课程名称︰微积分乙上 课程性质︰大一共同必修科 课程教师︰陈其诚 开课学院：公卫院 医学院 农化 生科系 考试日期︰2008/10/21 考试时限：两节课 是否需发放奖励金：需要 XD 试题 : 1. (5 points each) Calculate the follwing limits: (a) lim sin(3x)/x x→0 (b) lim (e^5x-1)/x x→0 2. (5 points each) Calculate the derivatives of the following functions: (a) f(x) = 10x^21 + 2x^3 + 8 (b) f(x) = cos(x)/√x (c) f(x) = ln(x^2 +1) (d) f(x) = g(x^2) with g'(x) = √(x^4 -1) (e) f(x) = (e^x)sinx (f) f(x) = arcsin(x) 3. (10 points) A person uses Newton's method to solve the equation x^3 + x + 1 = 0 by taking Xo = -0.7. Calculate X1. 4. (10 points) Find the tangent line to the curve x^7 + xy - y^6 = 1 at (1,1). 5. (10 points) An airplane is flying on a flight path that is kept to be 10km above the ground. Suppose the flight path will take it directly over a radar tracking station, and let s:= s(t) denote the distance between the plane and the radar station (at time t). If s is decreasing at a rate of 600km per hour when s = 15km. What is the speed of the plane? 6. (10 points) Find the intervals on which the fuction f(x) = x^3 - 9x is increasing or decreasing. 7. (10 points) Find the maximum of the function Q(t) = 100 + 200t/(100 + t^2) for t≧0. 8. (5 points) For θε(0,90), let ΔABC be an right triangle with ∠BAC = θ(degree) and ∠ACB = 90(degree), and let s(θ) = BC/AB. Find ds/dθ. 9. (5 points) Suppose that f(x) is a differentiable function on (-∞,∞) and the equation f'(x) = 0 has exactly 7 distinct roots. Is it possible that the equation f(x) = 0 has more than 10 distinct roots? Why? --

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