2018 微微微积积积分分分(((1)))-2 参参参考考考解解解答答答
一一一、、、计算题:(每题褵分,共褳褰分)
褱、求曲线x = cos t, y = sin t, z = t cos t上点(1, 0, 0)处的切线方程褮
解褺 对曲线方程关于t求导可得切向量为
(−sin t, cos t, cos t − t sin t). ·····················3分
代入点(1, 0, 0)对应的参数t = 0可得点(1, 0, 0)处的切向量为(0, 1, 1). 于是褬
切线方程为
x − 1
0
=
y
1
=
z
1
. ·····················2分
褲、求曲面z = xy在点(−2, −3, 6)处的切平面方程褮
解褺 曲面z = xy的法向量是
(−z
x
, −z
y
, 1) = (−y, −x, 1), ·····················3分
于是在点(−2, −3, 6)处的法向量为(3, 2, 1). 因此,所求切平面方程为
3(x + 2) + 2(y + 3) + z − 6 = 0,即
3x + 2y + z + 6 = 0. ·····················2分
褳、设D = {(x, y) ∈ R
2
| x + y 6 1, x > 0, y > 0},求
RR
D
xdxdy.
解褺
ZZ
D
xdxdy =
Z
1
0
dx
Z
1−x
0
xdy ·····················3分
=
Z
1
0
(x − x
2
)dx
=
1
2
−
1
3
=
1
6
. ·····················2分
褱
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