課程名稱︰微積分乙
課程性質︰大一共同必修
課程教師︰陳其誠
開課系所︰醫學院、公衛學院等
考試時間︰2005.11.29
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning. This exam contains eight problems
written in one page.
(1)(5 points each)Calculate the following limits:
(a)lim (e^2x-1)/x
x→0
(b)lim (11x^3+29x^2+2005)/(x^3+x-1)
x→∞
(c)lim (lnx)^100/x
x→∞
(d)lim (1+cosx)/sinx
x→π
(2)For each of the following functions, find the n'th Taylor polynomial
Pn(x) at x=a
(a)f(x)=lnx, a=1, n=3 (7 points)
(b)f(x)=sinx, a=π, n=10 (8 points)
(c)f(x)=e^x, a=0, n=1 (10 points)
(3)(10 points)A person uses Newton's method to solve the equation
x^3+x+1=0
by taking x0=-0.5 . Calculate x1
^ ^
下標 下標
(4)(10 points)Find the maximum of the function P(t)=1000+45/(25+x^2)
for t≧0
(5)(10 points each)Find the following dynamical systems, find every
euilibrium and determine the stability of each equilibrium:
(a)Mt+1 = Mt-0.5Mt/(1+0.1Mt) +1
^^^ ^ ^ ^
下標
(b)bt+1 = 2(1-bt/500)bt
^^^ ^ ^
下標
(6)(5 points)The dynamical system xt+1 = 3(1-xt)xt has a positive equilibrium.
^^^ ^ ^
下標
Is it stable? Why?
(7)(5 points)Suppose that f(x) is a differentiable function on (-∞,∞) and
the equation f'(x)=0 has less than 20 distinct roots. Is it possible for
the equation f(x)=0 to have more than 20 distinct roots? Why?
(8)(5 points)Is it possible that a discrete dynamical system has more than
one (locally) stable eqilibrium? If you say yes, give an example; if you
say no, give a reason.
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