※ 引述《EricDampier (蛋皮)》之銘言： : 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) x x ∫ f(t) dt = (x)(e^(2x)) + ∫ (e^(-t))(f(t)) dt 0 0 等號兩邊對 x 微分得 f(x) = e^(2x) + (2x)(e^(2x)) + (e^(-x))(f(x)) (1 - e^(-x))(f(x)) = (1 + 2x)(e^(2x)) (1 + 2x)(e^(2x)) f(x) = ------------------ 1 - e^(-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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