APPLICATION OF DERIVATIVES IN REAL
The derivative is the exact rate at which one quantity changes with respect to another. In calculus we have learnt that when y is the function of x , the derivative of y with respect to x i.e dy/dx measures rate of change in y with respect to x .Geometrically , the derivatives is the slope of curve at a point on the curve . The derivative is often called as the “instantaneous” rate of change. In the next few paragraphs, we will take a deep dig about the application of derivatives in real life.
The derivative of a function represents an infinitely small change the function with respect to one of its variation. The process of finding the derivatives is called as differentiation. Modern differentiation and derivatives are usually credited to “ Sir Issac Newton” and “ Gottfried Leibniz”. They developed the fundamental theorem of calculus in the 17th century. This related differentiation and integration in ways which revolutionized the methods for computing areas and volumes. However , Newton’s work would not have been possible without the efforts of Issac Brown who began early development of the derivative in the 16th century.
The bottom line is that nothing is useless. Whenever we say something is useless, it simply means that we don’t know how to use them. but I will bet somewhere someone knows how to use it.
And derivatives, which is the mathematical model of change and has amazing prediction powers, is extremely useful in our everyday life. You would need some practice to know how to use it well in everyday life. But once you master it, it helps greatly to weed out irrationality, clarify your choices and helps in forecasting.
Through derivatives we can easily find out maximum and minimum values of particular functions and find whether function is increasing or decreasing. There are countless areas where derivatives can be used.
Here are the some most important areas where derivatives are used.
Today’s usage of derivatives has seen the development of multiple strategies, into which companies incorporate derivatives. The use of hedging through derivatives is still highly prevalent. Companies, both in-and-out of the financial industry have begun to use derivatives a method of speculating and generating income. Arbitrage firms have also started to use derivatives as a method creating arbitrage opportunities.
The cases of Enron, John Paulson, Page 29 Orange County, Exchange Traded Funds, and Long Term Capital Management are all demonstrative of the methods of using derivatives in today’s market. Many companies also have started to turn to using derivatives for income generation since income from derivatives, even if being used for hedging is treated as ordinary income.
Enron is an excellent example of a firm that started to drift from the original business in favour of financial derivatives. Enron originated as an energy producer, but at the time of the firms collapse, the company had become a full-time energy derivatives dealer.
Derivatives with respect to time
In physics, we are often looking at how things change over time:
- Velocity is the derivative of position with respect to time: v(t)=ddt(x(t))v(t)=ddt(x(t)).
- Acceleration is the derivative of velocity with respect to time: a(t)=ddt(v(t))=d2dt2(x(t))a(t)=ddt(v(t))=d2dt2(x(t)).
- Momentum (usually denoted pp) is mass times velocity, and force (F) is mass times acceleration, so the derivative of momentum is dpdt=ddt(mv)=mdvdt=ma=Fdpdt=ddt(mv)=mdvdt=ma=F.
One of Newton’s laws says that for every action there is an equal and opposite reaction, meaning that if particle 2 puts force F on particle 1, then particle 1 must put force −F on particle 2. But this means that the total momentum is constant, since
This is the law of conservation of momentum.
Derivatives with respect to position
In physics, we also take derivatives with respect to xx.
- For so-called “conservative” forces, there is a function V(x) such that the force depends only on position and is minus the derivative of V, namely F(x)=−dV(x)dxF(x)=−dV(x)dx. The function V(x) is called the potential energy. For instance, for a mass on a spring the potential energy is 12kx212kx2, where k is a constant, and the force is −kx.
- The kinetic energy is 1/2mv2. Using the chain rule we find that the total energy is
This means that the total energy never changes. These are just a few of the examples of how derivatives come up in physics.
Derivatives are used in to model population growth, ecosystems, spread of diseases and various phenomena. The area that I will focus particularly is population growth. Suppose n =f(t) is the number of individuals of some species of animal or plant population at time t.
The change in the population size in the population size between n=f t1 and t2.
The average rate of growth then is:
Average rate of growth = (Δn / Δt)=( f (t2) – f(t1)) / (t2-t1 )
The instantaneous rate of growth is the derivative of the function n with respect to t i.e. growth rate = lim(Δt -> 0) ( n/. t) = (dn/dt)
The instantaneous rate of change does not make exact sense in the previous example because the change in population is not exactly a continuous process. However, for large population function by a smooth (continuous) curve.
- Example: suppose that a population of bacteria doubles its population, n, every hour. Denote by no the initial population
i.e. n(0)=no. In general, then, n(t)=2t no
– Thus the rate of growth of the population at time t is (dn/dt)=no2tln2
One use of derivatives in chemistry is when you want to find the concentration of an element in a product. Derivative is used to calculate the rate of reaction and compressibility in chemistry.
In recent years, economic decision making has become more and more mathematically oriented. Faced with huge masses of statistical data, depending on hundreds or even thousands of different variables, business analysts and economists have increasingly turned to mathematical methods to help them describe what is happening, predict the effects of various policy alternatives, and choose reasonable courses of action from the myriad of possibilities. Among the mathematical methods employed is calculus. In this section we illustrate just a few of the many applications of calculus to business and economics.
All our applications will centre on what economists call the theory of the firm. In other words, we study the activity of a business (or possibly a whole industry) and restrict our analysis to a time period during which background conditions (such as supplies of raw materials, wage rates, and taxes) are fairly constant. We then show how derivatives can help the management of such a firm make vital production decisions.
Management, whether or not it knows calculus, utilizes many functions of the sort we have been considering. Examples of such functions are C(x) = cost of producing x units of the product, R(x) = revenue generated by selling x units of the product, P(x) = R(x) − C(x) = the profit (or loss) generated by producing and (selling x units of the product.) Note that the functions C(x), R(x), and P(x) are often defined only for nonnegative integers, that is, for x = 0, 1, 2, 3,… .
The reason is that it does not make sense to speak about the cost of producing −1 cars or the revenue generated by selling 3.62 refrigerators. Thus, each function may give rise to a set of discrete points on a graph, as in Fig. 1(a). In studying these functions, however, economists usually draw a smooth curve through the points and assume that C(x) is actually defined for all positive x.
Of course, we must often interpret answers to problems in light of the fact that x is, in most cases, a non-negative integer. Demand function – an equation that relates price per unit and quantity demanded at that price. If ‘p’ is the price per unit of a certain product and x is the number of units demanded, then we can write the demand function as x=f(p) or p = g(x) i.e., price(p) expressed as a function of x.
Here are some examples of derivatives used in daily life:
Example 1 : Games
If you don’t understand derivatives, you will suck at many games.
Say FPS. When you shoot a moving enemy, it is very easy to miss. So you need to “predict” where they might be in the next moment, and then shoot there. That’s how you get them.
How would you predict? You predict by checking out the direction and speed of their movement, i.e. the derivative of their movement. Then when you predict their eventual location, you are unintentionally doing a mental integration. That’s how you “predict” their movement and get your shot.
Say you play COC. Do you spend your real-life dollars to buy in-game golds, or do you add another builder? I say you should add another builder, because that increases the derivative of your gold quantity.
Say you play D&D and a feat let you exchange attack for damage. How many attack should you turn into damage? Say you turn x attack into x damage. This means you are optimizing the DPR function (damage per round), which would be a degree 2 polynomial in x. With derivatives, you can find the optimal x for this DPR function. Then your fighter will always fight a little better than other fighters of the same level.
In short, any game with changing numbers, be it victory points, scores, or in-game money, or level or experience values or HP, must uses derivatives somehow. So derivatives can help you understand or invent strategies to crash people who naively thinks that math is useless. Why do you think there are so much more bad-ass Asian gamers than American gamers? I’d say it is because the American society fear and despise mathematics in general.
(For some extremely hard games, derivatives play an even deeper role. Say Hanoi tower with four peg. Its optimal solution involves triangle numbers. But why are triangle numbers involved? Because the Frame-Stewart algorithm breaks the whole pile into two, a top-pile and a bottom-pile. And when you add a disk to the whole pile, it might be added to either the top-pile or the bottom-pile.
For the optimal solution, the added difficulty should equal. In another words, the derivative of top-pile solution length and the derivative of bottom-pile solution length should equal. This is how one arrives at the triangle numbers.)
Example 2: Teaching
If you want to teach someone something, the ultimate goal is to increase their knowledge, say K. We can simply let them memorize some knowledge at a fixed speed. So K’ is constant and we are increasing K at a constant speed. So K grows like x.
Now, rather than teach them the material, you can try to motivate their interests. With more interests in the subject, they will learn faster. So motivating their interests is like increasing K’. So K’’ is constant (the amount of their current interests), and K’ is increasing at a constant speed, and K is increasing like a parabola, like x^2.
Now, rather than motivate them in some subject, you can teach them to self-motivate. To learn to nurture one’s own interests in something. This is like increasing K’’. So now K’’’ is constant (their ability to self-motivate), K’’ is increasing at a constant speed, K’ is increasing like a parabola, and K is increasing like a degree three curve, something like x^3.
The best way to teach start with giving them some knowledge, then motivate them little bit, and then teach them to self-motivate. Why? Compare x, x^2, x^3 and so forth. Starting at the origin, the function x grows the fastest. Then other gradually catch up, and eventually x^3 will become the fastest. So at the early stage of teaching, simple give them some knowledge is the best approach.
Then gradually, we should shift our teaching focus from providing knowledge to providing interests. And once some interests is established, we should eventually teach them to self-motivate.
At last, derivatives are constantly used in everyday life to help measure how much something is changing. They are used by government in population censuses , various types of sciences and various other areas. Knowing how to use derivatives, when to use them and how to apply them in everyday life can be a crucial part of any profession, so learning early is always a good thing.
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