課程名稱︰微積分甲上 課程性質︰必修 課程教師︰李白飛 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰100/11/11 考試時限（分鐘）：150 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1 1 1 1. Evaluate lim ( ------- + ------- + ... + ------- ) _____ _____ _____ n->∞ √n^2+1 √n^2+2 √n^2+n ___ ___ _ ___ 2. Evaluate lim √x^3 ( √x+1 - 2√x + √x-1 ) x->∞ 3. Find the lines that are tangent to both the graphs of y = x^2 and y = -x^2 + 4x - 10. 4. Let f(x) be a differentiable function on R(實數). Suppose that |f'(x)|<=1 for all 實數x. Show that there exists at most one value of x>1/2 such that f(x)=x^2. 5. Show that the equation x^2 = x*sin(x) + cos(x) has two and only two real roots. 6. Show that x > sin(x) for all x > 0, and cos(x) > 1 - x^2/2 for all x =/= 0. 3 _________ 7. Find the asymptotes of the graph of y = √x^2 (x-1). (x-1) (x-2) 8. Sketch the graph of y = -----------. x^2 9. An empty water tank is in the shape of an inverted right circular cone with depth 10 m and top radius 5 m. The water is poured into the tank at a rate of 2 m^3/min. How fast is the water surface expanding after 9 minutes? 10. Find the shortest distance from a given point (0,b) on the y-axis to the parabola x^2 = 4y. (The number b may have any real value.) --

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