Since the lines in polygons are straight in geometry, defining the region is simple for the students who are studying geometry. In certain cases, though, the lines on a graph may be curved rather than straight. These are the graph representations of functions.
When calculating the area of a polygon on a graph of a function, the formulas used to calculate the area of a polygon would not perform because they are imprecise when a curved line is involved. Calculus is used instead to determine the precise result for curved lines.
In this post, we will explain the fundamental theorem of calculus, its importance, applications, and why you should go for calculus in school. If you are a math geek or love to do calculus, you will find this post much more meaningful than a whole lecture in class.
Table of Contents
Calculus – Fundamental Theorem
The fundamental theorem of calculus must be used to find the area of space on a graph centered on a function. It is presumed in this theorem that there is a function on the graph and that there are two points denoting the start and end of the field to be evaluated. The integral is the distance between these two lines.
The theorem can be separated into two sections. The first part indicates what is needed, and the second part clearly indicates how to solve it. The theorem states that the integral of f(x)dx from a to b is equal to g(b)-g(a), where g is the antiderivative of f(x).
Simply stated, the region is determined by subtracting the antiderivative of the function solved for a from the antiderivative of the function solved for b.
It is important to find the antiderivative of a function in order to solve the theorem for that function. You can use an antiderivative calculator or integral calculator (same term used for integration) to find the antiderivative (integral) of a function. This is transformed into g (x). So, a and b are plugged into this and the dilemma is solved. Finally, the result for a is subtracted from b to obtain the needed area.
Solving for Antiderivative
Understanding how to locate the antiderivative of a function is one of the most important aspects of solving the basic theorem of calculus. The method for determining the antiderivative is determined by the form of function in the problem. The power law, for example, would apply to functions with an exponent. In certain instances, the reverse rules would be necessary to locate the function’s antiderivative.
Why Calculus Is Simple?
Calculus consists of only two components:
Differential calculus is specifically concerned with deciding derivatives, while integral calculus is concerned with determining integrals in general. Finding derivatives involves dividing and finding integrals involves multiplying. This is the fundamentals of calculus, and as long as a person knows the algebra and trig formulas used to do the above, they can find it simple to solve a wide range of calculus problems.
While it is straightforward to identify the area where all of the lines are parallel, this is not always feasible. Calculus and the fundamental theorem of calculus allow students to calculate the field of graphs that have a structure rather than a straight line.
One of the most common examples of this in practice is using the fundamental theorem to calculate how fast a vehicle was traveling at a given time, rather than just the average of how fast they were going during the journey. Since the car will accelerate and decelerate, the graph will have a curved line, so the fundamental theorem is needed since the methods used for polygons would not have a detailed result.
Applications of Calculus in real life
Calculus can be used in a variety of ways and applications.
- Medicine, physics, finance, statistics, and chemistry are only a few of the fields that use calculus.
- It is used to construct mathematical equations in order to find the best solution.
- Calculus is used in many physics terms. Dynamics, electricity, harmonics, motion, acoustics, astronomy, and heat are some of the physical concepts that use calculus.
- It is used in advanced physics principles such as electromagnetism and Einstein’s theory of relativity.
- It can be used to model functions of chemistry such as reaction rates and radioactive decay.
- In genetics, it is used to calculate rates such as birth and death rates.
- Calculus is used in economics to compute marginal cost and marginal income, allowing economists to estimate maximum benefit in a given situation.
- It is also used to validate responses for various mathematical subjects such as analytical geometry, probability, algebra, and statistics.
Why you should study Calculus?
As an undergraduate, you probably regard calculus as yet another illogical memorization of calculations that must be completed in order to graduate from high school. You believe you will never have to use any of the ‘hypothetical gibberish’ you have learned. As a result, you treat the issue with intellectual cynicism and prepare yourself to bear what you cannot heal.
Perhaps this ill-will stems from the idea that calculus is a vital weapon for your survival in the real world. Regardless of how you measure progress, learning calculus is not needed for your well-being. In reality, to say that it is required to become a scientist is an exaggeration.
After all, many professional engineers are not afraid to confess their lack of knowledge on the matter. The knowledge of calculus of such engineers is restricted to learning the equations and how to use them. If an engineer has a clear sense of what to expect based on previous experience, number crunching isn’t as relevant. The best of them just need the numbers crunched to affirm what they believe the outcome would be.
What can you do with Calculus?
Calculus gives you the opportunity to determine the impact of shifting circumstances on the methods under investigation. You will learn how to manipulate the system and make it do what you want by learning it. Calculus allows engineers incredible influence over the structural works by allowing them to model and manipulate structures.
The invention of this and its applications to physics and engineering is likely the most important influence in the advancement of scientific science outside Archimedes’ time. And it was Calculus that was responsible for the industrial revolution and all that came with it, including almost all of the great developments of the last few decades.
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