=[/tex]，[tex=6.071x1.286]8YDjIMsJw+Fl1yEpDeuNKIfze3JUgeXp6SjNr7JK09Y=[/tex]，并检验[tex=6.357x1.286]0OnkK91gsieklRPIkF9T5msFfzRg+66UXNiBLxjYk8c=[/tex]是否成立。

答案：

解：由下图所示，[tex=1.143x1.214]RYBMZUDv9OzdyPxfq8eUgA==[/tex]表示绕[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴右旋[tex=1.929x1.071]kqea7/BwZvM6J/RbreCo/Q==[/tex]的变换，因此 [tex=2.571x1.214]c5Cf4pRARaBipYntugL/3sWBxZDZ4OVydxjRW68H5mfT7ADfDWCDTPaAaTakDGpc[/tex]，同理，[tex=4.571x1.214]hT+oJ7OUfFv/vJSX4KjeNcf4l7hT9rhcV6PxFZ6GpR1s5BVxNn4ObUG76K2aUoWFdUfcbc69mhhWiAcuxhiNkA==[/tex]。用[tex=3.357x1.214]Zvo/mN9XByhE53UuBPtpQQTEJod+2qse/MGI3xcdEvoWs7yyoP4mZTID5fhpDSnv[/tex]分别表示[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴，[tex=0.5x1.0]yBR4oiFoTexGaFalQ7m8kg==[/tex]轴，[tex=0.5x0.786]C7x+w8+jOPZzxFrGGne6Dw==[/tex]轴上的单位向量，则[tex=4.429x1.357]ZeXmt4Og2leBNgS2vOaZDWsnStKWTLeQ/nIhtM7CSKyHntbIY/PZH9zPbsh8Os8+FKdnG702Ubg2MeJbVRkzaQ==[/tex]，[tex=4.429x1.357]ZeXmt4Og2leBNgS2vOaZDenRoanK/UYubSysvhZRdczzUNFW/4ikcROBLtuhvsA20Pc7KTnQp1QKDThOHHz7Dg==[/tex]，[tex=5.214x1.357]ZeXmt4Og2leBNgS2vOaZDc3nvqUpI/GMx9uxG0znNq+LGhb2KTFIEwBfugguUafmbAHtacFZcXnokLl2uIRweA==[/tex]，[tex=5.214x1.357]BtTDSwvshYw45F1hbtxo8gB6Y2dXJPCEPnUzgfPL3seQReipRiGSX8T9zki4RDIhhHraj86RRUboyUhHuJB0bg==[/tex]，[tex=4.429x1.357]BtTDSwvshYw45F1hbtxo8tXyLxKze/xOBmy54eSrTya72D3uhdYtgY/qCACW9XJYsNAYrjxsheoV6UPjzLJ35Q==[/tex]，[tex=4.429x1.357]BtTDSwvshYw45F1hbtxo8ok6dN/6w8Ad8fw+2uxlaZUOonZk4zf+hbN0UAulcVg4VRVuNDCFTh79NEhRUG2Q0A==[/tex]，，于是[tex=10.0x1.357]0idGSV3RW/tbV3escumNdPHEPku7a8z/CbMnfW2E7IwgrLAS2OqIAGKT91YMY4t7tSKIM4EOEroeKYgXX1iXu5oHguJnlOCdhfGL6734P+ipzFYPNk+4SZTKMyFmRB3OEVRFThFoD2Fk7hhQR8KTJtPITiUuC5Jb5ArnB+b9ZqU=[/tex]，[tex=10.0x1.357]0idGSV3RW/tbV3escumNdG+od0HFOsB35W2aMxkCmjNsUYsVv+BzdZTfoTZLp7UlfuDxuO2FjA06a0/HxRUzSd79Po966ihXHB/XVBvzJEjUz6MArK7DULBcDm/KSu30QHwtMpR5wXc5HyinD8gddYRarvG48UOF/BaGWMLhacQ=[/tex]，由于 [tex=8.643x1.286]kTuwmDoBGHuFN/tHUgvPoJmdtl78xY5/XS38Z+dW+Wup0BCwf4UBaR8tf+B06hYHvITHvr8/fhA7tHRVK7L04iWXEvmLmB4aqHIQkf+MMmE=[/tex]，因此[tex=4.357x1.286]7zPmGc5p06lu73qQqBk42KQ4K3CRxaGNqouTIk1qDg0=[/tex]。[tex=10.857x1.571]0idGSV3RW/tbV3escumNdL2ZmI/rBJ98VrehAlzRZGNeRH5eRqwuKnSEfOp1cJIAhwvOw4djamjgNN1y7sG1ETF9XQdHAJGJ3K+0YlqwsIlMiWyqdsyowk5zShOp5kzltrrObIIePmzO0omfZ9HQlHJjlhAC8loAPHXTrixsiYPtyY/GwD/c3IrtgvTaHIpq[/tex][tex=12.0x1.5]42h7DQG251K1pVIYG5MdSxy5VfRKAQ3ahUhEpYlme8lD15Qff22qD9KVMGgnIJklpuOuIEmgwvDPlHH/U15o/62t1XoPl1o7GRGWSVsf09Q7uw5K8PJnwOCTE0Im7SGUQtQHWJL34epQKTwr+QlZCvRuFdMoj9gAk0ozLnB2qOY=[/tex]，[tex=10.071x1.571]0idGSV3RW/tbV3escumNdL2ZmI/rBJ98VrehAlzRZGNeRH5eRqwuKnSEfOp1cJIAhwvOw4djamjgNN1y7sG1EdxnoHh/+AbbcXq2S2OxU/VIFGENGO1ccDcIFy7dLDnpWVA8xQxVQdkry5iHUepRfxw+HZj1kyUzI8BIehi3of++YC0wqNfrSWfRyfL/5vk9[/tex][tex=9.929x1.5]42h7DQG251K1pVIYG5MdS61yPv0dmQyz3g0DeYP4u/fFrCjoZkvGsBSeFKTB7rJV2CpHtABN/cUytCpqvI8C1HG3gSWbXplXXtRDsK0bteTdUNKy9I9XytRUqIFq9fRn[/tex]，[tex=10.071x1.571]0idGSV3RW/tbV3escumNdL2ZmI/rBJ98VrehAlzRZGNeRH5eRqwuKnSEfOp1cJIAhwvOw4djamjgNN1y7sG1EdLh+svWu72vgGch5wWtm+3YxlonwjP9rOAbMzaojVDbXcpNznrcT326O2nY9z6N80vjYBobO0kj77vKiXfDCMgD8UGEb4CzrtosA9qGonK2[/tex][tex=11.0x1.5]42h7DQG251K1pVIYG5MdSxy5VfRKAQ3ahUhEpYlme8nU7xPAWDxD+cnTz745+ftIvNrgX/0jZ1kt7NRVDGtGBVQTW/HnEGwwqmfRpywsq2lMZD4KAWJfhtI4vClNdgCz6+kvysOTCCPd2SdUozts88EtMDjxxKSsMAUmv4mMgrurPzBI2oiptv7mXPePZLy/[/tex]，[tex=13.0x1.571]0idGSV3RW/tbV3escumNdB1gcsTKk1+45a/KuugGf/1b2qQ27kjCcNihbvIGos1KqLI1W2Z9woIm/vv6eSyaC2JViZ96uc1Z2EaBoRNVH5LcEQKW1kIDcFc4n4ZENEOlwMWLIsVQYWVgSOJ5nd11aPZ/RDUzclPiNy/5SypFcwTc/Aukrmlo9Tw6khOSOdADC11WsOe1fTtLNqwo9tGnvg==[/tex]，[tex=12.5x1.571]0idGSV3RW/tbV3escumNdB1gcsTKk1+45a/KuugGf/1b2qQ27kjCcNihbvIGos1KqLI1W2Z9woIm/vv6eSyaC2lQLDNpOk7Mr0h4M3/PsLM7mL1Q2zR0aieN6BOdZ89WSdBqylnOHPZ+riC45muhi5YCbcu92eeuazVsmFIamZ1JsgvPaZ8iNrIpb9Q6cTrJ[/tex]，[tex=10.857x1.571]0idGSV3RW/tbV3escumNdB1gcsTKk1+45a/KuugGf/1b2qQ27kjCcNihbvIGos1KqLI1W2Z9woIm/vv6eSyaCwDHtrFCsiddxbCr51v2l/3IMtvDCfjtMEP9VqtDigmzuRsSV2EendiUcrBZnz9epJdb9Qb/EwaiR0qqg17QJzwGyioUlLOxUZMp4f6CY09T[/tex][tex=11.214x1.5]gVmion3+fO188iKq9EkaELuKjKxS8/6u6PUnSwAFH9qkVwaszX3QBAU+i79WDiUhYJHe6v23zl0nzbCBhDkZfFNp51V8sKvq/37vWorqYfXAHRJuApWzKAnves3l1RasPZAHYb4zVp/GyuoOCVtS5jGfYycNUPbgYJDpmLqmrIw=[/tex]，因此 [tex=11.286x1.571]v6h0emw+MX5fBgkKQJsSsGt+DAc0N0ZogZ64XT2PPRDKNALbcnmhFW5B2WNfJSuZpyqMl84qhtqQXU0pYmMVmKk5A6ZURWsolasO4eHq6p34YrTCvrBQe7FvSvqW7Q7rqA/mNy4dIR+jgQBlrziq91hG//EBM7Li+pJJ0DRS+ZP6ZTBFQ/klhv02GQDQm4H0euBhWfeXctOZP+JWWe5e9Q==[/tex]，[tex=3.571x1.214]vOXVGYH4JTHMb0pJ1Jg1zg==[/tex]。由于绕一条直线的旋转是线性变换，且[tex=3.357x1.214]Zvo/mN9XByhE53UuBPtpQQTEJod+2qse/MGI3xcdEvoWs7yyoP4mZTID5fhpDSnv[/tex]是[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]的一个基，因此[tex=5.857x1.214]317mMb/UfJBjZHDU7raSnp+9Y8fiUt7JDwIiO502PxaYEulOh42V3W3nzcCNKW8CMKIPJC/kcx4hDmYdE4H78/WHNZvw4bE6chsMY+wfLgI=[/tex]。由于[tex=12.071x2.0]iZkg9HotdykN5NmHnJ/droKGin0pemAbZgZ3JDVuXyWnyIGumKNzNAn0yynJvHY1ocVDMbunSAMFTWDfs/HTFfXN2hh9tZ1evreupKN+UgsZRq/MiIAUpvuh6AThff+Z6oCK9Y471hWMeyW0le1TyyP3EzzOBeXIig723PP1LOergRTMw7z9I1c81yw+4XXL2S1X4x0seS5+0couPk88hQ==[/tex][tex=15.714x1.357]F+NWXYIbqoBpD+FJOD6P6treRkfEPHbFovlEK/d0kdbvPzfyZUf/EeWMb75boNcpLenziN7nAGJKbZ2jI/OY+ZCoBXS/A9NHMIAJUTH9TolpaqShKwlIT17Sg33f7Ba926ob98+4m4qOVL6bBF7JvgWaBLuYkvx7A/dEJr+M0M53e6SEelPjkaYhdfQ+JEaahhJ46x6sn4GaLBwI3R1AY5h1ynkVNn+DJHY4WlkkYYQ=[/tex]，[tex=8.286x1.571]0idGSV3RW/tbV3escumNdL2ZmI/rBJ98VrehAlzRZGNeRH5eRqwuKnSEfOp1cJIAhwvOw4djamjgNN1y7sG1ETjqJTAHOKjphAqMWK8ecTJQE6qeNxEji2H+u/osZhPty0XHgVvkYHagvx5W19p37Q==[/tex]，因此[tex=6.357x1.286]EWus09csxacgQ1nVz8/atWaTd3uKLhFYivZonzM8W7M=[/tex]。[img=431x403]177aee386c0a693.png[/img]

举一反三

内容

0
若:(1)函数 f(x)在点[tex=3.714x1.357]7VByCIzkNySq3s2l9I6f5zccNJDeV+6SQrVr3iwjgB0=[/tex]有导数,而函数g(x)在点[tex=2.286x1.0]DSJKaWfJALImFxxTg/8qhA==[/tex]没有导数;(2)函数f(x)在点[tex=3.714x1.357]7VByCIzkNySq3s2l9I6f5zccNJDeV+6SQrVr3iwjgB0=[/tex]没有导数,而函数g(x)在点[tex=2.286x1.0]DSJKaWfJALImFxxTg/8qhA==[/tex]有导数;(3)函数f(x)在点[tex=3.714x1.357]7VByCIzkNySq3s2l9I6f5zccNJDeV+6SQrVr3iwjgB0=[/tex]没有导数及函数g(x)在点[tex=2.286x1.0]DSJKaWfJALImFxxTg/8qhA==[/tex]没有导数,则函数[tex=5.643x1.357]GmtX7Vop79exGU/rpqXUYw==[/tex]在已知点[tex=2.286x1.0]DSJKaWfJALImFxxTg/8qhA==[/tex]的可微性怎样?
1
求抛物线 [tex=4.071x1.429]hl4JpLynrxmqrmVdtohNfg==[/tex] 与它的通过坐标原点的切线及 [tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex] 轴所围成的图形绕 [tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex] 轴旋转所得的旋转体的表面积. 解设切线为 $y=k x$, 它与抛物线的交点 $(x, y)$ 满足$$y=\sqrt{x-1}, y=k x, \frac{1}{2 \sqrt{x-1}}=k$$
2
在空间取定直角坐标系 [tex=3.143x1.0]11uB8u/aa5tBPn0lc/2zwg==[/tex] 以 [tex=0.857x1.0]JGak6BG8IqnzqFUlidM8wQ==[/tex] 表示空间绕 [tex=1.643x1.0]FE9IowmQhmFQWxnEPOl1+w==[/tex] 轴由 [tex=1.571x1.286]Egwgc/sA4c91SnJo8hcqpw==[/tex] 向 [tex=1.5x1.0]eSux/eDx7IGn9pvVEa9VoA==[/tex] 方向旋转 [tex=1.429x1.071]0x1sflXOqrsdrJlmAbVenQ==[/tex] 的变换，以 [tex=0.714x1.0]UF6bSrGL+IxoE78FFIRkgA==[/tex] 表示绕 [tex=1.571x1.286]Egwgc/sA4c91SnJo8hcqpw==[/tex] 轴由 [tex=1.5x1.0]eSux/eDx7IGn9pvVEa9VoA==[/tex] 向 [tex=1.643x1.0]FE9IowmQhmFQWxnEPOl1+w==[/tex] 方向旋转 [tex=1.429x1.071]0x1sflXOqrsdrJlmAbVenQ==[/tex] 的变换，以 [tex=0.571x1.0]SVbk6JgMC3iD8o/A9O6b0g==[/tex] 表示绕 [tex=1.5x1.0]eSux/eDx7IGn9pvVEa9VoA==[/tex] 轴由 [tex=1.643x1.0]FE9IowmQhmFQWxnEPOl1+w==[/tex] 向 [tex=1.571x1.286]Egwgc/sA4c91SnJo8hcqpw==[/tex] 方向旋转 [tex=1.429x1.071]0x1sflXOqrsdrJlmAbVenQ==[/tex] 的变换. 证明 [tex=6.357x1.214]QjLC2nJKcX7740JWxb7SVYVDV4INV7DTGrErQg0c9Ma96Z8p32Faa6Ui+JZoHELwg1mZeTXcoKOOnlq0i1wSRg==[/tex]；[tex=1.5x1.0]UZbiW+2K0+ZPZjkSkbzXng==[/tex][tex=0.786x1.286]94yQi4lF9xfg5bsYhw/rNQ==[/tex][tex=7.429x1.429]yZhMZrduGAuMEaMVc7GXey8T+zBFw+czE3vLGL9hZKIyErupBUbSMnuBYYYTsaI9JaDU1Cz/2m3NKRD3CBYcRZS/wKKCevjvGx8tchTEU+onNAJX4HcPCIWSLg5HyJix[/tex] 并验证 [tex=5.643x1.5]dh1Uq5ornjD+EDdKub98YzX2Ays3n5vOPyvvTaxwhnlrecsk0nlOusxVp4BTW0QjgzBL5bkwUrV9vJlYyW1MhA==[/tex] 是否成立.
3
设抛物线[tex=7.5x1.429]PuOOiuXliw3SbXOlC3PxEg==[/tex]与x轴有两个交点x=a,x=b(a<b).函数f在[a,b]上二阶可导,f(a)=f(b)=0，并且曲线y=f(x)与[tex=7.5x1.429]PuOOiuXliw3SbXOlC3PxEg==[/tex]在(a,b)内有一个交点.证明：存在[tex=3.286x1.357]EV4pc+LBkNBOhd4NZUA5NQ==[/tex]，使得[tex=4.357x1.429]/FYTUVhgTPYa3RqQR+bSSXpHSralD3pTYi2H35Z8qsw=[/tex].
4
6个顶点11条边的所有非同构的连通的简单非平面图有[tex=2.143x2.429]iP+B62/T05A6ZTM0eeaWiQ==[/tex]个，其中有[tex=2.143x2.429]ndZSw3zT0QTOVLVdoUto1Q==[/tex]个含子图[tex=1.786x1.286]J+vVZa2YaMpc6mJBbqVvWw==[/tex]，有[tex=2.143x2.429]lmhx48evnQMhi03NovPXig==[/tex]个含与[tex=1.214x1.214]kFXZ1uR8GjycbJx+Ts2kyQ==[/tex]同胚的子图。供选择的答案[tex=3.071x1.214]3KinXFh3SXhZ7nIe1y9KEV6aadxhhJWeEy6Dij1iObdMUZkY6ZA5J2dVVjPSuhEf[/tex]：(1) 1 ;(2) 2 ;(3) 3 ; (4) 4 ;(5) 5 ;(6) 6 ; (7) 7 ; (8) 8 。