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作者: heterotroph () 看板: NTU-Exam
標題: [試題] 94上 陳其誠 微積分乙 第一次期中考
時間: Sat Nov 26 00:26:33 2005
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??好像還有別學院的??
考試時間︰2005.10.25
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning. This exam contains seven problems
written in two pages.
(1)(6 points) Calculate the derivatives of the following functions :
(a) f(x)= x^25 + x^10 + 2005x
(b) f(x)= (x^2 - 1)/(x^3 + 1)
(c) f(x)= e^x
(d) f(x)= ln(x^2 - 1)
(e) f(x)= √(x^2 + 1)
(2)(10 points) The concentration Mt of certain medication in the bloodstream
at time =t satisfies
M(t+1) = Mt-(0.5Mt)/(1+0.1Mt) + 1
^^^^^
這是下標，和Mt的t一樣
Find the equilibrium.
(3)The number of bacteria in a lab follows the discrete-time dynamical system
2bt if bt≦1
b(t+1)=｛
^^^^^ -0.5(bt-1)+2 if bt＞1
下標
where t is measured in hours and bt in millions of bacteria.
(a)(5 points) Graph the updating function (namely,the function f(x) such
that b(t+1) = f(bt)
^^^^^
下標
(b)(15 points)Find the equilibrium. Are they stable? Why?
(4)Let f(x)= 2/3*x^3 - 2x + 1
(a)(6 points)Determine the intervals where f is increasing and those where
f is decreasing.
(b)(4 points)Determine the intervals where f is concave up and those where
it is concave down.
(c)(3 points)Find all the critical points and every local maximum and local
minimum.
(d)(2 points)Determine all the inflection points.
(e)(5 points)Sketch the graph y=f(x)
(5)(8 points)Let f(x)=√x and put f(x)= 1 + 0.5(x-1) + (x-1)g(x).Find the
limit
lim g(x)
x→1
(6)(6 points)Assume that f'(0) exists and let g(x)= f(x^2). Then what is the
value of g'(0)?
(7)(6 points)Suppose that f is increasing and concave up. Assume the inverse
function g exists. What can you say about g?
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※ 編輯: heterotroph 來自: 140.112.247.43 (11/26 00:34)
※ 編輯: heterotroph 來自: 140.112.247.43 (11/26 08:27)
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