2015直通车高数基础题(5)
时间:2025-07-10
时间:2025-07-10
230 设u=f(r),而r
x2
y2
z2
,f(r)具有二阶连续导数,则 2u 2u u
x2 y2 z
2等于
(A)f (r)
1rf (r). (B)f (r) 2
r
f (r). (C)1r(r) 1r
f (r). (D)122f r2f (r) rf (r). [ ]
31 已知函数f(x+y,x-y)=x2-y2对任何x与y成立,则
f(x,y) f(x,y)
x
y
等于 (A)2x-2y.
(B)2x+2y.
(C)x+y.
(D)x-y. [ ]
32 设区域D={(x,y)|1≤x2+y2≤4},f(u)是区域D上的连续函数,则
f(x2 y2
)dxdy等于 D
(A)2π
2
rf(r2
1
)dr.
(B)2π[
2
0rf(r)dr 1
rf(r)dr]
(C)2π
2
rf(r)2
1
21dr.
(D)2π[
rf(r2)dr 0
rf(r)dr]
33 交换积分次序,则累次积分 2
dx
x2
f(x,y)dy
(A) 4dy
20y
f(x,y)dx.
(B) 40dy
y
0f(x,y)dx.
(C) 4dy 2
4
x
2f(x,y)dx.
(D) 0
dy
2
f(x,y)dx. [ ]
π
34 若
f(x,y)dxdy 2acos π)rdr,其中a>0为常数,则区域D是
D
d 2
f(rcos ,rsin (A){(x,y)|x2+y2≤a2}. (B){(x,y)|x2+y2≤ax}. (C){(x,y)|x2+y2≤a2,x≥0}. (D){(x,y)|x2+y2≤ay}. [ ] 35 设平面区域D={(x,y)|x2+y2≤1,y≥0};D1={(x,y)|x2+y2≤1,x≥0,y≥0},则
(A) xdxdy 2 xdxdy
(B)xydxdy 2D
D D
xydxdy
1
D1
(C)
|x|dxdy 2 xdxdy
(D)
y)dxdy 2D
D (x (x y)dxdy
1
D
D1
36 对于级数
( 1)
n 1
un,其中un>0(n=1,2,…)
,在下列命题中正确的是 n 1
(A)若
( 1)
n 1
un收敛,则
n 1 ( 1)n 1un必条件收敛.n 1
n 1
(B)若
( 1)
un收敛,则n 1 un必收敛.
n 1
(C)若
u
n
发散,则
( 1)
n 1
un必发散.
n 1
n 1
] ][
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