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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