课程名称︰微积分乙上 课程性质︰必修 课程教师︰张志中 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2013/11/19 考试时限（分钟）：110 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. (10%) Evaluate the limits. n x (a) (5%) n为自然数, lim ----- = ? x→∞ x e lnx (b) (5%) p>0, lim ----- = ? x→∞ p x 2. (12%) arctan(x/3) 1 9+x^2 Let h(x) = --------------- + ---ln(-------), x≠0. x 6 x^2 b 2 c d (a) Determine a,b,c,d 属於实数 such that h'(x) = ax (9+x ) (arctan(x/3)) . (b) Find the line tangent to the graph of y=h(x) at x=√3. 3. (30%) Evaluate the limits. x 1 (a) (10%) lim (--- - ---) = ? x→1 x-1 lnx 1 (b) (10%) lim (secx)^(---) = ? x→0 x^2 2/3 1 (c) (10%) lim x * sin--- * cosx = ? x→-∞ x 4. (10%) 3 x Show that the equation x + e = 0 has exactly one real root. 5. (38%) x^3-3x^2+7x+3 Answer the following questions and sketch the graph of y=f(x)=---------------, 3x^2 x≠0. (a) Evaluate f'(x) = ? (b) Evaluate f''(x) = ? (c) f(x) is decreasing on the interval(s): ? (d) f(x) is concave upward on the interval(s): ? (e) f(x) has local maximum/maxima at (x,y) = (if any) ? , and local minimum/minima at (x,y) = (if any) ? (f) f(x) has inflection point(s) at x = (if any) ? (g) f(x) has asymptote(s) (if any) = ? (h) Sketch the graph of y=f(x). --

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