課程名稱︰微積分甲上 課程性質︰暑修 課程教師︰朱樺 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017/07/17 考試時限（分鐘）：120 試題 : (1) (35%) Find the limit. √(1+x)+√(1-x) (a) lim --------------- x→0 √(4-x)-2 (b) lim ln x tan(πx/2) x→0+ (c) lim (4x+1)^cotx x→0+ (d) Let f(x) = √(x^2-ax)-√(x^2-bx). Find lim f(x) and lim f(x). x→∞ x→-∞ [x]-2 (e) Let f(x) = ---------. Find lim f(x) and lim f(x). [x^2]-4 x→-1+ x→2- secx -1 2/3 (2) (7%) Find the derivative of f(x) = log∣2 sin (x )∣. (x^4)sin(1/x) x≠0 (3) (7%) If f(x) = { find f'(0) and f''(0). 0 x = 0, x (4) (7%) Let f(x) = x+e and g(x) be the inverse function of f(x). Find g'(1). 1 (5) (7%) Find the nth derivative of the function f(x) = -----. x^2-x (6) (7%) If x^y = y^x , find equations of the tangent line and the normal line to the curve at the point (a,a), a>0, a≠e. (7) (7%) Find two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y = x^2. (8) (7%) Let n be an positive even integer. Show that the equation x^n+2x^2+4x-1 = 0 has exactly two real solutions. (You may try the case n=4 first.) (9) (7%) A kite 50 m above the ground moves horizontally at a speed of 2m/sec. At what rate is the angle between the string and the horizontal decreasing when 100 m of the string has been let out? (10) (7%) A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the larger cone. Find the height h such that the inner cone has maximum volume. -1 x-1 (11) (7%) Sketch the gragh of y = tan (-----). x+1 Find all asymptotes, the intervals of increase or decrease, the intervals of concavity, and the local maximum and the local minimum values, the inflection points. --

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