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# System prepended metadata

title: 書華的數學作業

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# 書華的數學作業
$\begin{align}\end{align}$
## 0131
1. $\begin{align}\int\sec^3 x\tan^5 x dx\end{align}$

2. $\begin{align}\int\sin^4 xdx\end{align}$

3. $\begin{align}\int\frac{1}{\sqrt{1-(x+1)^2}}dx\end{align}$

4. $\begin{align}\int\frac{1}{x\sqrt{4x^2-1}}dx\end{align}$

5. $\begin{align}\int\sec xdx\end{align}$

6. $\begin{align}\int\frac{1}{x^2-3x+1}dx\end{align}$

6. $\begin{align}\int\frac{x+1}{x^2-3x+1}dx\end{align}$

7. $\begin{align}\int\frac{1+\ln x}{2+(x\ln x)^2}dx\end{align}$

8. $\begin{align}\int\sin^{-1}xdx\end{align}$

8. $\begin{align}\int\tan^{-1}xdx\end{align}$

8. $\begin{align}\int\sec^{-1}x dx\end{align}$

9. $\begin{align}\int\frac{\tan^{-1}\sqrt{x}}{\sqrt{x}(1+x)}dx\end{align}$

10. $\begin{align}\int\frac{1}{\sin 2x\cos x}dx\end{align}$

11. $\begin{align}\int\frac{\sin x}{1+\sin x}dx\end{align}$

12. $\begin{align}\int\cos 3x\cos 2x dx\end{align}$

13. $\begin{align}\int\frac{\ln\tan x}{\sin x\cos x}dx\end{align}$

14. $\begin{align}\int\frac{1-\ln x}{(x-\ln x)^2}dx\end{align}$

15. $\begin{align}\int\frac{1}{\sqrt{x}+\sqrt[3]{x}}dx\end{align}$

17. $\begin{align}\int\frac{1}{(x^2+1)^2}dx\end{align}$

18. $\begin{align}\int\frac{1}{1+\sin x}dx\end{align}$

19. $\begin{align}\int\frac{1}{2+\cos x} dx\end{align}$

20. $\begin{align}\int\frac{1}{x^2\sqrt{x^2-1}}dx\end{align}$

21. $\begin{align}\int\sqrt{1+\sqrt{x}}dx\end{align}$

22. $\begin{align}\int\frac{3x+1}{\sqrt{x^2+2x-5}}dx\end{align}$

23. $\begin{align}\int\frac{x}{4+x^4}dx\end{align}$

24. $\begin{align}\int\frac{x^2+2}{(x+1)^3}dx\end{align}$

25. $\begin{align}\int\frac{x^5}{\sqrt{1-x^2}} dx\end{align}$

26. $\begin{align}\int\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1} dx\end{align}$

27. $\begin{align}\int\frac{\ln(\ln x)}{x\ln x} dx\end{align}$

28. $\begin{align}\int e^x\sin x dx\end{align}$

30. $\begin{align}\int\frac{e^\sqrt{x}\sin\sqrt{x}}{\sqrt{x}}dx\end{align}$

## 0125
![](https://i.imgur.com/SA2xR35.jpg)

1. $\begin{align}\int\frac{1}{5x+3} dx\end{align}$

2. $\begin{align}\int e^{2x+3} dx\end{align}$

3. $\begin{align}\int xe^{x^2}dx\end{align}$

4. $\begin{align}\int x\sqrt{1-x^2}dx\end{align}$

5. $\begin{align}\int \frac{1}{x^2}\sin\frac{1}{x} dx\end{align}$

6. $\begin{align}\int\frac{e^{3\sqrt{x}}}{\sqrt{x}}dx\end{align}$

7. $\begin{align}\int\frac{1}{x(1+x^6)}dx\end{align}$

8. $\begin{align}\int\cos 2x dx\end{align}$

9. $\begin{align}\int\frac{\sin x}{\sqrt{5+\cos x}} dx\end{align}$

10. $\begin{align}\int\tan^4 xdx\end{align}$

11. $\begin{align}\int\frac{e^{2x}}{1+e^x}dx\end{align}$

12. $\begin{align}\int\frac{1}{1+e^x}dx\end{align}$

13. $\begin{align}\int\frac{1}{x\ln^2x}dx\end{align}$

14. $\begin{align}\int\frac{1}{x(1+2\ln x)}dx\end{align}$

15. $\begin{align}\int\frac{1}{a^2\cos^2 x+b^2\sin^2 x}dx\end{align}$

16. $\begin{align}\int\frac{1}{a^2+x^2}dx\end{align}$

17. $\begin{align}\int\frac{1}{a^2-x^2} dx\end{align}$

18. $\begin{align}\int\frac{1}{\sqrt{a^2-x^2}}dx\end{align}$

19. $\begin{align}\int\sin^3 xdx\end{align}$

20. $\begin{align}\int\sin^5xdx\end{align}$

## 0103
![](https://i.imgur.com/lbQAC3Y.jpg)
![](https://i.imgur.com/sLX5aJs.jpg)

## 1227 
![](https://i.imgur.com/sOEPmyc.jpg)

## 1223 
![](https://i.imgur.com/ufwacLF.jpg)

## 1213

![](https://i.imgur.com/iYun7ag.jpg)
![](https://i.imgur.com/h7fzmNt.jpg)
![](https://i.imgur.com/E4L2KdO.jpg)
![](https://i.imgur.com/0TRXHGL.jpg)

## 1206
![](https://i.imgur.com/CN4NoHp.jpg)
![](https://i.imgur.com/jEFsmZ5.jpg)


## 1123
Find the first 3 non-zero terms in the Maclaurin series for the follwing functions：

Using the fact $\begin{align}\frac{1}{1-x}=1+x+x^2+x^3+x^4+\cdots\end{align}$
by a simply substitution：

(1) $\begin{align}\frac{x}{1-4x}\end{align}$
(2) $\begin{align}\frac{x^4}{9+x^2}\end{align}$
(3) $\begin{align}\frac{1+2x}{1-x}\end{align}$
(4) $\begin{align}\frac{1}{x^2-5x-6}\end{align}$
(5) $\begin{align}\frac{1}{(1-x)^2}\end{align}$

![](https://i.imgur.com/goh4DyP.jpg)
![](https://i.imgur.com/aQtR7ga.jpg)


Using the definition of Maclaurin series
$\begin{align}f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots\end{align}$

(6) $\begin{align}\sin^2(x)\end{align}$
(7) $\begin{align}\sqrt{4-x}\end{align}$
(8) $\begin{align}\sin(x)\cos(x)\end{align}$
(9) $\begin{align}e^{-x}\end{align}$
(10) $\begin{align}\frac{2x^3-3x^2+1}{x+1}\end{align}$
答案附上計算過程，拍照後上傳到hackmd來.

![](https://i.imgur.com/A1Ylmpd.jpg)
![](https://i.imgur.com/n3rJCVk.jpg) 

以上為草稿計算

## 1026考試 （11/1重考）
###訂正
![](https://i.imgur.com/JDLBZT2.jpg)

![](https://i.imgur.com/LqjMm1c.jpg)
![](https://i.imgur.com/CULMN6X.jpg)

## 1004 (4.2)
![](https://i.imgur.com/Vh8A19p.jpg)

## 0927 (4.1)
作業![](https://i.imgur.com/HQNMkdI.jpg)

## 0926 (0922的考卷)
1. Use the definition of the derivative to find the deritative of f(x) = 4x^2-3 at x=2 
2. Sketch the possible graph of a continuous functions f that has domain [-3, 3], where f(-1) =1 1 and the group of y = f'(x) is shown below. 
==shown below 阿圖咧？==
![Uploading file..._vm93x86c5]()

4. Which of the following describes the behavior of y = cube root of (x+2) at x = -2 
(a) Differentiable (b) corner (c) cusp (d)vertical tangent (e) discontinuity 
4. let f(x) = int x ; find NDER (f(x), 3,). Is your answer to part a meaningful estimate of a derivative of f(x)? Explain. 
5. Find all values of x for which the functions f(x) - (x^2)+10x+25) / (x^2)-6x+8) 

7. Does the curve y = x^3+ 5x^2#+ 6 have any horizantoal tangents? If so, where? 

13. The number of gallons of water in a tank t minutes aftre the tank has started to drain is Q(t) = 250(40-t)^2. 
(a) How fast is the water running out at the end of 10 minutes? 
(b) What is the average rate at whcih the water flows out druing the first 10 minutes? 

14. Find dy/dy if y = cos (x) / (1+tan x) 
==dy/dy？==
16. Find tghe points on the graph of y = sec x, 0 <= x <= 2 pi, where the tangent iis parallel to the line 3y-2x =4
 
## 0920 (0922第三章考試) 

 
## 0912 (上週因為中秋連假，考完試後學校沒有上數學課，明天開始上3-2)
今天從第四章開始上~

## 0905 (學校作業) (Page 108-110 22,26,27,28,31,32,36-41) 
![](https://i.imgur.com/qGpjEkv.png)
![](https://i.imgur.com/hds6tku.png)
![](https://i.imgur.com/RoE2xqG.png)
![](https://i.imgur.com/9ujlATV.png)
![](https://i.imgur.com/a8wb3zW.png)


## 0830(考卷) 
![](https://i.imgur.com/EELAO9H.jpg)
![](https://i.imgur.com/hN16WdN.jpg)

 
## 0829
![](https://i.imgur.com/NW4mnBq.jpg)
![](https://i.imgur.com/tNt4a2x.jpg)
![](https://i.imgur.com/WPaRKwx.jpg)
![](https://i.imgur.com/v8CMhEs.jpg)

## 0725 
p.170 27-30 p.178 1-8 
![](https://i.imgur.com/IaSVZkD.jpg)
![](https://i.imgur.com/p58LOSL.jpg)


![](https://i.imgur.com/sxoHscp.jpg)

## 0718 
![](https://i.imgur.com/HtgB5iH.jpg)
![](https://i.imgur.com/qiIZVnm.jpg)
![](https://i.imgur.com/WrH3PaT.jpg)
![](https://i.imgur.com/k0uKkf4.jpg)
![](https://i.imgur.com/TT1nfRr.jpg)

## 0714
 1. $\lim_{n\to\infty}\frac{(1^2+2^2+3^2+\cdots+n^2)^2}{(1+2+3+\cdots+n)^3}$

 1. $\lim_{n\to\infty}(\frac{n^2+n+1}{n+1}-\frac{n^2+3n-5}{n+2})$

 1. $\lim_{n\to\infty}\frac{1^2+2^2+3^2+\cdots+n^2}{n^3+n^2+1}$

 1. $\lim_{n\to\infty}\frac{3n^4+2n^2+5}{1^3+2^3+3^3+\cdots+n^3}$

 1. $\lim_{n\to\infty}(\frac{n^2-2n}{n+3}-\frac{n^3+n^2}{n^2+n})$

 1. $\lim_{n\to\infty}(1-\frac{1}{4})(1-\frac{1}{9})(1-\frac{1}{16})\cdots(1-\frac{1}{n^2})$

 1. $\lim_{n\to\infty}\frac{\sqrt{9n+7}-\sqrt{9n+4}}{\sqrt{n+5}-\sqrt{n+1}}$

 1. $\lim_{n\to\infty}\frac{\sqrt{8n-1}+3}{\sqrt{2n+5}}$

 1. $\lim_{n\to\infty}\frac{5^{n+2}-2^{2n}}{5^{n-1}+3^{n+1}}$

 1. $a=\frac{\pi}{3}, b=\frac{\pi}{4},求\lim_{n\to\infty}(\frac{pa^n}{3+4a^n}+\frac{qb^n}{5+6b^n})$, $p, q$ 為常數

 1. $\lim_{n\to\infty}\sqrt[n]{3^n+5^n+8^n}$

 1. $\lim_{n\to\infty}(2^n+3^n)^\frac{2}{n}$

 1. $\lim_{n\to\infty}(\sqrt{n^2+2n-1}-\sqrt[3]{n^3+2n^2-1})$
![](https://i.imgur.com/KlCLOHo.jpg)
![](https://i.imgur.com/r2bU1Z7.jpg)
![](https://i.imgur.com/eiHZ4sa.jpg)

## 0704
 1. $\lim_{n\to\infty}\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}$
 $\frac{n}{\sqrt{n^2+n}}<\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}<\frac{n}{\sqrt{n^2+1}}$
 $\lim_{n\to\infty}\frac{n}{\sqrt{n^2+n}}=\lim_{n\to\infty}\frac{n}{\sqrt{n^2+1}}=1$
 什麼時候想到要用夾擠定理？黎曼和？
 
 1. $\lim_{x\to0}\frac{\sqrt{1+x\sin x}-\sqrt{\cos x}}{x\tan x}$
 $=\lim_{x\to0}\frac{1+x\sin x-\cos x}{x\tan x(\sqrt{1+x\sin x}+\sqrt{\cos x})}$
 看到 $1-\cos x=2\sin^2\frac{x}{2}$、$1+\cos x=2\cos^2\frac{x}{2}$ 半角公式
 $=\lim_{x\to0}\frac{2\sin^2\frac{x}{2}+x\sin x}{x\tan x}\cdot\frac{1}{\sqrt{1+x\sin x}+\sqrt{\cos x}}$
 $=\lim_{x\to0}\frac{\frac{2\sin ^2\frac{x}{2}}{x^2}+\frac{\sin x}{x}}{\frac{\tan x}{x}}\cdot\frac{1}{\sqrt{1+x\sin x}+\sqrt{\cos x}}=\frac{3}{4}$
 $\lim_{x\to 0}\frac{\sin^2 x}{x^2}=\lim_{x\to 0}\frac{\sin x}{x}\cdot\frac{\sin x}{x}=1\cdot 1=1$
 方程式 $\sqrt{x}=2\Rightarrow x=2^2$
 函數式 $\sqrt{x}$
 
 1. $\lim_{x\to0}\frac{1+\sin x-\cos x}{1+\sin px-\cos px}$，$p$ 為常數
 $=\lim_{x\to 0}\frac{\sin x+2\sin^2\frac{x}{2}}{\sin px+2\sin^2\frac{px}{2}}$
 $=\lim_{x\to 0}\frac{\frac{\sin x}{x}+2\frac{\sin^2\frac{x}{2}}{x}}{\frac{\sin px}{x}+2\frac{\sin^2\frac{px}{2}}{x}}=\frac{1}{p}$
 $\lim_{x\to 0}\frac{\sin ^2 x}{x}$
 $=\lim_{x\to 0}\frac{\sin x}{x}\cdot\sin x=0$
 
 1. $\lim_{x\to0}\frac{\sqrt{2-2\cos x}}{x}$
 $=\lim_{x\to 0}\frac{\sqrt{2\cdot2\sin^2\frac{x}{2}}}{x}$
 $=\lim_{x\to 0}\frac{2|\sin\frac{x}{2}|}{x}$
 左右極限不相等, 極限值不存在。
 1. $\lim_{x\to0}\frac{\sqrt{1+\tan x}-\sqrt{\sin x+1}}{x^3}$
 $=\lim_{x\to 0}\frac{\tan x-\sin x}{x}\cdot\frac{1}{x^2(\sqrt{1+\tan x}+\sqrt{\sin x+1})}$
 $=\lim_{x\to 0}\frac{\sin x}{x}(\frac{1-\cos x}{\cos x})\cdot\frac{1}{x^2(\sqrt{1+\tan x}+\sqrt{\sin x+1})}$
 $=\lim_{x\to 0}\frac{\sin x}{x}\cdot\frac{2\sin^2\frac{x}{2}}{x^2\cos x}\cdot\frac{1}{(\sqrt{1+\tan x}+\sqrt{\sin x+1})}$
 $=1\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$
 1. $\lim_{x\to a}\frac{(2x-a)^m-a^m}{x^n-a^n}$，$m，n$ 為自然數
 因式分解 $a^p-b^p=(a-b)(a^{p-1}+a^{p-2}b+a^{p-3}b^2+\cdots+ab^{p-2}+b^{p-1})$
 $=\lim_{x\to a}\frac{(2x-2a)[(2x-a)^{m-1}+(2x-a)^{m-2}\cdot a+\cdots +a^{m-1}]}{(x-a)[(x^{n-1}+x^{n-2}\cdot a+\cdots+a^{n-1})]}$
 $=2\cdot\frac{m\cdot a^{m-1}}{n\cdot a^{n-1}}=\frac{2m}{n}\cdot a^{m-n}$
 
 1. $\lim_{x\to0}\frac{(1+2x)^5-(1+4x)^3}{x}$
 
 1. $\lim_{x\to0}\frac{2x}{\sqrt{x+5}-\sqrt{5}}$
 
 1. $\lim_{x\to2}\frac{\sqrt{5x-1}-\sqrt{2x+5}}{x^2-4}$
 $=\lim_{x\to 2}\frac{3x-6}{(x^2-4)(\sqrt{5x-1}+\sqrt{2x+5})}$
 $=\lim_{x\to 2}\frac{3}{(x+2)(\sqrt{5x-1}+\sqrt{2x+5})}=\frac{1}{8}$
 
 1. $\lim_{x\to2}\frac{\sqrt[3]{3x+2}-2}{x-2}$
 立方差 $a^3-b^3=(a-b)(a^2+ab+b^2)$
 $=\lim_{x\to2}\frac{3x-6}{(x-2)(\sqrt[3]{(3x+2)^2}+2\cdot\sqrt[3]{3x+2}+4)}$
 $=\lim_{x\to2}\frac{3}{\sqrt[3]{(3x+2)^2}+2\cdot\sqrt[3]{3x+2}+4}$
 $=\frac{1}{4}$
 
 1. $\lim_{n\to\infty}n\left(\sqrt[3]{\frac{n-1}{n+2}}-1\right)$
 $=\lim_{n\to\infty}\frac{\frac{-3n}{n+2}}{\sqrt[3]{(\frac{n-1}{n+2})^2}+\sqrt[3]{\frac{n-1}{n+2}}+1}$
 $=-1$
 1. $\lim_{n\to\infty}n\left(1-\sqrt{\frac{2n-1}{2n}}\right)$
 $=\lim_{n\to\infty}n\frac{\sqrt{2n}-\sqrt{2n-1}}{\sqrt{2n}}$
 $=\lim_{n\to\infty}\frac{n}{\sqrt{2n}(\sqrt{2n}+\sqrt{2n-1})}=\frac{1}{4}$
 
 1. $\lim_{n\to\infty}\frac{\sqrt{n^4+3n^3-6}-(n-1)(n+1)}{n}$
 $=\lim_{n\to\infty}\frac{n^4+3n^3-6-n^4+2n^2-1}{n(\sqrt{n^4+3n^3-6}+(n^2-1))}$
 $=\frac{3}{2}$
 
 1. $\lim_{n\to\infty}\left[\sqrt{n^2+4n+5}-(n-1)\right]$
 $=3$
 
 1. $\lim_{n\to\infty}\sqrt{n}(\sqrt{n+2}-\sqrt{n+1})$
 $=\frac{1}{2}$
 
 1. $\lim_{n\to\infty}\left[\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\cdots+\frac{1}{(2n-1)(2n+1)}\right]$
 分項消去 $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$
 $\frac{1}{n(n+2)}=\frac{1}{2}(\frac{1}{n}-\frac{1}{n+2})$ Telescope
 $=\lim_{n\to\infty}\frac{1}{2}(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\cdots\frac{1}{2n-1}-\frac{1}{2n+1})$
 $=\lim_{n\to\infty}\frac{1}{2}\cdot\frac{2n}{2n+1}=\frac{1}{2}$
 
 ![](https://i.imgur.com/v9sRdCk.jpg)
![](https://i.imgur.com/vemI772.jpg)

 1. $\lim_{x\to 0}\frac{\tan x-\sin x}{\sin^3 x}$
 $=\lim_{x\to 0}\frac{1-\cos x}{\sin^2 x\cos x}$
 
 1. $\lim_{x\to 0}\frac{\tan(\frac{x}{2})}{x}=$？
 背：(1) $\lim_{x\to 0}\frac{\sin x}{x}=1$
  (2) $\lim_{x\to 0}\frac{1-\cos x}{x}=0$
  (3) $\lim_{x\to 0}\frac{\tan x}{x}=1$
 $=\frac{1}{2}$
 1. $\lim_{x\to\infty} x(\sqrt{x^2+1}-x)=$？
 $=\frac{1}{2}$
 
 1. $\lim_{x\to \infty}(\sqrt{4x^2+3x+1}-\sqrt{4x^2-3x-2})=$？
 $=\frac{3}{2}$
 
 1. $\lim_{x\to 1}\frac{x+x^2+\cdots+x^n-n}{x-1}$
 $=\lim_{x\to 1}\frac{(x-1)+(x^2-1)+(x^3-1)+\cdots+(x^n-1)}{x-1}$
 $=1+\cdots +n=\frac{n(n+1)}{2}$
 
 1. $\lim_{x\to 1}\frac{x^k-1}{x-1}=$？
 $=\lim_{x\to 1}(x^{k-1}+\cdots 1)$
 $=k$
 
 1. $\lim_{x\to 0}\frac{\sqrt[3]{x+1}-1}{x}=$？
 $=\lim_{x\to 0}\frac{(\sqrt[3]{x+1}-1)(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}+1)}{x\cdot(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}+1)}$
 $=\frac{1}{1+1+1}$

 1. $\lim_{x\to\infty}\frac{(2x-3)^{20}(3x+2)^{30}}{(5x+1)^{50}}=$？
 $=\frac{2^{20}3^{30}}{5^{50}}$
 
 1. $\lim_{x\to 1}(\frac{1}{1-x}-\frac{3}{1-x^3})=$？
 $=\lim_{x\to 1}\frac{1+x+x^2-3}{1-x^3}$
 $=\lim_{x\to 1}\frac{x+2}{x^2+x+1}$
 $=1$
 
 1. $\lim_{x\to \infty}(\frac{x^3}{2x^2-1}-\frac{x^2}{2x-1})=$？
 $=\lim_{x\to\infty}\frac{2x^4-x^3-2x^4+x^2}{(2x^2-1)(2x-1)}$
 $=\frac{-1}{4}$
 
 1. $\lim_{x\to 1}\frac{\sqrt{3-x}-\sqrt{1+x}}{x^2-1}=$？
$=\lim_{x\to 1}\frac{2-2x}{(x-1)(x+1)(\sqrt{3-x}+\sqrt{1+x})}$
$=\frac{-1}{2\sqrt{2}}$

 1. $\lim_{x\to 4}\frac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt{2}}=$？
$=\lim_{x\to 4}\frac{(2x-8)(\sqrt{x-2}+\sqrt{2})}{(x-4)(\sqrt{2x+1}+3)}$
$=\frac{2\sqrt{2}}{3}$