推 Jman208 :農化已經收入 10/26 01:16
課程名稱︰微積分乙上
課程性質︰大一共同必修科
課程教師︰陳其誠
開課學院：公衛院 醫學院 農化 生科系
考試日期︰2008/10/21
考試時限：兩節課
是否需發放獎勵金：需要 XD
試題 :
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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