This Is The Rate Change Of Position: Fill & Download for Free

This answer is going to change your life…I am about to tell you something superb and flawless.Consider the population of Deers in a forest.It is affected by many factors like habitat, competitors, predators (say Tigers), and the availability of food.We can see changes in their population (increase or decrease) with a certain period of time. This rate of change can be affected by nature.At first, let us assume that there are no Tigers in the jungle.Now, there’s no other factor that can reduce the population of the Deers.So, the rate of change of the population of Deers (with time) will depend only on their current population.Note: You don’t need to be a math student to understand the equations, which I will describe further. I will explain the meaning of all equations in this answer literally too.We represent the change in the population of any species (with time) by this differential equation:This equation suggests that the rate of change of population of the Deers in the forest depends on their population itself.(That is, the rate of increase in the number of deers depends on the number of deers present).If the population is high, the increase in their population is high too.If the value of constant k is positive, it suggests that the rate is positive (population increases with time).And if k is negative, population decreases with time.So, according to that equation, this is how the population of Deers changes with time:The number of deers increases exponentially.But… this isn’t what happens in reality.If a forest has deers, it ought to have tigers too.This is what keeps nature balanced.Now, let us add Tigers to this forest of our thought experiment.You may suggest that once we add Tigers to the forest, the Deer population will keep decreasing until all Deers wipeout. But this is not how nature works.I will explain how the scenario changes in an interesting manner.The growth of the species, when we have a Predator (say tiger) and a Prey (say deer), is be explained by the Lotka-Volterra Model.This model was made to describe how the population of predators and preys affect each other. It describes the rate of change of population of both species, with time.According to this model, the population change of each species depends on the current population of both.I will explain you this model with the example of Deers and Tigers:This is how the differential equation for the rate of change of deer population changes due to the extra term:Deer Population.Don’t panic after seeing such a large equation at once.If you see the output graph of this equation, mosquitoes will stop being attracted towards you. (You will sweat less)[That was an easter egg for my previous answer]I will first explain each term in the equation and then describe the meaning of that equation.The new equation that we got from the Lotka-Volterra Model suggests that the rate of change of the Deer population depends on the current population of both Deers and Tigers.When you look towards the equation, it will speak out to you, “If the population of Tigers is high, the population of Deers decreases (since the coefficient will be negative. And if the population of Tigers is less, the population of Deers will increase (since the coefficient will be positive).We can similarly describe an equation for the rate of change of the Tiger population:Tiger populationThe most important thing to note here is…Both equations of Deer and Tiger are interrelated.This suggests that,Population change of both Deers and Tigers, with time, is dependent on each other.This is the output graph of both differential equations:A graph that shows the population of both Deers and Tigers with time. Their population fluctuates periodically.Within a certain period of time, things keep repeating.This is how nature maintains balance on its own.Are things getting easy?No?Don’t worry.I will explain the graph with the words:Initially, the no. of Deers is less → So, tigers don’t get food and start dying.Due to less no. of Tigers → Deer population starts increasing.Deer population keeps increasing.At a certain point, no. of Tigers start increasing due to the plenty no. of Deers to eat.As Tigers keep increasing → no. of Deers starts to decline.Tigers keep increasing and eating Deers.At a point, due to less no. of Deers → some Tigers start dying.So, with the decrease in no. of Deers → no. of Tigers decrease too.Now, due to less no. of Tigers → Deers are safe and begin increasing their population.The cycle continues…That is so surprising, indeed!This is how animals maintain the balance of each other’s population.Now that I have taught you the concept,I am about to give you a life lesson from that.“Son, you should not overthink on a situation. You will end up being messed up and depressed”, your parents keep telling you all the time.Why do you keep overthinking every time?How does overthinking lead you towards a mess?This is how it happens:I am changing the terms inside the differential equations present in the Lotka-Volterra Model:Let us call this Lotka-Volterra-Sunny Model. Lol.These equations will give this graph as the output:Now, you will understand the familiarity of this graph with your life when I describe the graph with my words: (look at the graph while reading this)Initially, you think less about your condition → this leads to many problems in your life.Due to a problematic situation → You start to think and sort it out.Your problems start to decrease → As your thinking level is increasing.Till a certain point of thinking your problems keep decreasing and reach a minimum level.But, you don’t stop thinking about the situation. You keep doing that further.This is where you, my friend, start to overthink!The problems in your life begin increasing → As you start overthinking.The peak point of your thinking arrives → Here you realize that the rate of increase in the problems in your life has increased.So, you kind of give up and don’t want to think about the situation.As you start reducing the number of thoughts → The problems reach a peak and start decreasing too.When you realize that the problems in your life are decreasing → You gain the confidence to think back!The cycle continues…This is how your behaviour towards anything works!I would suggest that you must sort out everything and think about a problem only until a critical point of thinking in the graph is reached.Don’t think further and leave it. Otherwise, you will end up creating more problems.Thinking till the critical point on the graph is good. Further, it is a mess.What if I tell you,This is why we call ‘critical thinking’… ‘critical’Sunny DhondkarPost credits:It is interesting to note that this model works universally.You can swap the terms like this:N1 = CreativityN2 = BoringThis suggests that there is a limit to the creativity you show for making any artistic work (like painting, story, etc.).The higher is your creativity, less boring shall your work seem. But if you cross the critical point of creativity, it starts to look boring and less appealing.A similar effect can arise when you swap the terms with these:N1 = GoodN2 = BadMore is the number of good people in the world, bad deeds will decrease.But, once the power of good crosses a certain critical point, bad deeds start emerging.I can explain this with an example.In the movie The Dark Knight,Batman asks the Joker, “Why don’t you kill me?”To this, the Joker responds, “What would I do without you? You complete me.”The Joker (bad power) rises due to Batman (good power).The city of Gotham deals with usual crimes (before the rise of Batman) and these crimes looked by the cops of that city. Due to the presence of cops and other organisations, bad deeds are decreased.But… when Batman comes into action, enemies with a higher power (like the Joker) start rising too.This is why the real world doesn’t and shouldn’t have superheroes.If the power of good is increased by such a high level, enemies with great power would rise on the contrary. (This is what my model suggests too).Now I conclude my explanation.

Choose any continuously differentiable function [math]f(x)[/math]. The Fundamental Theorem of Calculus tells you that[math]\displaystyle \int_a^b f'(x) [/math][math][/math][math]\ dx = f(b) - f(a). \tag*{}[/math]What this says is that the integral of the rate of change of a quantity in some parameter is the same as the net change of that quantity. Now, you might naturally ask: what quantity? Which parameter? However, that is the wonderful thing about the Fundamental Theorem of Calculus: it doesn’t matter. Choose any quantity that varies in terms of some parameter. Bam! You automatically have an application of integrals. Here is a short list of examples:The rate of change of position with respect to time is velocity; ergo, integrating the velocity with respect to time gives the change in position.The rate of change of velocity with respect to time is acceleration; ergo, integrating the acceleration with respect to time gives the change in velocity.The rate of change of momentum with respect to time is force; ergo, integrating the force with respect to time gives the change in momentum.The rate of change of work with respect to position is force; ergo, integrating the force with respect to position gives the work done on an object.The rate of change of voltage with respect to position is the electric field strength; ergo, integrating the electric field strength with respect to position gives the voltage change.The rate of change of volume in a tank with respect to time is the flow speed; ergo, integrating the flow speed with respect to time is the change in volume.The integral of the rate of change of a population with respect to time gives the population growth.The integral of the rate of change of a stock with respect to time gives the net gains/losses over that period of time.The integral of the rate of concentrate change of a chemical with respect to time gives the net gain/loss of concentration over that period of time.In most calculus classes, you learn to evaluate integrals using the Fundamental Theorem of Calculus, by finding explicit anti-derivatives of a given function. This is sometimes useful, but for the vast majority of integrals that you can write down, you can’t actually find an anti-derivative that can be written down in terms of functions that you are well-familiar with. So, instead you approximate the integral, by using the fact that you know that it can be understood as an area under a curve. If you are unfamiliar with what sort of tools are available to do something like that, I gave some basic examples in my answer to How are we so sure that the Chudnovsky algorithm gives the right value of π up to billions of decimal places?, where I showed how you can use integration to give a provably accurate approximation of [math]\pi[/math].I should stress that this isn’t some sort of theoretical sleight-of-hand to apparently conjure up applications, even though no one actually thinks about it this way. To wit, here is an excerpt from Shigley’s Mechanical Engineering (which was just the first engineering textbook that I could find).

The trajectory alone does not tell you about velocity, acceleration or jerk. You also need to know its time dependence.Velocity is the rate of change of position over time. Acceleration is the rate of change of velocity over time. And jerk is the rate of change of acceleration over time. For all three, knowing the time coordinate is essential. So if the trajectory itself is three dimensional, what you need is the four-dimensional path (basically, the worldline) of the object through space and time.