監試人員一時糊塗...竟然說考卷給你們帶回去當禮物
後來第二節他去問才知道試卷也要收回
所以就被拿回來當禮物了...
甲.填充
1. if f is a continuous function such that
x x
∫f(t)dt = x．exp(2x) + ∫exp(-t)f(t)dt for all x,
0 0
find an explicit formula for f(x)
2. in what direction is the derivative of
(x^2+y^2)
f(x,y) = ───── at P(1,1) equal to zero?
(x^2-y^2)
3. find the maximum value of x^2 + y^2 subject to the constraint
x^2 - 2x + y^2 - 4y = 0
4. suppose that f(0) = -3 and f'(x) <= 5 for all values of x
how large can f(2) possibly be?
5. find the tangent plane of the surface
cos(πx) - x^2．y + exp(xz) + y．z = 4
x-2y
6. evaluate ∫∫ ─── dA , R is the parallelogram enclosed by the lines
R 3x-y
x-2y = 0 , x-2y = 4 , 3x-y = 1 , 3x-y = 8
7. find the area of surface cut from parabloid x^2 + y^2 - z = 0 by the
plane z = 2
8. evaluate ∮(6y+x)dx+(y+2x)dy , C : (x-2)^2 + (y-3)^2 = 4
C
乙.計算,證明
1. evaluate the following limits
tan(2x) n √(n^2 - j^2)
(a) lim （tan x） (b) lim （Σ ───────）
x→(π/4)- n→∞ j=1 n^2
∞ n ln(n)
2. (a) test the series Σ (-1) ──── for convergence or divergence
n=1 n-ln(n)
∞ x^n
(b) let f(x) = Σ ── find the intervals of convergence for f' & f''
n=1 n^2
3. evalute
a/√2 √(a^2-y^2)
(a) ∫ ∫ exp(x^2+y^2) dxdy
0 y
8 2 dydx
(b) ∫ ∫ ────
0 x^(1/3) y^4 + 1
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