## 2014

### 2014-04-11

TWBBS.orgDNS 伺服器異常，導致本站域名無法解析。因此我決定把本站又重定向回 jdh8.no-ip.org 以求穩定。

## 反導函數與雅量

y = cos x

\tan x = \frac{\sin x}{\cos x} = -\frac{y'}{y}
\int \tan x\,dx = -\ln \left| y \right| = \ln \left| \sec x \right|.

y = sin x

\tan x = \frac{\sin x}{\cos x} = \frac{\cos x \sin x}{\cos^2 x} = \frac{yy'}{1 - y^2}

\begin{align*}
\int \tan x\,dx &= \int \frac{y}{1 - y^2} dy \\
&= -\frac{\ln \left( 1 - y^2 \right)}2 \\
&= -\frac{\ln \left( \cos^2 x \right)}2 \\
&= \ln \left| \sec x \right|.
\end{align*}


y = tan x，則 y′ = y2 + 1。


\begin{align*}
\int \tan x\,dx &= \int \frac{y}{y^2 + 1} dy \\
&= \frac{\ln \left( y^2 + 1 \right)}2 \\
&= \frac{\ln \left( \sec^2 x \right)}2 \\
&= \ln \left| \sec x \right|.
\end{align*}


## 導函數

### 牙醫系第 2 題

Use the definition of the derivative  \displaystyle f' \left( x \right) = \lim_{h \to 0} \frac {f \left( x + h \right) - f \left( x \right)} h $\displaystyle f' \left( x \right) = \lim_{h \to 0} \frac {f \left( x + h \right) - f \left( x \right)} h$ given that to find the derivative of f(x) = cos(x).

### 其他系第 2 題

given that to find the derivative of f(x) = x3 + 7x.

 f' \left( x \right) = \lim_{h \to 0} \frac {\left( x + h \right)^3 + 7 \left( x + h \right) - \left( x^3 + 7x \right)} h 

(x + h)3x3 = 3x2h + 3xh2 + h3

(x + h)3 + 7(x + h) − (x3 + 7x) = (3x2 + 7)h + 3xh2 + h3

### 牙醫系第 5 題

Logarithmic differentiation. Let $\displaystyle f \left( x \right) = \frac {\left( x^3 - 1 \right)^4 \sqrt{3x - 1}} {x^2 + 4}$ and find f′(x)

## 導數

### 第 3 題

Given $\displaystyle f \left( x \right) = \frac {x^2 \left( 1-x \right)^3} {1+x}$, find f′(3).

## 切線與法線

### 牙醫系第 4 題

Find the equation of the tangent line of the family of curves y3 + xy = sec(xy2) + c at (1, 1).

##### Hint

3y2y′ + y + xy′ = sec(xy2) tan(xy2) (y2 + 2xyy′)

(3y2 + x) y′ + y = sec(xy2) tan(xy2) (y2 + 2xyy′)

4y′ + 1 = sec 1 tan 1 (1 + 2y′)

cos2 1 (4y′ + 1) = sin 1 (2y′ + 1)

cos2 1 − sin 1= (2 sin 1 − 4 cos2 1) y

### 其他系第 4 題

Find the equation of the normal line of the family of curves y3 + xy = xy2 + c at (3/16, 2).

3y2y′ + y + xy′ = y2 + 2xyy

(3y2 − 2xy + x) y′ = y2y

(12 − 9/16) y′ = 2

y′ = 32/183

## 應用題

### 牙醫系第 6 題

Electron speed. An electron with a whose mass of is 9.1 × 10-31 kg and a charge of is -1.6 × 10-19 C travels in a circular path with no loss of energy in a magnetic field of 0.05 T that is orthogonal to the path of the electron. If the radius of the path is 0.002 m, what is the speed of the electron?

mv2 / r = q |v × B|

|v × B| = vB

mv2 / r = qvB

v = qrB / m

v = (1.6 × 10-19 C) (0.002 m) (0.05 T) / (9.1 × 10-31 kg)

v ≈ 1.76 × 107 m/s

### 其他系第 6 題

In the late 1830s, French physiologist Jean Poiseuille discovered the formula we use today to predict how much the radius of a particular clogged artery decreases the normal volume of flow. His formula,

V = kr4

says that volume of fluid flowing through a small pipe or tube in a unit of time at a fixed pressure is a constant times the fourth power of the tube’s radius r. How dose does a 10% decrease in r affect V?

#### 官方解法

Δr = 0.05r

ΔV ≈ 0.2kr4 = 0.2V

V 增加 20%

#### 直接解法

V0 = kr04

r = 1.05r0

V = k (1.05r0)4 = 1.054 V0

## 線性回歸

### 其他系第 1 題

In a study of 12 subjects, a clinical researcher obtained these data relating the ages of a subject pool (in years) to that of their systolic blood pressure (in mmHg):

No. Age (years) BPsystolic (mmHg)
125120
237139
320118
449150
527125
623115
734146
856170
924128
1036137
1122126
1242148

x y x2 xy
25120 6253000
3713913695143
20118 4002360
4915024017350
27125 7293375
23115 5292645
3414611564964
5617031369520
24128 5763072
3613712964932
22126 4842772
4214817646216
39516221446555349