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作者: prolegend (淡淡忧郁味) 看板: NTU-Exam
标题: [试题] 94上 张镇华 微积分乙第一次考试 (一学期共三次)
时间: Fri Nov 11 03:18:38 2005
课程名称︰微积分乙
课程性质︰必修
课程教师︰张镇华
开课系所︰医学院各系、生命科学系、农业化学系、公共卫生系等
考试时间︰2005.10.25(二) 三四节 (约两小时)
试题 :
Examination 1 (Calculus B, first semester)
[Each problem weights 10 point] 2005-10-25 (Gerald J. Chang)
Who is the author of the textbook? What is the Chinese name of the instruction?
1. Find the center and radius of the circle given by the equation
2 2
2x + 2y + 4x - 8y - 8 = 0.
2. Which of the following function is one to one? Which is not? Why?
2 2
(a) f(x) = 2x + 4x + 2, x≧0. (b) f(x) = 2x + 4x + 2, x属于R.
x + 2 3x+4
(c) f(x) = ————, x≧0. (d) f(x) = e , x属于R.
x + 1
3 3
(e) f(x) = ———————, x≠0. (f) f(x) = ———————, x＞0
2 2
2x + 4x + 2 2x + 4x + 2
3. A strain of bacteria reproduces asexually every 45 minutes. That is, every
45 minutes, each bacteria cell splits into two cells. If initially, there
are 5 bacteria, how long will it take until there are 1280 bacteria?
28
4. Let a = — for n≧0, If lim a exists, find the limit. Determine for
n+1 a n→∞ n
n
which value of a the limit lim a exists, and for which value of a the
0 n→∞ n 0
limit lim a does not exist.
n→∞ n
5. Suppose a sequence N , N , N ,... is defined by N = N = 1 and
0 1 2 0 1
Nt
N = 2 N + N for t = 1, 2, 3,.... Assume that lim —— exists. Find
t+1 t t-1 n→∞ Nt-1
Nt+1 Nt+2
the limit lim —— and lim ——.
n→∞ Nt n→∞ Nt
6. Determine whether each of the following statements is true or not. If it
is false, explain why or give an example that show it is false.
(a) If both f(c) and g(c) are defined and lim f(x) = lim g(x), then
x→c x→c
f(c) = g(c).
1 1 1 1 1
(b) lim —— , lim —— , lim —— , lim —— , lim —— all exist.
x→-∞ sinx x→0- sinx x→0+ sinx x→0 sinx x→∞ sinx
7. Find constant a and b such that the following function is constinuous on
the entire real line. 3 2
┌ x + 2x + 1, for x≦1
│
f(x) =│ ax + b , for 1＜x＜3.
│ 2
└ x - 2x + 3, for x≧3
sinx 1
8. (a) Prove that cosx ≦ —— ≦ ——.
x cosx
sinx
(b) Use (a) to determine lim ——.
x→0+ x
sinx sinx
(c) Determine lim —— and lim ——.
x→0- x x→0 x
3 2
9. How many real roots are there for the polynomial p(x) = x -10x -20x + 30?
Why?
10. Use the ε-δ definition of a limit to prove that lim 2√(x+5) = 6.
x→4
简答:
1. Center: (-1,2) ; Radius = 3.
2. (a) One to One (b) Not One to One (c) One to One
(d) One to One (e) Not One to One (f) One to One.
3. 360 minutes
4.(1) lim a = √28 or -√28.
n→∞ n
(2) When a = √28 or -√28, limit exists.
0
Nt+1
5.(1) lim —— = 1 + √2.
n→∞ Nt
Nt+2
(2) lim —— = 3 + 2√2.
n→∞ Nt
6.(a) False (b) False.
7.(a,b) = (1,3)
8.(a)略
(b) 1
(c) 1;1
9. 3 real roots.
10.略