課程名稱︰微積分乙
課程性質︰共同必修
課程教師︰陳其誠
開課系所︰管院各系、地理、經濟等系
考試時間︰2006.10.24
是否需發放獎勵金：需要^^"
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning. This exam contains nine problems
written in two pages.
(1) Calculate the derivatives of the following functions(6 points each):
(a) f(x) = x^24 + x^10 + 2006π
(b) f(x) = (x^2 + 1) / (x^3 - x)
(c) f(x) = cosx
(d) f(x) = (x + 1)^100 * x^99
(e) f(x) =√(x^2 + 1)
(2) Determine the slope of (the tangent line to) the graph of
3(x^2 + y^2)^2 = 100xy at the point (1,3). And find the tangent line
at the point. (10 points)
(3) A man 6 feet tall walks at a rate of 6 feet per second toward a street
light that is 15 feet above the ground. At what rate is the tip of his
shadow moving when he is 10 feet from the base of the light? (8 points)
(4) Let f(x) be a function satisfying the following conditions:
(1) f(-5) = -5 , f(0) = 0 , f(3) = 4 , f(4) = 0
(2) f'(x) > 0 for x < -5 , f'(-5) = 0 , f'(x) > 0 for -5 < x < 3 ,
f'(3) = 0 and f'(x) < 0 for x > 3
(3) f"(x) > 0 for -5 < x < 0 or 5 < x , f"(x) < 0 for x < -5 or 0 < x < 5
(4) lim f(x) = -∞ and lim f(x) = -5
x→∞ x→∞
Use the above data to answer the following:
(a) Determine the intervals where f is increasing and those where it is
decreasing. (4 points)
(b) Determine the intervals where f is concave up and those where it is
concave down. (4 points)
(c) Find all the critical points and every local maximum and local
minimum. (4 points)
(d) Determine all the inflection points. (4 points)
(e) Sketch the graph y = f(x) (7 points)
(5) Apply Newton's method to solve the equation x^2 - 2006x + 12000 = 0
and take x1 = 2001. Find x2 = ? (7 points)
(6) An off shore oil well is 2km off the coast. The refinery is 4km down the
coast. Laying pipe in the ocean is twice as expensive as on land. What
path should the pipe follow in order to minimize the cost? (8 points)
(7) Find the tangent line approximation of f(x) = 1 + 2(sinx)^2 at the point
(π/4 , 2). Namely, find two numbers A and B so that among all linear
functions, the function g(x) = A + B(x - π/4) is the best approximation
of f(x). (5 points)
(8) Use ε-δ terminology to prove lim 3x + 1 = 1. (5 points)
x→∞
(9) Assume that f'(1) exists and let g(x) = f((x - 1)^2). Then what is the
value of g'(0) ? (5 points)
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