When a person coughs, the trachea (windpope ) 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(R − r)
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.
首先，變數 r 的自然限制是 0 ≤ r ≤ R。此外，v 是 r 的多項式。所以 v 在 [0, R] 的最大值只會出現在端點或平坦處。
f(0) = f(R) = 0
接著，我們考慮平坦處。為了方便計算，我們把 f(r) 展開。
f(r) = kRr2 − kr3
f′(r) = 2kRr − 3kr2
解得平坦處在 0 及 2R/3 二處。我們已經討論過 0 這個端點了，而
f(2R/3) = k (2R/3)2 (R/3) > 0
所以最大值發生在 r = 2R/3。
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
What was the cost of inoculating the first 50% of the population?
What was the cost of inoculating the second 50% of the population?
What percentage of the population had been inoculated by the time 37.5 million dollars had been spent?
C(50) = 50
C(100) − C(50) = 100
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?
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?
V = πr2h
我們希望只留下單一變數，r 或 h 其一。平方項好欺負！
r2 = d2 − h2
V = π (d2h − h3)
其中 h 的自然限制是 0 ≤ h ≤ d。代入端點會使得 V 為 0。接著我們對 h 微分得
導數為 0 的點有 ，但負不合。將 代入得
並且造成 V > 0，故此處確定是最大值。所以可以安心作答。
醫管系第 10 題
Based on a study conducted in 1997, the percent 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 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 age 65 year
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 ×
Find the population after 1, 2, and 3 hours.
Write the discrete-time dynamical system.
= 2.0 (bt − 1.0 × 106)
= 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.
The number of wild-type bacteria that mutate.
The number of wild-type bacteria and the number of mutants after mutation.
The number of wild-type bacteria and the number of mutant after reproduction.
The total number of bacteria after mutation and reproduction.
The fraction of mutants after mutation and reproduction.
(Begin with 1.0 × 106 wild type and 1.0 × 105 mutants.)
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 much there. More generally,
new concentration = old concentration
− fraction used × old concentration
Suppose the fraction used =
0.5 / (1 + 0.1Mt).
Write the discrete-time dynamical system and solve for the equilibrium.
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
Find the equation of the least-squares line y = Ax + B (to two decimal points .)
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我建議 CC 當局廢除 ND，而廢除 ND 是很容易的。畢竟它禁止改作，所以早已是死巷，當局只要不再推出新的 ND 條款即可。