# 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

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

## 1227

## 1223

## 1213


## 1206

## 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) $\begin{align}\frac{1}{(1-x)^2}\end{align}$


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.


以上為草稿計算

## 1026考試 （11/1重考）
###訂正


## 1004 (4.2)

## 0927 (4.1)
作業

## 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)

## 0830(考卷)

## 0829

## 0725
p.170 27-30 p.178 1-8


## 0718

## 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. $\lim_{n\to\infty}(\sqrt{n^2+2n-1}-\sqrt[3]{n^3+2n^2-1})$

## 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
$=\lim_{n\to\infty}\frac{1}{2}(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\cdots\frac{1}{2n-1}-\frac{1}{2n+1})$
$=\lim_{n\to\infty}\frac{1}{2}\cdot\frac{2n}{2n+1}=\frac{1}{2}$


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