## 導函數

#### 第 2 題

Given f(x) = x3 − 3x2 + 3, use the definition $\displaystyle f' \left( x \right) = \lim_{h \to 0} \frac {f \left( x + h \right) - f \left( x \right)} h$ to find f′(x).

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

(x + h)2x2 = 2xh + h2

## 切線與法線

#### 公、保、管第 3 題

Find the equation to the tangent of the curve (x2 + 3) (x − 3)1/2 at x = 4.

#### 牙醫系第 3 題

… at x = 5.

f(x) = (x2 + 3) (x − 3)1/2，則

#### 牙醫系第 7 題

Find the equation of the tangent line to the curve x2 + 3xy + y2 = 5 at (1, 1).

#### 公、保、管第 7 題

… the normal line ….

2x + 3y + 3xy′ + 2yy′ = 0

5 + 5y′ = 0

y − 1 = -(x − 1)

y − 1 = x − 1

## 導數、二階導數

#### 公、保、管第 4 題

Consider a curve $\displaystyle f \left( x \right) = \frac {x + 2} {\left( x - 3 \right)^{0.5}}$. Find f′(5).

… and f″(5).

#### 公、保、管第 6 題

Given f(x) = 2x2+1, find f′(2).

y = x2 + 1，由鏈式法則

f′(x) = (ln 2) x 2y+1 = (ln 2) x 2x2+2

f′(2) = 128 ln 2

#### 牙醫系第 6 題

Given f(x) = ln(sec4(x) tan2(x)), find f′(π/4).

f′(x) = y′ / y

y′ = 4 sec4(x) tan3(x) + 2 sec6(x) tan(x)

f′(π/4) = 8

## 函數圖形

#### 第 5 題

Sketch the graph of $\displaystyle f \left( x \right) = \frac {3x^5 - 20x^3} {32}$ and also find the relative extreme points and inflection points at the interval of [-1, 1].

## 線性近似

#### 公、保、管第 8 題

Use the differentials to approximate the quantity $\sqrt{5.6}$ to two decimal points places.

#### 牙醫系第 8 題

$\sqrt{5.5}$ ….

f(x) ≈ f(a) + f′(a) (xa)

f(5.6) ≈ 2 + f′(4) (5.6 − 4)

## 應用題

#### 牙、公、保第 9 題

When a person coughs, the trachea (windpope windpipe) contracts, allowing air to be expelled at a maximum velocity. It can shown that during a cough the velocity v of airflow is given by the function v = f(r) = kr2(Rr) , where r is the trachea’s radius (in centimeters) during a cough, R is the trachea’s normal radius (in centimeters), and k is a positive constant that depends on the length of the trachea. Find the radius r for which the velocity of airflow is greatest.

f(0) = f(R) = 0

f(r) = kRr2kr3

f′(r) = 2kRr − 3kr2

f(2R/3) = k (2R/3)2 (R/3) > 0

#### 醫管系第 9 題

Suppose that during a nationwide program to immunize the population against certain from of influenza, public health officials found that the cost of inoculating x% of the population was approximately $\displaystyle C \left( x \right) = \frac {150x} {200 - x}$ million dollars.

1. What was the cost of inoculating the first 50% of the population?
2. What was the cost of inoculating the second 50% of the population?
3. What percentage of the population had been inoculated by the time 37.5 million dollars had been spent?

1. C(50) = 50
2. C(100) − C(50) = 100
3. 這實質上是解 解得 x 為 40，答案為 40%。

#### 牙醫系第 10 題

A rain gutter is made from sheets of metal 9 in wide. The gutters have a 3-in base and two 3-in sides, folded up at an angle θ (see figure). What angle θ maximizes the cross-sectional area of the gutter?

[本圖受著作權保護，請勿轉載。]

A = 9 (sin θ) (1 + cos θ)

#### 公、保第 10 題

Several mathematical stories originated with the second wedding of the mathematician and astronomer Johannes Kepler. Here is one: While shopping for wine for his wedding, Kepler noticed that the price of a barrel of wine (here assumed to be a cylinder) was determined solely by the length d of a dipstick that was inserted diagonally through a hole in the top of the barrel to the edge of the base of the barrel (see figure). Kepler realized that this measurement does not determine the volume of the barrel and that for a fixed value of d. The volume varies with the radius r and height h of the barrel. For a fixed value of d, what is the ratio r/h that maximizes the volume of the barrel?

[本圖受著作權保護，請勿轉載。]

V = πr2h

r2 = d2h2

V = π (d2hh3)

#### 醫管系第 10 題

Based on a study conducted in 1997, the percentage of the U.S. population by age affilcted with Alzheimer’s disease is given by the function

P(x) = 0.0726x2 + 0.7902x + 4.9623　where　0 ≤ x ≤ 25

where x is measured in years, with x = 0 corresponding to age 65 yr. Show that P is an increasing function of x on the interval (0, 25). What does your result tell you about the relationship between Alzheimer’s disease and age for the population that is aged 65 year and or older?

P 在 [0, 25] 等於多項式，所以在 (0, 25) 可微。我們觀察它的導函數。

P′(x) = 0.1452x + 0.7902　where　0 ≤ x ≤ 25

## 第 1 題

Suppose a population of bacteria doubles every hour, but that 1.0 × 106 individuals are removed before reproduction to be converted into valuable biological by-products. Suppose the population begins with b0 = 3.0 × 106 bacteria.

1. Find the population after 1, 2, and 3 hours.
2. Write the discrete-time dynamical system.

bt+1 = 2.0 (bt − 1.0 × 106)

bt+1 = 2.0 bt − 2.0 × 106

b1 = 4.0 × 106

b2 = 6.0 × 106

b3 = 1.0 × 107

## 第 2 題

In one simple scenario, mutations occur in only one direction (wild type tum into mutants but not vice versa), but wild type and mutants have different levels of per capita production. Suppose that a fraction 0.1 of wild type mutate each generation, but that each wild type individual produces 2.0 offspring while each mutant produces only 1.5 offspring. In each case, find the following.

1. The number of wild-type bacteria that mutate.
2. The number of wild-type bacteria and the number of mutants after mutation.
3. The number of wild-type bacteria and the number of mutants after reproduction.
4. The total number of bacteria after mutation and reproduction.
5. The fraction of mutants after mutation and reproduction.

(Begin with 1.0 × 106 wild type and 1.0 × 105 mutants.)

0.1 (1.0 × 106) = 1.0 × 105

1.0 × 106 − 1.0 × 105 = 9.0 × 105

1.0 × 105 + 1.0 × 105 = 2.0 × 105

2.0 (9.0 × 105) = 1.8 × 106

1.5 (2.0 × 105) = 3.0 × 105

1.8 × 106 + 3.0 × 105 = 2.1 × 106

## 第 3 題

Consider again a lung that has a volume of 6.0 L and that replaces 0.6 L each breath with ambient air. Suppose that we are tracking oxygen, with an ambient concentration of 21%. Assume that the actual oxygen concentration in exhaled air is approximately 15%. What fraction of oxygen is in fact absorbed?

(6.0 − 0.6) 15% α = 0.6 (21% − 15%)

81α = 3.6

α ≈ 0.044

## 第 4 題

The model describing the dynamics of the concentration of medication in the bloodstream,

Mt+1 = 0.5Mt + 1.0,

becomes nonlinear if the fraction of medication used is a function of the concentration. In the basic model, half is used no matter row how much [the concentration] is there. More generally,

new concentration = old concentration − (fraction used) × (old concentration) + supplement.

Suppose the fraction used = 0.5 / (1 + 0.1Mt). Write the discrete-time dynamical system and solve for the equilibrium.

0.5M = 1 + 0.1M

M = 2.5

## 第 5 題

In an excitable heart model. Let Vc be the threshold potential. The discrete-time dynamical system can be describe as

With the parameter values u = Vc = 1, compute the conditions on c for the existence of an equilibrium.

cV > 1，則 c = 1V > 1。但是 c 是衰減係數，應小於 1，故不合。

cV ≤ 1，則

V = cV + 1

c ≤ 0.5

## 線性回歸

#### 第 1 題

In a study of five industrial areas, a researcher obtained these data relating the average number of units of a certain pollutant in the air and the incidence (per 100,000 people) of a certain disease:

 Units of pollutant Incidence of disease 3.4 4.6 5.2 8 10.7 48 52 58 70 96

Find the equation of the least-squares line y = Ax + B (to two decimal points places.)

Given that $A = \frac {n \sum xy - \sum x \sum y} {n \sum x^2 - \left( \sum x \right)^2}$ and $B = \frac {\sum y - A \sum x} n$.

x y x2 xy
3.4 48 11.56 163.2
4.6 52 21.16 239.2
5.2 58 27.04 301.6
8 70 64 560
10.7 96114.491027.2
31.9324238.252291.2

## 導函數

#### 第 2 題

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

## 切線與法線

#### 第 3 題

Find the equation of the tangent line and the normal line of the curve x2y3 + xy = 10 at (1, 2).

3x2y2y′ + 2xy3 + xy′ + y = 0

(3x2y2 + x) y′ + 2xy3 + y = 0

13 y′ + 18 = 0

## 函數圖形

#### 第 4 題

Sketch the graph of f(x) = 6x5 − 5x3 and also find the relative extreme points and inflection points.

f′(x) = 30x4 − 15x2

f″(x) = 120x3 − 30x

f‴(x) = 360x2 − 30

6x5 − 5x3 = x3 (6x2 − 5)

## 線性近似

#### 第 5 題

Use the differentials to approximate the quantity $\sqrt{0.089}$ to four decimal points places.

f(x) ≈ f(a) + f′(a) (xa)

f(0.089) ≈ 0.3 + f′(0.09) (0.089 − 0.09)

## 牛頓法

#### 第 6 題

Use Newton’s method to find the real root r of f(x) = x3x − 1 to two decimal points places, given that initial point x0 = 1.5.

nxn
01.5
11.347826086956522
21.325200398950907
31.324718173999054

x ≈ 1.32

## 問題的問題

1. 第一個答案是 A 的問題是哪一個？
1. 1
2. 2
3. 3
4. 4
2. 唯一的連續兩個具有相同答案的問題是
1. 5, 6
2. 6, 7
3. 7, 8
4. 8, 9
3. 本問題答案和哪一個問題的答案相同？
1. 4
2. 9
3. 8
4. 2
4. 答案是 A 的問題的個數是
1. 5
2. 4
3. 3
4. 2
5. 本問題答案和哪一個問題的答案相同？
1. 1
2. 2
3. 3
4. 4
6. 答案選 A 的問題的個數和答案選什麼的問題的個數相同？
1. C
2. C
3. D
7. 按照字母順序，本題答案與下一題相差多少？（A 與 B 之間，或 B 與 A 之間均相差 1）
1. 3
2. 2
3. 1
4. 0
8. 十道題中答案為母音的題數為
1. 0
2. 1
3. 2
4. 3
9. 十道題中答案為子音的題數
1. 合數
2. 質數
3. 小於 5
4. 平方數
10. 本題答案為
1. A
2. B
3. C
4. D

### 解

• 第 5 題為 A，因為第 2 題非 B。
• 因 7、8 二題相同，所以第 7 題為 D

Stewart Brand

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2. 它無法阻止戲仿拼貼
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• SA 允許其他人分享我的作品。
• 若沒有 SA，別人會控告我拿我的作品的衍生作品營業。

CC 的律師也表示 NC 已經蘊涵 SA，即有 NC 的作品的衍生著作必須也含有 NC。因為若衍生著作沒有 NC，則這個衍生著作反而可商用，產生矛盾。NC 阻止作品進入自由領域，即 BY 和 BY-SA，所以 BY-NC-SA 與 BY-SA 毫無關聯。BY-NC-SA 的名字容易讓人產生錯誤聯想。

BY

BY-SA

BY-NC

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