課程名稱︰微積分乙上 課程性質︰必修 課程教師︰李聰成 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2013年11月7號 考試時限（分鐘）：160分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Evaluate the limit. x^1/3 －1 lim ─────── x→1 √x －1 2.Evaluate the limit. ╴╴╴╴ lim (√x^2+x+1 +X) x→-∞ 3. Find a function f and a number a such that sin(3+x)^2－sin9 lim ──────── ＝f'(a) x→0 x and evaluate the limit. 4. Sketch the graph oh the given function, making use of any suitable information you can obtain from the function and its first and second derivatives. x f(x)＝─── x^2＋1 5. Show that the equation x^3＋3x^2＋6x＋1＝0 has exactly(恰好) one root. 6. Find an equation of the tangent line(切線) to the curve x^3＋y^3＝6xy at the point (3,3). 7. Find the values of m and b such that f is differentiable on R, where ╭ x^2, if x<＝2 ; f(x)=│ ╰ mx＋b, if x>2 . 8. Find all critical(臨界) points and the absolute maximum and absolute minimum values of f(x)＝│x^2＋4x－12│ on the interval [－4,4]. 9. A man walks along a straight path at a speed of 1.5 m/s. A searchlight is located on the ground 6 m from the path and is kept focused on the man. At waht rate is the searchlight rotating when the man is 8 m from the point on the path closest to the searchlight? 10. Find the dimensions(長,寬) of the rectangle of largest area(面積) that can be inscribed(內接) in a circle of radius r. --

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