微積分唔識做，請各位大哥幫幫忙﹗﹗﹗

PinkyPYC

唔識做微積分，請大哥們幫下我﹑指導下

Question 1:
Suppose f(x) = x2 – 2x + 3
Evaluate lim h → 0 [f(x + h) – f(x)]/h

Question 2:
Evaluate the following limits:
(a) lim x → ∞ (x2 + 3x)1/2 – x
(b) lim x → 3 (x3 – x + 26)/(x2 + 3)

Question 3:
Do the following limits exist? If so, evaluate them
(a) lim x → 0- 1/ |x|
(b) lim x → 3+ 1/| x - 3 |

Question 4:
(a) What is the domain of the following function?
f(x) = x/(x2 – 6x + 8)
(b) Suppose f(x) = 2x2 – 1
Find [f(x + 0.01) – f(x)]/0.01

Question 5:
Consider the following compound function h(x):
h(x) = x6 if x > 0
           - x5 if x < 0
(a) Evaluate lim x → 0- h(x) and lim x → 0+ h(x)
(b) Does lim x → 0 h(x) exist? If so, find it

AYuMe

Q1:
lim h → 0 [f(x + h) – f(x)]/h
=lim h → 0 [x^2 + 2xh + h^2 - 2x - 2h + 3 - x^2 + 2x - 3]/h
=lim h → 0 [2xh + h^2 - 2h]/h
=lim h → 0 [2x + h - 2]
=2x-2

Q2:
(a)我唔知1/2係指數定乘數= =所以幫唔到你
(b)因為分母代x=3唔會變成0 , 所以可以直接代x=3入公式 就會得到答案

[ 本帖最後由 AYuMe 於 11-2-8 10:39 PM 編輯 ]

PinkyPYC

Q.2 part a個1/2係指數來的

AYuMe

11-2-11 10:08 PM 上傳

下載附件 (39.16 KB)

答案係3/2
希望看得明..很少用小畫家寫字

PinkyPYC

我唔係好明你Q.2 a)中間打後既step係點來？
越計越怪咁

[ 本帖最後由 PinkyPYC 於 11-2-13 01:57 PM 編輯 ]

Naozumi

我諗佢係用 L'Hospital rule

不過可以第一步係分子分母同時乘 (x^2 + 3x)^(1/2) + x,
令到題目個 expression變成:

[(x^2 + 3 x) - x^2]/[(x^2 + 3x)^(1/2) + x]
再簡化下
3x/[(x^2 + 3x)^(1/2) + x]

分子分母同時除x, 如果 x->+infinity,不妨有 x = (x^2)^(1/2)
3/[(1 + 3/x)^(1/2) + 1]

最後x->+infinity時 3/x -> 0

[ 本帖最後由 Naozumi 於 11-2-13 11:38 PM 編輯 ]

PinkyPYC

我堅唔識做Q3同5，至於第四條個做法唔係好肯定...

Naozumi

Q3兩part都冇 limit,但我唔知你學過乜,所以唔知點教你做
Q5咪考慮x > 0時 lim x->0+ x6 = ???
x < 0時 lim x->0- (-x5) = ???
如果兩個???的數一樣咪lim x->0 h(x)存在lor

第4題我唔明你有乜野唔肯定,因為都好直接喎

PinkyPYC

咁第三題應該用咩方法做呀？
第四題個答案係咪
a) f(x) = x/ (x-2)(x-4)
b) 0.04

Naozumi

第3題我會用反證
假設lim x → 0- 1/ |x|存在並等於 L

咁
lim x → 0- 1/ |x| lim x → 0-   |x| = L*0
lim x → 0- [1/|x| *|x|] = 0
lim x → 0- 1 = 0
1 = 0
有矛盾

Q4
a 你知唔知乜野係domain?你個答案都唔係domain喎
b [2(x + 0.01)^2 - 1 - [2x^2 -1]]/0.01
  =[ 0.04x + 0.00002]/0.01
  = 4x + 0.02

[ 本帖最後由 Naozumi 於 11-2-15 08:13 AM 編輯 ]

PinkyPYC

我唔識咩係domain...
我都係唔係好明第三題

Naozumi

睇番maths書,一般form 4有提

PinkyPYC

我冇maths書，依家我地都唔使用書...

PinkyPYC

堅唔識做Q3 & Q4a)，有冇人可以詳細教下我點樣做呀？

[ 本帖最後由 PinkyPYC 於 11-2-19 02:22 PM 編輯 ]

PinkyPYC

份功課已經完成了，謝謝各位幫忙﹗﹗﹗