Application of Derivative
Group 5: • Nguyễn Khắc Thế (leader) • Huỳnh Việt Thống • Trần Hưng Trí • Nguyễn Hoàng Nhật • Trần Đại Ngọc Hải • Đỗ Hoàng Phúc
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Table of Contents 1. Introduction 2. The Taking-off Process 3. The Internal Combustion Engine 4. The Filling-up Process 5. The Developing Rate of Bacteria 6. The Drug Sensitivity 7. Main Result
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Introduction Students often feel lost when studying Math, and thus, they do not have the motivation to learn this key subject. One of the reasons for this problem is that, students can deal with the theoretical problems very well, they can do all the questions in a test, but they can not find the ways to apply that knowledge to daily life. Therefore, in this report, we will exhibit the practical aspects of Calculus as well as Derivative, how can we apply the theorem we learned at the university, or how it can affect our lives tremendously. For instance, we will take five examples in the fields of Physics and Biology: the taking-off process, relation between pressure and volume, filling up the tank with
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water, the reproduction of bacteria and drug sensitivity.
Firstly, we can use derivative to determine the instantaneous velocity at a time. We represent the orbit of the objet as a function, then we use the Derivative to calculate the function of the instantaneous velocity. This method is widely used to find out the velocity of a horizontally-moving object. But can we apply it to an object that go upward, such as the taking-off process of a plane? We will consider this one in the first example.
Fig 1.1: The special struture of airplane’s wings, with make it be able to fly.
A plane can lifts off the ground because of the difference of pressure to the upside and downside of the wings due to its structure. This process is affected very much by the density of atmosphere around the plane. At low altitudes, where the air is more dense, the plane can climb well, whereas it goes higher, the climbing rate decreases over time. The graph, which represents the changing in height of the plane from the time it takes off until they keep their Fig 1.2: The actual orbit of elevation as a constant, is similar to the graph the taking-off process of an of an equation of natural logarithm according to airplane. y-axis. Therefore, we can determine the height of the plane by using the equation: H=A.ln(f(x)) For example, an Airbus A380 takes off from an airport at sea level and its altitude (in feet) at time t (in minutes) is given by: h = 1000.ln(x + 2) Find the rate of climb at time x = 2 min. To find the rate of climb (vertical velocity), we need to find the first derivative: Fig 1.3: The model of function y =ln (x), which looks d/dt 1000.ln(x+2)=1000/(x+2) similarly to the orbit of an At t = 2, we have v = 1000/4 = 250 feet/min. plane. So the required rate of climb is 2500 feet/min.
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Secondly, we will come to the topic of pressure. We know the famous equation, the Ideal Gas Law: P.V=n.R.T where P is pressure, V is volume, T is temperature, n is the number of gas molecules, and R is the gas constant. The internal combustion engine (ICE) is an engine in which the combustion of a fuel occurs with an oxidizer in a combustion chamber that is an integral part of Fig 2.1: Robert Boyle (1627 the working fluid flow circuit. – 1691), an Irish mathematician, who laid the founA mechanic wants to know the increase of Pressure dations for Ideal Gas Law.
inside the engine to check the stabilization. As we considered before, with everything that has a change in value, we can calculate the rate of change with respect to another value. Because the temperature inside the engine is often at constant, so we only consider the Pressure and Volume.
The engine will crack if the pressure surpass 400 kPa. At a certain instant, the volume is 480 cm3, the pressure is 160 kPa, and the volume is decreasing at a rate Fig 2.2: The graph of P with of 45 cm3 / min. At what rate is the pressure increasrespect to V, in the case ing at this instant? that T and n are constants.
We have P.V=C, by alternating the given values, we have a new equation, P.V = 76800. We want the rate at which the pressure is increasing, so solve for P =76800/V. Then differentiate both sides with respect to time. dP/dt = -76800/V^2 (dV/dt)
Back to the equation, substitute the given rates. dV/dt = -76800/((480)^2) * (-45) Fig 2.3: The construction of dV/dt = 15 (positive means increasing) a typical 4-stroke engine. By realizing this factor, the mechanic can compute the increasing rate of pressure inside the engine, and can prepare for bad situation like cracking or breaking.
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Now, after solving some complicated issue, we back to some daily situation. Suppose that we are filling a tank with water. There are two things that are increasing: the volume of water in the tank, and the height of the water level in the tank.
Furthermore, the rate at which each of those quantities is increasing is related. Thus, if we know the Fig 3.1: The volume of a rate that the volume is increasing, we can determine cone is calculated by the the rate at which the height of the water level is in- equation: V=1/3.h.Ď€.r2 creasing. In related rates problems, one can compute the rate of change of one quatity, then use Derivative to calculate another.
Assume that now we are covering an inverted conical tank with the dimensions: height equals to 8 feet, base radius equals to 6 feet. Hypothesize that we are filling the tank with water at the rate of 2 ft3/minute. How fast is the height of the water changing when the water level is 4 feet?
Let V, r and h consecutively be the volume of the water, the radius of the base, and the height at time t, where t is measured in minutes. At the water level is 4 feet, the surface area equal to 1/4 area of the base, or equal to 9 π. Fig 3.2: The height of the We have that: h=3V/S water level is not so simple
dh/dt=3/S.(dV/dt) substitute dV/dt by the given value (2 ft3/min), we can compute the rate of water level chaging: dh/dt=3/(9 π).2=0.21 feet/minute. Thus, the rate of water level changing is 0.21 feet/ minute
as we thought.
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Fig 4.1: The bactiria are too small to be saw by naked eyes, we must use micro-scope instead.
Fig 4.2: The image of E. Coli (15000 times larger).
Fig 4.3: The Bacteria’s Growth Curve. In this example we only consider in ideal enviroment.
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Come to Biology topics, bacteria are considered as the smallest organism on the world. However, they are also the quickest reproducing creature. One reason lead to this issue is their special duplicating process: Asexual Reproduction, not Sexual Reproduction like the other organism. In typical asexual reproduction, the parent living thing is replaced by two daughter organisms, because it literally divides in two. It makes them breed very quickly. A biologist want to determine the instantanneous developing rate of Escherichia coli bacilli (or E. coli), that are raised in ideal enviroment. Suppose that the initial amount of they is no. After t cycle, the number of E. coli is: N= 2n.no Derivative both sides with respect to t, we can have the instantaneous rate of growth of their population: d/dt N= ln(2). no. 2n we can substitute t with the specific value to calculate the rate at one time, for example, after the second cycle with 100 bacteria is: d/dt N= ln(2).no.22 = 277.25 bacteria/cycle
To medical field, the drug-using skill of a doctor is very important. If the amount of the drug is a little bit higher or shorter, it would cause terrible impact to the patient’s body. Therefore, it is extremely important for doctors to understand the strength of the drugs they prescribe to patients. The strength of the drug is is a function of M, in which M measures the dosage. Dosage is the amount of medicine absorbed in the blood. In addition, the sensitivity of the patient’s body to the drug is the derivative of R with respect to M.
Fig 5.1: A wrong use of pills can lead to an awful disaster.
For the Aspirine, the drug strength is described by the equation: R (M) = 2.M.√(10 + 0.5M) where M is given in milligrams. For a doctor who want to find to sensitivity to a dose of 50mg, he has to find the Derivative of drug strength in order to compute the sensitivity of the patient’s body: R’ = 2 √(10 + 0.5M) + 1/(2. √(10 + 0.5M)) This is the function for patient’s sensitivity. In this Fig 5.2: A doctor must case, by substitute the given value of M, he can know very well about the sensivity of each kind of determine the effect of drug to his patient: drug. R’=70.01
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MAIN RESULT Those are our 5 examples about physic and biology used in daily life. After reading our assignment I think our group members will be able to fully understand how are derivatives used in some field in our real life. Furthermore they can learn some useful way how to use derivatives to solve the most difficult task not only in physics but also in biology. 10
Summary Those are 5 example of using Derivative in daily life. In short, each example is about: the climb rate of an airplane, the changing rate of pressure and volume inside an Internal Combustion Engine, the change depend on time of water level in a inverted conical tank, the developing rate of E Coli and the drug sensitivity. We can use Derivative to calculate the changing rate of these value.
Reference
This report use sources from: • Two e-books by Stewart, section 3.3 in text book (pages 210 & 211) • Wikipedia - The Free Encyclopedia • http://www.intmath.com/ 11