When a foreign object lodged in the trachea (windpipe) forces a personto cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through . For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the air-stream, the greater the force on the foreign object. X rays show that the radius of the circUJlar tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing,the velocity vof the air-stream is" related to the radius r of the trachea by the equation v(r) = k(r0 - r)r2 1/2 r0 ≤ r ≤ r0where k is a constant and r0 is the normal radius of the trachea. The restriction on r is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than 1/2 r0 is prevented (otherwise the person would suffocate).(a) Determine the value of rin the interval [1/2 r0 r0] at which v has an absolute maximum. How doesthis compare with experimental evidence?(b) What is the absolute maximum value of v on the interval?(c) Sketch the graph of v on the interval [0, r0].
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
When a foreign object lodged in the trachea (windpipe) forces a person
to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through . For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the air-stream, the greater the force on the foreign object. X rays show that the radius of the circUJlar tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing,the velocity vof the air-stream is" related to the radius r of the trachea by the equation
v(r) = k(r0 - r)r2 1/2 r0 ≤ r ≤ r0
where k is a constant and r0 is the normal radius of the trachea. The restriction on r is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than 1/2 r0 is prevented (otherwise the person would suffocate).
(a) Determine the value of rin the interval [1/2 r0 r0] at which v has an absolute maximum. How doesthis compare with experimental evidence?
(b) What is the absolute maximum value of v on the interval?
(c) Sketch the graph of v on the interval [0, r0].
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