The derivative and integral changed the world. They made computers, smartphones, airplanes and many other modern machines possible. Without them, we would be living in a very different world.
The derivative and integral are the building blocks of the mathematical universe. Yet they are probably one of the least understood and most avoided topics. In fact, both concepts are quite simple. Newton in England and Leibniz in Germany independently laid the foundations of these concepts. As the concept of calculus began to develop, the derivative and integral, which form its basis, became involved. Now let’s focus on the answer to the question “what are derivative and integral?” in this article.
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What is a Derivative? What is Integral?
In fact, both are essentially about change. However, they do it differently. The derivative is about how fast something changes, while the integral is about quantity, how much something changes. If you are a doctor, you need the integral to understand how big a patient’s tumor will grow after a while.
The total amount of electricity you use, the chances of your favorite candidate being re-elected, how much a support beam can bend… It doesn’t matter. No matter what, when we want to know how much something has changed, we use integrals.
What are Derivatives and Integrals Good For?
Imagine you work as a mathematician in a car factory. Your job is to make the cars as safe as possible. You can measure this by burning cars, crushing them, crashing them into things. But you can do it much cheaper by using math. In a car accident, the biggest risk is a head injury. The forward or backward movement of the head during a collision can be vital. The speed of the vehicle is very influential on the severity of this movement. You can use differential equations to measure this speed.
A differential equation is an equation relating one or more functions and their derivatives. With the help of these equations, you can see how fast the head is moving at each moment of the collision. This allows you to recommend the necessary safety equipment for the vehicle.
As a mathematician, you have calculated how fast the head is moving at a series of moments. But you still don’t know how dangerous it is to collide with the speed at which the head is moving. That’s why you now need the integral. To define the danger, you need to find how much the head moves in total during the collision. Let’s say you find that it moved back and forth twice in one second. The accident lasted a total of 4 seconds. You can’t say 2 times 4 is 8 movements. What we need to do is actually simple. One second is a big time frame for an accident.
However, if we look at much smaller time units instead of seconds, determine the movement in each time unit and then take their sum, our solution will be realistic. By the way, this example was not made up by us. Indeed, car manufacturers use this calculation in the safety tests of their cars. Now, what is a derivative? What is integral? And you should have an idea of what they are for, but it’s not over yet. You may remember that the integral is also related to geometry.
Derivative and Integral in Geometric Meaning
Let the graph below show how something changes over time. To see this change, we need to examine two points that are very close to each other. These points are so close that we can think of them as forming a line. So we can judge how short this line is by looking at the shape of the tangent. The “steeper” the tangent, the faster the change, the more horizontal, the less change. This is the geometric equivalent of the derivative. What about the integral?
Actually, if you look carefully at the image, it shows how to calculate the area under a curve in a series of steps. The smaller the steps, the less space is left in the area under the wavy shape. In short, if you divide the area under the line into small rectangles. You calculate the area of each rectangle and add them all up. The smaller the rectangles, the more accurate the calculation. You can calculate the volume in the same way. However, this will be a little more difficult. You need to work both vertically and horizontally, although the principle is the same.
Derivative and Integral are in Every Area of Our Lives
Whether you realize it or not, the integral and the derivative are everywhere, in your car, in your coffee machine or in the thermostat of your central heating system. Everywhere there are changes that need to be calculated, and it is almost impossible to do so without these two. In the natural world, everything is constantly changing, so we use the same logic to study it. But do you have to learn derivatives and integrals? Actually, it depends a bit on the profession you choose.
In our daily life, we enjoy the advantages of this duo, but we don’t have to do the calculations ourselves. In this sense, unless you choose a profession where you need to use mathematics, you do not need to have a detailed knowledge of integrals and differentials. Even if you choose a related profession, the calculations are now done by computers.
Nevertheless, this does not mean that you should complain about what you have been taught. The idea behind it may be hard to understand at first, but as we have tried to explain briefly, it is not crazy. It is much easier to make sense of it if we get away from the frightening appearance of its symbolic representation and try to grasp the logic of it.
Understanding the basic idea is crucial if you want to know how the world around you works. Integrals and differentials have changed the world. They made computers, smartphones, airplanes and many other modern machines possible. Without them, we would be living in a very different world.
