※ [本文轉錄自 NTU-Exam 看板]
作者: prolegend (淡淡憂鬱味) 看板: NTU-Exam
標題: [試題] 94上 張鎮華 微積分乙第一次考試 (一學期共三次)
時間: Fri Nov 11 03:18:38 2005
課程名稱︰微積分乙
課程性質︰必修
課程教師︰張鎮華
開課系所︰醫學院各系、生命科學系、農業化學系、公共衛生系等
考試時間︰2005.10.25(二) 三四節 (約兩小時)
試題 :
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屬於R.
x + 2 3x+4
(c) f(x) = ————, x≧0. (d) f(x) = e , x屬於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
簡答:
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