# 大一奇數微積分共筆 (E6 A204)(Ch.7)

:::info
授課教師：高華隆
:::
- 輸入數學式的操作說明可以在[這裡](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference)找到

- 有的中文沒抄到，所以是找[國家教育研究院的數學名詞中英對照](http://www.mathland.idv.tw/scene/tables/mathchieng.pdf)
- 上為簡易版，[詳細版可按此下載](http://terms.naer.edu.tw/terms/manager_admin/new_file_download.php?Pact=FileDownLoad&button_num=g5&source_id=284&Pval=1585)

- 授課速度有點快，歡迎幫忙校稿或補充

- 方便複製：
- $\lim \limits_{h \to 0}$
- $\dfrac {d}{dx}$
- $\lim \limits_{x \to a} \dfrac {f(x)}{g(x)}$
- $ [-\frac \pi 2, \frac \pi 2]$

# 微積分 Chapter7：Transcendental Functions

## 7-1 Inverse functions and their derivatives

- 應該是其他版的，正確版本沒問題→~~前面相關單元： 1-6、(2-5)、3-8~~

### One-to-one funtions

- **Defintion**

- A function $f(x)$ is **one-to-one** on a domain $D$ if $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$ in $D$.

$g(f(_j)) = x_j, \forall j$
$f(g(a_j)) = a_j, \forall j$
$g \equiv f^{-1}$

- **Example**


:::success
1. $f(x) = \sqrt x : [0, \infty) \to \Bbb R \Rightarrow f$ is one-to-one on $[0, \infty)$
2. $f(x) = x^2 : \Bbb R \to \Bbb R \Rightarrow f$ is not one-to-one on $\Bbb R \because f(-1) = f(1) = 1$
3. $f(x) = \sin x : [0,\pi] \to \Bbb R \Rightarrow f$ is not one-t-one on $[0,\pi] \because f(0) = f(\pi) = 0$
4.
:::

- **Note:**

:::warning
1. Some function are not one-to-one on their natural domain,
but we can resrtict the function to a smaller domain
s.t. the function is one-to-one

2. $f$ is one-to-one on $D \Leftrightarrow$ each horizontal line intersects the grapf $y=f(x)$ at most one point.
:::

### Inverse functions

- **Definition:**

:::info
$f : D \to f(D)$ is one-to-one on a domain $D$ with range $f(D).$
The inverse function $f^{-1}:f(D) \to D$ of $f$ is defined by
$$f^{-1}(b) = a \quad \mathsf{if} \; f(a) = b, \quad \forall b \in f(D)$$

:::

- **Note:**
:::warning
1. "$f^{-1}$" reads "$f$ inverse"
2. $f^{-1} (f(x)) = x,\quad \forall x \in D$
$f(f^{-1}(y)) = y,\quad \forall y \in f(D)$
:::

### Finding inverses

- 圖形以 $x = y$為對稱軸，將x y對調
- 只有 one-to-one function 會擁有 inverse function


- **Note**

:::warning
The graph $y = f^{-1}(x)$
is obtained by refleting the graph $y=f(x)$ about the line $y=x$

:::

- **Step**

:::info
1. $y=f(x) \Rightarrow x = g(y)$
ex: $y=f(x) - x^2 \Rightarrow x = \sqrt y =\equiv g(y)$

2. Interchange, $x$ and $y$
i.e. $y=g(x) \equiv f^{-1}(x)$
i.e. $y= \sqrt x = f^{-1}(x)$
:::

- **Example**


Find the inverse of $f= \frac x 2 +1$,
expressed as a function of $x$

:::success
**Sol:**
$y = \frac x 2 +1 \Rightarrow x = 2y-2$
$\Rightarrow x = f^{-1}(y) = 2y-2$
$\Rightarrow y = f^{-1}(x) = 2x-2$

:::

### Derivatives of inverses of differentiable functions

- **Recall**

If $f$ and $f^{-1}$ are diff.
:::info
$$ f(f^{-1}(x)) = x$$
:::
$1 = \frac d {dx} = \frac d {dx} f(f^{-1}(x)) = f'(f^{-1}(x)) \bullet \frac d {dx} f^{-1}(x)$
$\Rightarrow \frac d {dx} f^{-1}(x) = \frac a {f'(f^{-1}(x))}$

- **Theorem**


$I \subset \Bbb R$, on interval, $f:I \to f(I)$ is one-to-one

1. If f is conti. on $I \Rightarrow f^{-1}$ is conti. on $I$
2. If $f$ is diff. on $I$ and $f'(x) \neq 0, \quad \forall x \in I$
3. $\Rightarrow f^{-1}$ is diff. on I and,
:::info
$$\frac d {dx} f^{-1}(x) = \frac 1 {f'(f^{-1}(x))}$$
:::
- **Proof**


1. Check: $\lim \limits_{y \to y_0} f^{-1}(y_0), \quad \forall y_0 \in f(I)$
:::info
**Pf:**
$y_0 = \lim \limits_{y \to y_0} y = \lim \limits_{y \to y_0} f(f^{-1}(y)) = f(\lim \limits_{y \to y_0} f^{-1}(y))$
$\Rightarrow f^{-1}(y_0) = \lim \limits_{y \to y_0} f^{-1}(y)$
$\because f$ is one-to-one $\Rightarrow f^{-1}(y_0) = \lim \limits_{y \to y_0} f^{-1}(y)$
:::

2. Check: $\lim \limits_{y \to y_0} \frac {f^{-1}(y) f^{-1}(y_0)} {y - y_0}$ exist
:::info
**Pf:**
$\forall y, y_0 \in f(I)$
$\Rightarrow \exists ! x, x_0 \in I$ s.t. $f(x) = y, f(x_0) = y_0$
$\lim \limits_{y \to y_0} \frac {f^{-1}(y) - f^{-1}(y_0)}{y - y_0} = \lim \limits_{y \to y_0} \frac {x - x_0}{f(x) - f(x_0)}$
:::
沒抄到

- **Example**
:::success
$f(x) = x^2, x>0 \Rightarrow f^{-1}(x) = \sqrt x, x>0$
$\Rightarrow f'(x) = 2x$
$f^{-1}(4) =$
:::

- **Example**
:::success
$f(x) = x^3 - 2, x>0$
Find $f^{-1}(x)$
**Sol:**
$f'(x) = 3x^2 \Rightarrow f'(2) = 12$
$f'(6) =$
:::

## 7-2 Natural logarithms

- 應該是其他版的，正確版本沒問題→~~1-6(定義)、3-8(微分)、8-3(三角函數積分)~~

### logarithmic functions


### Natural logarithms

- **Defintion**

The natural logarithms is the function defined by
:::info
$$\ln x = \int_1^x \frac 1 t dt, \quad \forall x >0$$
:::


- **Note**

:::warning
1. $\ln 1 = 0$
2. $\ln x > 0, \quad \forall x >1$
3. $\ln x <0, \quad \forall 0 < x < 1$
4. By F.T.O.C. $\Rightarrow \ln x$ is conti.
and diff. on $(0, \infty)$, and $\frac d {dx} \ln x = \frac 1 x$
5. $\ln 2 < 1 < \ln 3$
By I.V.T. $\Rightarrow \exists$ a number $e \in (2,3)$ s.t.
$1 = \ln e = \int_1^e \frac 1 t dt$
6. if $f(x) >0$
By chain rule $\Rightarrow \frac d {dx} \ln f(x) = \frac {f'(x)}{f(x)}$
:::

- **Example**
:::success
1. $\frac d {dx} \ln 2x = \frac 2 {2x} = \frac 1 x, x>0$
2. $\frac d {dx} \ln (x^2 + 3) = \frac{2x}{x^2 +3}$
3. $\frac d {dx} \ln |x| = \frac 1 {|x|} \cdot \frac x{|x|} = \frac x {x^2} = \frac 1 x$
4. $\frac d {dx} \ln bx = \frac b {bx} = \frac 1 x, \quad \forall x,b >0$
:::

- **Properties**


:::info
1. $\ln bx = \ln b + \ln x, \quad \forall x, b >0$
2. $\ln \frac b x = \ln b - \ln x, \quad \forall x, b >0$
3. $\ln \frac 1 x = - \ln x, \quad \forall x >0$
4. $\ln x^r = r \ln x, \quad \forall x >0, \forall r \in \Bbb Q$
:::

### 多項式微分證明

- $$\frac d {dx} x^n = n x^{n-1}$$

- Case 1: $n \in \Bbb N$
:::success
**Proof:**(Using M.I.T.)
$n = 1 \Rightarrow \frac d{dx} x=1 = 1x^{1-1}$

assume
$\frac d{dx} x^{n-1} = (n-1) x^{(n-1)-1}$
holds

$\frac d {dx} x^n = \frac d {dx} x \bullet x^{n-1} = x^{n-1} +x(n-1)x^{n-2}$
$= nx^{n-1}$
- Case 2: $\frac d {dx} x^{-n} = (-n) x^{-n-1}, \quad \forall n \in \Bbb N$
:::success
**Proof:**

:::
- Case 3: $\frac d {dx} x^{\frac 1 m}, \quad m \in \Bbb N$
:::success
**Proof:**

- $\frac d {dx} x^n = n \cdot x^{n-1}, n \in \Bbb Q$
- Case:
**Proof**

- $\ln x^r = r \ln x,\quad \forall r \in \Bbb Q$
- **Proof**

暫時省略

- **Note**
:::warning
$\ln x:(0, \infty) \to (- \infty, \infty)$ is
one-to-one,
onto,
incresing,
concave down
:::

### Integrals

- $$\frac d {du} \ln |u| = \frac 1 u$$
$\Rightarrow \int \frac 1 u du = \ln |u| + c$

If $u = f(x),$ diff. and $f(x) \neq 0, \quad \forall x$
$\Rightarrow du = f'(x) dx$
$\Rightarrow \int \frac{f'(x)}{f(x)}dx = \ln |f(x)| +c$

- **Example**

$\int_{-\frac \pi 2}^{\frac \pi 2} \dfrac{4 \cos \theta}{3 + 2 \sin \theta}du$,

:::success
$u=3 + 2 \sin \theta$
$du= 2 \cos \theta d \theta$

$\int_1^5 \frac 2 u du = 2 \ln |u| |_1^5 = 2(\ln 5 - \ln 1) = 2 \ln 5$
:::

### The integrals of tan x, cot x, sec x, csc x

- **Prop.**

:::info
1. $\int \tan u \; du = \ln |\sec u| + C$
:::
**Proof:**
$\int \tan x \; dx = \int \dfrac {\sin x}{\cos x} dx$
Let $u = \cos x ,\quad du = -\sin x \; dx$
$= -\ln |u| + C$
$= \ln \frac 1 {|\cos x|} + C$
$= \ln |\sec x| +C$
**Q.E.D**
:::info
2. $\int \cot x \; dx = \ln |\sin x| +C$
:::
**Proof:**
$\int \cot x \; dx = \int \dfrac {\cos x}{\sin x} dx$
Let $u = \sin x, \quad du = \cos x \; dx$
$=\ln |u| + C$
$=\ln |\sin x| +C$
**Q.E.D**
:::info
3. $\int \sec x \; dx = \ln |\sec x + \tan x| +C$
:::
**Proof:**
$\int \sec x \; dx$
$= \int \dfrac {\sec ^2 x + \sec x \tan x}{\sec x + \tan x} dx$
Let $u = \sec x + \tan x,\quad du = (\sec^2 x + \sec x \tan x)dx$
$=\ln |u| + C$
$=\ln |\sec x + \tan x|+C$
**Q.E.D**
:::info
4. $\int \csc x \; dx = - \ln |\csc x + \cot x| + C$
:::
**Proof:**
$\int \csc x \; dx$
$= \int \dfrac{-\csc^2 x+\csc x \cot x}{\csc x + \cot x}$
Let $u = \sec x + \cot x,\quad du = (-\csc^2 x + \csc x \cot x)dx$
$= - \ln |u| +C$
$= - \ln |\csc x + \cot x| + C$
**Q.E.D**
- **Note**

:::warning
$y = \dfrac {f_1(x) f_2(x) ... f_k(x)}{g_1(x)g_2(x)...g_m(x)}$

$\ln y = \ln f_1(x) + \ln f_2(x) + ... + \ln f_k(x) - \ln g_1(x) - \ln g_2(x) -...- \ln g_m(x)$

$y' = (\dfrac {f_1'(x)}{f_1(x)} + ... + \dfrac {f_k'(x)}{f_k(x)} - \dfrac {g_1'(x)}{g_1(x)} - ... - \dfrac {g_m'(x)}{g_m(x)})y$
:::

## 7-3 Exponential functions

- Recall
$\ln x = \int_1^x \dfrac 1 t dt:(0, \infty) \to (-\infty, \infty)$, one-to-one, onto

$\Rightarrow \ln ^{-1} x : \Bbb R \to (0, \infty)$ exists
Define ==$\exp (x) \equiv \ln ^{-1} x$==, $\quad \forall x \in \Bbb R$

$\Rightarrow$
1. $\ln e = 1 \Rightarrow \exp(1) = e = e^1$

2. $\ln e^r = r \cdot \ln e = r \Rightarrow \exp(r) = e^r, \quad \forall r \in \Bbb Q$

$\Rightarrow$

### e

The inverse function of $\exp (x)$ is $\;\ln (x)$
The inverse function of $\;\;\; e^x \;\;\;$ is $\;\log _e x$

-
$$\ln x = \log_e x$$

- **Note**
:::warning
1. $\ln x = \int_1^x \frac 1 t$
2. $e^x = \exp (x)$
3. $\ln x = 10$
4. $e^{\ln x} = x$
5. $\ln (e^x) = x$
:::

- **Prop.** (General power rule for derivative)

For $x \in \Bbb R, \quad n \in \Bbb R \Rightarrow$
:::info
$$\dfrac d {dx} x^n = n x ^{n-1}$$
:::
wherever $x^n$ and $x^{n-1}$ exist.

- **Proof**

Case 1: $x>0$
$x^n = e^{n \ln x}$

$\dfrac d {dx} x^n = e^{n \ln x} n \dfrac 1 x = n x^n \dfrac 1 x$

$= nx^{n-1}$

Case 2: $x<0$
Let $u=-x >0$

$\dfrac d {dx} x^n = \dfrac d {dx} (-1)^n u^n = (-1)^n n u^{n-1} (-1)$

$n(-u)^{n-1} = nx^{n-1}$

Case 3: $x=0$

$\lim \limits_{h \to 0} \frac {(0+h)^n - (0)^n}{h}= \lim \limits_{h \to 0} h^{n-1} = 0 = nx^{n-1}|$

- **Example**

For $x>0$

$\dfrac d {dx} x^x$

$= \dfrac d {dx} e^{\ln x^x}$

$= \dfrac d {dx} e^{x \ln x}$

$= e^{x \ln x}(\ln x +1)$

$= x^x(\ln x +1)$

### The number e

- **Theorem 4**
:::info
$e = \lim \limits_{h \to 0}(1+x)^{\dfrac 1 x} \equiv \lim \limits_{h \to \infty}(1 + \dfrac 1 t)^t$
:::

- **Proof**
Let $f(x) = \ln x \Rightarrow f'(x) = \dfrac 1 x$
$\ln e = 1 = f'(1) = \lim \limits_{h \to 0} \dfrac{f(1+x) - f(1)} x$
$= \lim \limits_{h \to 0} \frac 1 x \ln(1+x) = \lim \limits_{h \to 0} \ln (1+x)^{\frac 1 x}$
$\therefore \ln x$ is one-to-one
$\Rightarrow e = \lim \limits_{h \to 0} (1+x)^{\dfrac 1 x}$

- **Theorem**

- **Properties:**

$\dfrac {d}{dx} a^{f(x)}, \quad a>0$
:::info
1. $\dfrac {d}{dx} a^x = (\ln a) \cdot a^x, \quad \forall a >0$
**Proof:** $\dfrac {d}{dx} a^x = \dfrac {d}{dx} e^{\ln a^x} = \dfrac {d}{dx} e^{x \ln a}$
2. $\int a^x dx = \dfrac 1 {\ln a} a^x +c, \quad a>0$

3. $\dfrac {d}{dx} a^{f(x)} = \ln a \cdot a^{f(x)} \cdot f'(x)$

**Proof:** $\dfrac {d}{dx} e^{\ln a ^{f(x)}} = \dfrac {d}{dx} e^{(\ln a) \cdot f(x)}$

$= e^{\ln a \cdot f(x)} = \dfrac {d}{dx} e^{(\ln a) \cdot f(x)}$

$= e^{(\ln a) \cdot f(x)} \cdot \ln a \cdot f'(x)$

$= a^{f(x)} \cdot \ln a \cdot f'(x)$

4. $\int a^u du = \dfrac 1 {\ln a} a^u + c$

**i.e.** $\int a^{f(x)} \cdot f'(x)dx = \dfrac 1 {\ln a} a^{f(x)} +c$

5. If $a>1$
$\Rightarrow \ln a > 0 \Rightarrow \dfrac {d}{dx} a^x = \ln a \cdot a^x >0$
$\Rightarrow a^x$ is increasing, one-to-one

6. If $0<a<1 \Rightarrow \ln a <0$
$\Rightarrow \dfrac {d}{dx} a^x <0, \dfrac {d^2}{dx^2} a^x >0$
$\Rightarrow a^x$ is decresing, one-to-one, concave up
:::
- **Example**

1. $\dfrac {d}{dx} 3^x = (\ln 3) \cdot 3^x$
2. $\dfrac {d}{dx} 3^{-x} = -(\ln 3) \cdot 3^{-x}$
3. $\dfrac {d}{dx} 3^{\sin x} = (\ln 3) \cdot 3^{\sin x} \cdot \cos x$
4. $\int 2^{\sin x} \cos x dx$
$(u = \sin x, \quad du = \cos x dx)$
$= \int 2^u du$
$= \dfrac 1 {\ln 2} 2^{\sin x} +c$

### Logarithms with base $a: \log_a x$

- **Definition**

For $a > 0, a \neq 1, \log_a x$ is the inverse function f $a^x : \Bbb R \to (0, \infty)$

- **Note**

$a>0, a \neq 1$
$a^x : \Bbb R \to (0, \infty)$
$\log_a x:(0,\infty) \to \Bbb R$ ：
圖

- **Properties**

:::info
1. $a^{log_a x} = x, \quad x>0$

2. $\log_a a^x = x, \quad x \in \Bbb R$

3. $\log_a xy = \log_a x + \log_a y$

4. $\log_a \frac x y = \log_a x - \log_a y$

5. $\log_a x^y = y \cdot \log_a x$

6. $\log_a x = \dfrac {\ln x}{\ln a} = \dfrac{\log_e x}{\log_e a}$

7. $\dfrac {d}{dx} \log_a x = \dfrac {d}{dx} \dfrac 1 {\ln a} \cdot \ln x =\dfrac 1 {\ln a} \cdot \dfrac 1 x$

8. $\dfrac {d}{dx} \log_a f(x) = \dfrac {d}{dx} \dfrac{\ln f(x)} {\ln a} = \dfrac 1 {\ln a} \cdot \dfrac {f'(x)}{f(x)}$
:::

- **Example**

1. $\dfrac {d}{dx} \log_{10} (3x+1) = \dfrac 1 {\ln 10} \cdot \dfrac 3 {3x+1}$

2. $\int \dfrac {\log_2 x } x dx = \int \dfrac {\ln x} {x \cdot \ln 2} dx$

$(u = \ln x, \quad du = \frac {dx} x)$

$= \dfrac 1 {\ln 2} \int udu$

$= \dfrac 1 {2 \ln 2}(\ln x)^2$

### 用兩堂課講的四個重點微分公式

$$\mathsf{ 原式 \qquad 微分}$$

$$\ln x \qquad \frac 1 x$$

$$e^x \qquad e^x$$

$$a^{f(x)} \qquad e^{f(x) \ln a}$$

$$\log_a f(x) \qquad \frac {\ln f(x)}{\ln a}
$$

## 7-5 Indeterminate forms and L'Hôpital's rule 羅必達法則

### L'Hôpital's rule 羅必達法則

- Find
:::info
$$\lim \limits_{x \to a} \dfrac {f(x)}{g(x)} = \lim \limits_{x \to a} \dfrac {f'(x)}{g'(x)}$$

:::

Case 1:
$$\dfrac {f(a)}{g(a)} = \dfrac 0 0 or \dfrac \infty \infty$$

- **Theorem 5 L’Hôpital’s rule**

$a \in (\alpha, \beta) \subseteq \Bbb R, \quad f, g:(\alpha, \beta) \to \Bbb R$ are diff. on $(\alpha, \beta)$

$f(a) = g(a) = 0$ and $g'(x) \neq 0, \quad \forall x \in (\alpha, \beta)$ \\$\{a\}$

$\lim \limits_{x \to a} \dfrac {f(x)}{g(x)} = \lim \limits_{x \to a} \dfrac {f'(x)}{g'(x)}$ provied the limit of **RHS** exists

- **Example**

1. $\lim \limits_{x \to 0} \dfrac {}{}$

2. $\lim \limits_{x \to 0} \dfrac {}{}$

3. $\lim \limits_{x \to 0} \dfrac{}{}$

4. $\lim \limits_{x \to 0} \dfrac {}{}$

5. $\lim \limits_{x \to 0} \dfrac {}{}$

6. $\lim \limits_{x \to 0} \dfrac {}{}$

7. $\lim \limits_{x \to 0} \dfrac {}{}$

### Cauchy's Mean Value Theorem 柯西均值定理

- **definition**

$f, g: [a,b] \to \Bbb R$ are conti. on $[a,b]$, diff. on $(a,b)$
$\Rightarrow \exists c \in (a,b)$
$\qquad s.t. (f(b)-f(a)) g'(c)=(g(b)-g(a))f'(c)$
$\qquad i.e. \dfrac{f(b)-f(a)}{g(b)-g(a)}=\dfrac{f'(c)}{g'(c)}$

- **Proof**

Let
$h(x) = (f(b)-f(a))(g(x)-g(a))-(g(b)-g(a))(f(x)-f(a))$

$\Rightarrow h$ is conti. on $[a,b]$, diff. on $(a,b)$

and $h(a) = 0 = h(b)$

By Rolle's thm.
$\Rightarrow \exists c \in (a,b)$ s.t.
$0 = h'(c) = (f(b)-f(a))g'(c)-(g(b)-g(a))f'(c)$

- **Note**

$\exists c \in (a,b)$ s.t.

$\dfrac {f(b)-f(a)}{g(b)-g(a)} = \dfrac {f'(c)}{g'(c)}$

if $g(a) \neq g(b)$

- **Proof of thm 5**

略?還是現在下面的是在講證明

$\lim \limits_{x \to a} \dfrac {f(x)}{g(x)} = \lim \limits_{x \to a} \dfrac {f'(x)}{g'(x)} \quad$ if $\dfrac {f(a)}{g(a)} = \dfrac 0 0$
$\dfrac{f(a)}{g(a)} = \dfrac{\pm \infty}{\pm \infty}, \quad f(a) \cdot g(a) = 0 \cdot \infty, \quad f(a) - g(a) = \infty - \infty$

- **Note**

If $f(x), g(x) \to \pm \infty$ as $x \to a$
$\lim \limits_{x \to a} \dfrac {f(x)}{g(x)} = \lim \limits_{x \to a} \dfrac {\frac 1 {g(x)}}{\frac 1 {f(x)}} = \lim \limits_{x \to a} \dfrac {\frac {+ g'(x)} {g^2(x)}}{\frac {+f'(x)} {f^2(x)}} = \lim \limits_{x \to a} \dfrac {f(x)}{g(x)} \cdot \lim \limits_{x \to a} \dfrac {g'(x)}{ f'(x)}$
$\lim \limits_{x \to a} \dfrac {g'(x)}{ f'(x)}= \lim \limits_{x \to a} \dfrac {f(x)}{ g(x)}$

- **Example**
1. $\lim \limits_{x \to \frac \pi 2} \dfrac {\sec x}{ 1 +\tan x} = \lim \limits_{x \to \frac \pi 2} \dfrac {1}{ \cos x + \sin x} = \dfrac 1 {1+0} = 1$

2. ($\frac \infty \infty$)
$\lim \limits_{x \to \frac \pi 2} \dfrac {\sec x}{ 1 +\tan x} = \lim \limits_{x \to \frac \pi 2} \dfrac {1}{ \cos x + \sin x} = \dfrac 1 {1+0} = 1$

3. ($\frac \infty \infty$)
$\lim \limits_{x \to \infty} \dfrac {\ln x}{ 2 \sqrt x} = \lim \limits_{x \to \infty} \dfrac {\dfrac 1 {x}}{ \frac 1 {\sqrt x}} = \lim \limits_{x \to \infty} \dfrac {1}{ \sqrt x} = 1$

4. ($\infty \cdot 0$) 待補
$\lim \limits_{x \to \frac \pi 2} \dfrac {\sec x}{ 1 +\tan x} = \lim \limits_{x \to \frac \pi 2} \dfrac {1}{ \cos x + \sin x} = \dfrac 1 {1+0} = 1$

5. ($0 \cdot \pm \infty$) 待補但我發現我沒拍到

6. ($\infty - \infty$) 待補但我發現我沒拍到

- **Note**
$1^{\infty}, 0^0, \infty^0$
$\lim \limits_{x \to a} f(x) = \lim \limits_{x \to a} e^{\ln f(x)} = e^{\lim \limits_{x \to a} \ln f(x)}$

- **Example**
1. $\lim \limits_{x \to 0^+} (1+x)^{\frac 1 x} = \lim \limits_{x \to 0^+} e^{\frac {\ln (1+x)} x} = e ^{\lim \limits_{x \to 0^+} \frac {\ln (1+x)} x} = e^{\lim \limits_{x \to 0^+} \frac {\frac 1 {1+x}} 1} = e^1 = e$

2. $\lim \limits_{x \to 0^+} x^{\frac 1 x} = \lim \limits_{x \to 0^+} e^{\frac {\ln x} x} = e ^{\lim \limits_{x \to 0^+} \frac {\ln x} x} = e^{\lim \limits_{x \to 0^+} \frac {\ln x} x} = e^{\lim \limits_{x \to 0^+} \frac { \frac 1 x} 1} = e^0 = 1$

## 7-6 Inverse trigonometric functions 反三角函數

- **Definition**

==反三角函數的值是一個角度==

$\sin x:[-\frac \pi 2, \frac \pi 2] \to [-1, 1]$ is one-to-one
$\Rightarrow \sin^{-1} x : [-1,1] \to [-\frac \pi 2, \frac \pi 2]$

 


### The Arcsine and Arccosine Function

- **Definition**


- **Example**


$\sin ^{-1} (1) = \dfrac \pi 2$
算了我懶得打我相信大家都知道！！！

$\cos ^{-1} (1) = 0$
$\cos ^{-1} (\dfrac {\sqrt 3} 2) = \dfrac \pi 6$
$\cos ^{-1} (\dfrac {\sqrt 2} 2) = \dfrac \pi 4$
$\cos ^{-1} (\dfrac 1 2) = \dfrac \pi 3$
$\cos ^{-1} (0) = \dfrac \pi 2$
$\cos ^{-1} (-1) = \pi$
$\cos ^{-1} (- \dfrac {\sqrt 3} 2) = \dfrac 5 6 \pi$
$\cos ^{-1} (- \dfrac {\sqrt 2} 2) = \dfrac 3 4 \pi$
$\cos ^{-1} (- \dfrac 1 2) = \dfrac 2 3 \pi$

> [name=教授] (前略)......然後某一天我就發現我好像能反著背英文字母了，你們知道為什麼我要跟你們講這個小故事了吧

- 對sin有夠熟就知道arcsin在幹嘛了

- **Note**
1. $\sin ^{-1} (-x) = -\sin ^{-1} x$, ==**odd function**==

2. $\cos ^{-1} + cos^{-1} (-x) = \pi$

**Proof**
Let $\theta = \cos^{-1} (x) \Rightarrow \cos \theta = x$
$\cos(\pi - \theta) = - \cos \theta = -x$
$\Rightarrow \cos^{-1} (-x) = \pi - \theta = \pi - cos^{-1} x$

3. $\sin^{-1} x + \cos^{-1} x = \dfrac \pi 2, \quad \forall x \in [0,1]$

4.

- **Example**

我要睡著了 所以我要先睡覺

### Inverses of tan x, cot x, sec x, and csc x


- 睡著了所以tan x, cotx, sec x都沒打

$\csc x:[- \dfrac \pi 2, 0) \cup (0, \dfrac \pi 2] \to (-\infty, -1] \cup [2, \infty)$ is one-to-one
$\Rightarrow \csc^{-1} x:(-x,-1] \cup [1, \infty) \to [-\dfrac \pi 2, 0) \cup (0, \dfrac \pi 2]$

- **Note**
$\sec^{-1} x + \csc^{-1} x = \dfrac \pi 2$

- **Derivatives**

$(\dfrac d {dx} f^{-1}(x) = \dfrac 1 {f'(f(x))})$

1. $\dfrac d {dx} \sin ^{-1} x = \dfrac 1 {\sqrt {1-x^2}}$

### 反三角函數的定義域、值域、微分

- 定義域和值域

$\sin^{-1} x : [-1, 1] \to [- \dfrac \pi 2, \dfrac \pi 2]$

$\cos^{-1} x : [-1,1] \to [0, \pi]$

$\tan^{-1} x : \Bbb R \to (-\dfrac \pi 2, \dfrac \pi 2)$

$\cot^{-1} x : \Bbb R \to (0, \pi)$

$\sec^{-1} x: \Bbb R \setminus (-1,1) \to [0, \pi] \setminus \{\dfrac \pi 2\}$

$\csc^{-1} x: \Bbb R \setminus (-1,1) \to [- \dfrac \pi 2, \dfrac \pi 2] \setminus \{0\}$

- **Derivatives**

1. $\dfrac {d}{dx} \sin^{-1} x = \dfrac 1 {\sqrt {1-x^2}}, \quad |x|<1$

2. $\dfrac {d}{dx} \cos^{-1} x = \dfrac {-1}{\sqrt{1-x^2}}, \quad |x|<1$

3. $\dfrac {d}{dx} \tan^{-1} x = \dfrac 1 {1+x^2}, \quad x \in \Bbb R$
- **Proof:**
$\dfrac {d}{dx} \tan^{-1} x = \dfrac 1 {\sec ^2(tan^{-1} x)} = \dfrac 1 {1+\tan^2 (tan^{-1} x)} = \dfrac 1 {1+x^2}$

4. $\dfrac {d}{dx} \cot ^{-1} x = \dfrac {-1}{1+x^2}, \quad x \in \Bbb R$

5. $\dfrac {d}{dx} \sec ^{-1} x = \dfrac 1 {|x| \sqrt{x^2 - 1}}, \quad |x|>1$
- **Proof:**
$\dfrac {d}{dx} \sec^{-1} = \dfrac 1 {\sec(\sec ^{-1} x) \cdot \tan(\sec ^{-1} x}$
$= \dfrac 1 {|x| \sqrt {x^2 -1}}, \quad |x| > 1$

6. $\dfrac {d}{dx} \csc ^{-1} x = \dfrac {-1}{|x| \sqrt {x^2 -1}} , \quad |x| > 1$

- **Example**

1. $\dfrac {d}{dx} \sin ^{-1} x^2 = \dfrac {2x} {\sqrt {1-x^4}}$
2. $\dfrac {d}{dx} \sec ^{-1} (5x^4) = \dfrac {20x^3} {5x^4 \sqrt {25x^8 -1}} = \dfrac 4 {x \sqrt {25x^8 - 1}}$

- **Integral**

For $a>0$

1. $\int \dfrac 1 { \sqrt {a^2 - u^2}} du = \sin ^{-1} ( \dfrac u a) +c \quad$, For $|u|<a$
2. $\int \dfrac 1 { a^2 + u^2} du = \dfrac 1 a \tan ^{-1} ( \dfrac u a) +c \quad$, $\forall u$
3. $\int \dfrac 1 { u \sqrt {u^2 - a^2}} du = \dfrac 1 a \sec ^{-1} ( |\dfrac u a|) +c \quad$, $\forall |u| > a$

- **Example**
1. $\int ^{\frac{\sqrt 3}{2}}_{\frac{\sqrt 2}{2}} \dfrac{1}{\sqrt{1-x^2}}$

$=\sin^{-1}x ^{\frac{\sqrt 3}{2}} _{\frac{\sqrt 2}{2}}$

$= \frac{\pi}{3}-\frac{\pi}{4} = \frac{\pi}{12}$

2. $\int \dfrac {dx} {\sqrt{3 - 4x^2}}$

($u = 2x, \quad du = 2dx$)

$= \dfrac 1 2 \int \dfrac 1 {\sqrt {3 - u^2}} du$

3. $\int \dfrac 1 {\sqrt {e^{2x} -6}} dx$

$= \dfrac 1 {\sqrt 6} \sec ^{-1} \dfrac {e^x} {\sqrt 6} + c$

4. $\int \dfrac 1 { \sqrt {4x-x^2}}$

$= \int \dfrac 1 {\sqrt {4 - (4-4x+x^2)}} dx$

$= \int \dfrac 1 {\sqrt {4 - (x-2)^2}} dx$

($u = x-2, \quad du = dx$)

$= \int \dfrac 1 {\sqrt {4-u^2}} du$

$= \sin ^{-1} (\dfrac {x-2} 2) +c$

## 7-8 Relative rates of growth

(衝到 $\infty$ 的速度比較)
($\ln$最慢，而且是越多越慢)


- **Definition**

$f(x), g(x) \to \infty$ as $x \to \infty$


1. $f$ grows faster than $g$ ( $g$ is of smaller order than $f$ )
as $x \to \infty$ (write $g = o(f)$) if $\lim \limits_{x \to \infty} \dfrac {g(x)} {f(x)} = 0$
(i.e. $\lim \limits_{x \to \infty} \dfrac {f(x)} {g(x)} = \infty$

2. $f$ and $g$ grow at the same rate as $x \to \infty$ (write $f \sim g$)

3. $f$ is of at most order of $g$ as $x \to \infty$ if $\exists M>0$ s.t. $\dfrac{f(x)}{g(x)} ≤ M$ for large $x$.(write $f = O(g)$)

- **Note**

1. $h = o(g), g=o(f) \Rightarrow h=o(f)$

2. $f \sim g \Rightarrow g \sim f$

3. $h \sim g, g \sim f \Rightarrow h \sim f$

4. $x^n = o(x^m), m>n$

5. $x^n = o(e^x), \forall n>0$

6. $\ln x = o(x^n), \forall n >0$

7. $p(x) = a_nx^n + ... + a_0, a_n > 0$
$q(x) = b_m x^m + ... + b_0, b_n > 0$
$\Rightarrow$ 1. $p = o(q)$ if $m>n$
2. $p \sim q$ if $m = n$

- **Example**
1. $2^x = o(3^x)$$(since \lim \limits_{x \to \infty} \frac{2^x}{3^x} = 0)$
2. $a^x = o(b^x)$ if $0<a<b$
3. $a>1, b>1$ $\Rightarrow log_a x \sim log_b x$

(第八章另開檔案)