解下列几何问题:设抛物线[tex=5.786x1.429]hSurE+yrHHCxNtYWaQESYBwpITHTdCrB6QlVgfLyM2I=[/tex]通过原点[tex=2.286x1.357]sVCzP1QNUT517zJi7AAZqw==[/tex],且当[tex=3.286x1.357]P4bFrq1Y2Xf09lUts8bgeg==[/tex]时,[tex=2.357x1.214]xHnJkeOGjFTMqo4oqvoS4UWtWxUIzK8KQyPv/hPd5ac=[/tex].试确定[tex=2.286x1.214]/Uu9jgxB4g+DifSL38NMLQ==[/tex]的值,使得抛物线[tex=5.786x1.429]hSurE+yrHHCxNtYWaQESYBwpITHTdCrB6QlVgfLyM2I=[/tex]与直线[tex=4.071x1.214]68krnql5xkP9/gVPXfBtrg==[/tex]所围图形的面积为[tex=0.786x2.357]wpsXRIj0ceEvaPizZjXh1A==[/tex],且使图形绕[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴旋转而成的旋转体体积最小.

2022-05-30

解下列几何问题:设抛物线[tex=5.786x1.429]hSurE+yrHHCxNtYWaQESYBwpITHTdCrB6QlVgfLyM2I=[/tex]通过原点[tex=2.286x1.357]sVCzP1QNUT517zJi7AAZqw==[/tex],且当[tex=3.286x1.357]P4bFrq1Y2Xf09lUts8bgeg==[/tex]时,[tex=2.357x1.214]xHnJkeOGjFTMqo4oqvoS4UWtWxUIzK8KQyPv/hPd5ac=[/tex].试确定[tex=2.286x1.214]/Uu9jgxB4g+DifSL38NMLQ==[/tex]的值,使得抛物线[tex=5.786x1.429]hSurE+yrHHCxNtYWaQESYBwpITHTdCrB6QlVgfLyM2I=[/tex]与直线[tex=4.071x1.214]68krnql5xkP9/gVPXfBtrg==[/tex]所围图形的面积为[tex=0.786x2.357]wpsXRIj0ceEvaPizZjXh1A==[/tex],且使图形绕[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴旋转而成的旋转体体积最小.

答案：

解:由抛物线通过原点[tex=2.286x1.357]+Xa8fX15aepROWOQ80NZaw==[/tex],可知[tex=1.714x1.0]KsdsXiz3TIoOwg7hgNuJBA==[/tex],则抛物线方程为[tex=4.571x1.429]qutcE+rAGOImro5S0qD45g==[/tex],见图[tex=2.286x1.143]VJhJPAK6c8CWx6ExJizP5w==[/tex],则面积[tex=10.643x2.786]J5oVoFaVJaU69nx8ne2TBDFVHwRzs4rdA/Ars+gVqLcOqTl9gko576EDuDJs1rEzqQ4uZm+aAv11Dx9Hw81O9w==[/tex],即[tex=4.5x2.429]OpoByehklq9bq7zDnRG1hnVMn0VD22KNSvDUhBt9oPMIpHxRb2carhsOYKNR/TTW[/tex],绕[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴旋转的旋转体体积为[tex=23.0x2.786]zliBvpoa88TYgWZmewqdRuPnIxECO2MpL895XngM2zGWQvarNX5DrhREd7qMih42wNqEky/6YflqC/C1w5Ta4loX7pCjtQJynquC1xTRouub/eLA8rQwiEbbDvkzP5oKpgf1SISRyD9aUBuFEnPm27+3bowDIjMMjU7WImcMopI=[/tex][tex=8.857x2.357]3K/SZKC0IW84b96wPe8SWOcCUcV+86L2pl2BIKfPVBPkfnX3qmnjmOURFbAjiHkbIsNYj46FcQcjYwMdKVUizAoc8AjVa6BT6sn45M4PjeQ=[/tex].由[tex=4.5x2.429]OpoByehklq9bq7zDnRG1hnVMn0VD22KNSvDUhBt9oPMIpHxRb2carhsOYKNR/TTW[/tex]得[tex=4.714x2.357]kyxH3TbnYJ1uARzg5HXRpLo/VX0EHN3QyU8xwrfNcBY=[/tex],代人上式,得[tex=12.143x2.786]0mUmeQKqWdJxLwGpiNd8pDoItq5xzSznOKyAYgdUwFBn7X6KsNPAzLsGpp/f+Vf5dNbv4kPli2qhmdEI3qsNFDvw422Ss274OaXaPDuoMzg=[/tex],令[tex=10.571x2.786]mIjF+YCv2rkOkkeQ3vgiUEXmPCV5pFVIJX9mTdLKeRCAAQ2nENV7UK2Nnxb0ENoH13vPOFNRazTt0fiFzO6hvtFnnnxk1MJTQfO9vGOjHwY=[/tex],得[tex=3.0x2.357]DwBEpaVuRO8vWAHXZgW9uGHKctZM9y9/98WY+LeTS5s=[/tex].又[tex=7.214x2.5]Dq75QY5qIlYn0y0WLfeKeKgzyMFYGciTl2jHOfL00Ivj+Rnu6oleHmK+Ll/7uwh83xUYziBLKOejmJOK4kpMJg==[/tex],可知当[tex=3.0x2.357]3UiCrT0RmC5Mh0pngPN97n2bLrKSaj5wwWpFWvhPORM=[/tex]时,[tex=0.643x1.0]jro2X/cRz2SsmjZvcOdvsQ==[/tex]最小.此时,[tex=1.714x1.0]KLbk3ciBTpxgyOmkIDG2cQ==[/tex].故[tex=7.786x2.357]byZCqI/Rt2scQNFUeAouIWDMSXzulP26MLf53aiphNQ=[/tex][img=235x228]179358dbb6c5bac.png[/img]

举一反三

内容

0
[1989 年 2] 设抛物线[tex=5.786x1.429]hSurE+yrHHCxNtYWaQESYBwpITHTdCrB6QlVgfLyM2I=[/tex]过原点，当[tex=4.286x1.143]NTKJxb4sPu53TmmNfb9BbxCmD0njfTiqPrZqtEXQ55w=[/tex]时，[tex=2.357x1.214]aG47+7SEhmqHzrr0TZ8Tsg==[/tex]，又该抛物线与直线[tex=1.857x1.0]fwov+ZzREJJP/GTCJbKvrw==[/tex]及[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴围成平面图形的面积为[tex=1.5x1.357]7GCFN+wpKhWuwbrANtmgNg==[/tex]。求[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]，[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]，[tex=0.5x0.786]EL0hSqs6jZBGdsmH7TMShQ==[/tex]使此图形绕[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴旋转一周而成的旋转体体积最小。
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
设抛物线[tex=7.143x1.286]7yFMwM/4Nd+lfHNMTRRu+ac7hLI+DKw4KXRhJb/AHio=[/tex]通过点(0,0)，且当[tex=3.643x1.286]J2AjFpkP+hpGpzwZ3DOuKA==[/tex]时，[tex=2.357x1.286]KBZIJbskVVrycDOoD9RU26AVc5tr4kgvfe08o5WindY=[/tex]。试确定[tex=0.571x1.286]mRKL/orzOudCEARA8qn3Kw==[/tex]，[tex=0.5x1.286]PGyKeLDo0qv9T0n29ldi6w==[/tex]，[tex=0.5x1.286]m/VGGUpsnKNFGYXigdTc/A==[/tex]的值，使得抛物线[tex=7.143x1.286]7yFMwM/4Nd+lfHNMTRRu+ac7hLI+DKw4KXRhJb/AHio=[/tex]与直线[tex=2.357x1.286]jgIRiGqlkdCMqO92sJAASg==[/tex]，[tex=2.357x1.286]+lfyPLkaB2aZzha73p3Bvg==[/tex]所围图形的面积为4/9，且使该图形绕[tex=0.571x1.286]XubEW9+1+hkJqH7jXe5MrA==[/tex]轴旋转而成的旋转体的体积最小。
3
设拋物线 [tex=7.214x1.429]hSurE+yrHHCxNtYWaQESYBwpITHTdCrB6QlVgfLyM2I=[/tex]过原点,当[tex=4.286x1.143]NTKJxb4sPu53TmmNfb9Bb2yqhi+Jm/xG2jRm5Ftj9Js=[/tex]时[tex=3.071x1.214]2LSPaE5QBfGyldZNUIOPa54MTxSeL3/CPRmDm+BOVvU=[/tex] 又已知该抢物线与直线[tex=1.857x1.0]V1A/AhQDpaOkHSAYDdyRCQ==[/tex] 及[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex] 轴围成图形的面积为[tex=0.786x2.357]IwJCUxQJz+qfVDVP2eUlNg==[/tex], 求 [tex=2.571x1.214]JB9plX+DTdF1S9Y9u+/vQA==[/tex] 使得此图形绕[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex] 轴旋转一周而成的旋转体的体积最小.
4
求函数[tex=3.286x1.429]kdT+eIE7CHPynuN6CaN40g==[/tex]（抛物线）隐函数的导数[tex=1.071x1.429]BUw1BPFU3fsJlAl/vt9M9w==[/tex]当x=2与y=4及当x=2与y=0时,[tex=0.786x1.357]Hq6bf3CacUy07X+VImUMaA==[/tex]等于什么?