# 2012 期中考

## 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