求抛物线y=x^2在x=1与x=2处的切线方程
\u6c42\u629b\u7269\u7ebfy=x^2\u5728x=1\u4e0ex=2\u5904\u7684\u5207\u7ebf\u65b9\u7a0b\u89e3\uff1a
y '=2x
\u2460\u5f53x=1\u65f6\uff0cy=1\uff0c\u659c\u7387K=y '=2
\u6545\u5207\u7ebf\u65b9\u7a0b\uff1ay-1=2(x-1)\uff0c\u5373y=2x-1
\u2461\u5f53x=2\u65f6\uff0cy=4\uff0c\u659c\u7387K=y '=4
\u6545\u5207\u7ebf\u65b9\u7a0b\uff1ay-4=4(x-2)\uff0c\u5373y=4x-4
\u4e5f\u53ef\u4ee5\u8bbe\u4e00\u6b21\u51fd\u6570y=kx+b\u548cy=x²\u8054\u7acb\u6d88\u53bby\uff0c\u5f97\u5230\u5173\u4e8ex\u7684\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\uff0c\u518d\u4ee4\u25b3=0\uff0c\u4e5f\u53ef\u4ee5\u6c42\u89e3\u3002
1\u3002\u56fe\u8c61\u6cd5\uff0c
\u8fc7\u70b9\uff081\uff0c1\uff09\u4f5c\u5207\u7ebf\uff0c\u5927\u81f4\u53ef\u5f97
\u5207\u7ebf\u4e3ay=2x-1
2\u3002\u5b9a\u4e49\u6cd5\uff0c
\u4f5c\u629b\u7269\u7ebf\u7684\u5272\u7ebfPQ\u4ea4\u629b\u7269\u7ebf\u4e8eP\uff08x1\uff0cy1\uff09[\u5de6\u4e0b\u65b9]\u548cQ\uff08x2\uff0cy2\uff09[\u53f3\u4e0a\u65b9]
\u663e\u7136\uff0c\u76f4\u7ebfPQ\u7684\u659c\u7387k=tan\u03b1=\uff08y2-y1\uff09/\uff08x2-x1\uff09
\u56e0\u4e3ax2=x1\uff08\u5373x2\u6bd4x1\u5927\u25b3x\uff09\uff0cy1=x1²\uff0cy2=\uff0c
\u4e8e\u662f\uff0cy2=\uff08x1\uff09²\uff0cy2-y1=2x1\u25b3x+\u25b3x²
\u6240\u4ee5k=\uff082x1\u25b3x+\u25b3x²\uff09/\u25b3x
\u5373k=2x1+\u25b3x
\u5f53P\u3001Q\u65e0\u9650\u903c\u8fd1\u70b9\uff081\uff0c1\uff09\u65f6\uff0c2x1=2\uff0c\u25b3x=0
\u6240\u4ee5k=2\uff0c\u5207\u7ebf\u4e3a
3\u3002\u6c42\u5bfc\u6cd5\u3002
\u56e0\u4e3a\u659c\u7387k=\uff08x²\uff09\u2032=2x
\u5f53x=1\u65f6\uff0ck=2
\u6240\u4ee5\u8fc7\u70b9\uff081\uff0c1\uff09\u7684\u5207\u7ebf\u4e3ay=2x-1
切线的斜率为函数在该点的导数
当 x=1时
K1=y′=2x=2
把 x=1代入函数得y=1
设切线方程为
y=K1x+b
1=2+b
b=-1
切线方程为y=2x-1
当 x=2时
K2=y′=2x=4
把 x=2代入函数得y=4
设切线方程为
y=K2x+b
4=8+b
b=-4
切线方程为y=4x-4
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