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Real life Applications of limits in calculus

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hadeel ali

on 7 January 2017

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Transcript of Real life Applications of limits in calculus

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Given these two facts then every application of calculus requires limits of one sort or another.
How are limits (Calculus limits) used or applied to daily life? Or applied to the real world problems?
Real-life limits are used any time you have some type of real-world application approach a steady-state solution.  

As an example, we could have a chemical reaction in a beaker start with two chemicals that form a new compound over time.  The amount of the new compound is the limit of a function as time approaches infinity.

Similarly, if you drop an ice cube in a glass of warm water and measure the temperature with time, the temperature eventually approaches the room temperature where the glass is stored.  Measuring the temperature is a limit again as time approaches infinity.

Limits are also used as real-life approximations to calculating derivatives.  It is very difficult to calculate a derivative of complicated motions in real-life situations.  So, to make calculations, engineers will approximate a function using small differences in the a function and then try and calculate the derivative of the function by having smaller and smaller spacing in the function sample intervals.  For example, when designing the engine of a new car, an engineer may model the gasoline through the car's engine with small intervals called a mesh, since the geometry of the engine is too complicated to get exactly with simply functions such as polynomials.  These approximations always use limits.
You cannot have calculus without limits!
Limits are needed to define differential calculus and so every application of differential equations assumes that the limits defining the terms in the equations exist.
Limits are needed in integral calculus because an integral is over some range of variables and these form the limits in the integrations.
Infinity in what sense? Limits at infinity are super important for some of the most fundamental aspects of calculus, which is certainly useful in real world applications.
When we talk about infinite limits we are talking about how numbers behave as they get larger (or in the case of a series as we add more and more terms). Let's say we take the fraction x / (x+a) where 'a' is a small constant. If we take the limit as x -> Inf we would find this function goes to 1. Now at first you might look at this and say "So what?". What this basically says is that as x becomes large x and x+a become so close that their fraction is essentially 1 or that x+a can be approximated by x.
Does the concept of infinity have any practical applications?
I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world would go under long before an infinite amount of loops had been reached.
Applications of Infinity in real life.
Real life Applications of limits involving infinity in calculus.
Lesson 5 Section 5.1
By Hadeel and Thajba
a) the strength of electric, magnetic or gravitational fields.
b) the areas, forces, eights of objects.
Again you might say "So what?" Let's put this in a real world example. Let's say I have $1,000,000,000.48 (1 billion dollars and 48 cents). If you asked anyone how much money I have they might say "You have a billion dollars." Why is that? Because we say that the 48 cents is such a small amount it does not make much of a different. This is actually saying the exact same thing that the math above says. If 'x' (1 billion) is a huge number compared to 'a' (48 cents) then we can essentially treat x + a (1 billion dollars and 48 cents) as simply x (1 billion dollars).

One of the major applications of infinite limits is that it allows us to take big complex functions and figure out which pieces of information contribute the most to the answer. This allows us to simplify problems to the point we can actually solve them. Rounding is essentially the every day application of infinite limits that also has major applications in math and science.
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