Calculus: The Fundamental Theorem Of Calculus

Improved Essays
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus evaluate an antiderivative at the upper and lower limits of integration and take the difference. This theorem is separated into two parts. The first part is called the first fundamental theorem of calculus and states that one of the antiderivatives of some function may be obtained as the integral of the function with a variable bound of integration. The second part of the theorem, called the second fundamental theorem of calculus, states that the integral of a function over some interval can be computed by using any one of its infinitely many antiderivatives.
The fundamental theorem of calculus along curves states that if has a continuous infinite integral in a region containing
…show more content…
The mathematician who discovered what we call the fundamental theorem of calculus is Isaac Newton. For him, this theorem was virtually self-evident, as Victor Katz states on the book “A History of Mathematics”.
Isaac Newton was born in England on January 4,1643 and is a known as physicist mathematician, astronomer, natural philosopher, alchemist and theologian. He is considered as one of the most influent men in history by developing the principles of modern physics.
On 1661, Newton entered University of Cambridge’s Trinity College in a program similar to a work-study program, as his uncle was pushing him to do so. He spent all his free time reading from the modern philosopher and the result was amazing.
On 1665, Newton approached mathematics with a new perspective: infinitesimal calculus. He started developing the theory of calculus that Englishmen John Wallis and Isaac Barrow was working on earlier. Thus, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us. Even though he had to follow a tough path he was able to publish Philosophiae Naturalis Principia Mathematica (Principia) in 1687.This book contains information on all of the essential concepts of physics, except
…show more content…
On 1705, Newton’s career was known by everyone and he was knighted by Queen Anne of England.
He died on March 31, 1727. Back to the Fundamental Theorem of Calculus. This theorem is a simple theorem that establishes the relationship between the derivative and the integral and as we said before has two parts. The first part that talks about the relationship between the derivative and the integrals and the second part that tells us how to calculate a definite integral.
The easiest way to explain how this theorem works is:
• The First Fundamental Theorem of Calculus:
First, let’s say we have a function f(x)
Now
We take two points on the x axis: a and x. And we consider the area under the curve from a to x:

If we take a smaller , we will get a smaller area:

And if we take a greater , then we’ll have a bigger area:

So we can define our area function as A(x)->(depends on what curve we’re using)
We know that the area under the graph of a function f(t) from a to x is: and our function will be:
So the first part of the fundamental theorem of calculus

Related Documents

  • Superior Essays

    This is what sparked scientists, like Newton, to search for the truth about the natural world through science. Newton first became interested in the sciences during his time studying at The Trinity College at Cambridge. He was able to explore philosophy, chemistry, and mathematics there, and he found interest in all of these subjects. (Chalquist, 2009). He became familiar with the Roman ideas about Geometry and what little was known about Calculus.…

    • 1071 Words
    • 5 Pages
    Superior Essays
  • Superior Essays

    Sir Isaac Newton was one of the most influential scientists of all time. His ideas became the basic principles for physics according to http://www.livescience.com/46558-laws-of-motion.html. Newton is known for his work in gravity and the motion of planets. Isaac Newton based his discoveries off of Descartes and changed the modern era of scientific thinking. In mathematics, Newton created integral and differential calculus, and in optics, he created the first telescope.…

    • 1189 Words
    • 5 Pages
    Superior Essays
  • Decent Essays

    Andy Nguyen Mrs Lomax AP Calculus October 13, 2017 Limit and continuity Journal Entry Question 1: “Explain the importance of limits and continuity. Make sure to explain how limits and continuity are related. (Hints or topics to include: one sided limits, definition of continuity, important theorems, IVT)” The concept of the limit is one of the most essential things in order to prepare for Calculus. First, a limit is a certain number that one function can approach as an input comes closer and closer to one number. Limit can be found in many ways such as using substitution, graphical investigation, numerical approximation, algebra, or some combination of these.…

    • 880 Words
    • 4 Pages
    Decent Essays
  • Decent Essays

    I found it easier in English because mathematics is very complicated in Arabic, and the rest of us know that. Many students like to take this class to know more information about their major before attending the university. When they take this class, they will be ready for the calculus class in the university and everything will be easier. Scientific Method The Scientific method is an essential thing that you do before you publish your research. It contains six steps, and you have to follow the order…

    • 852 Words
    • 4 Pages
    Decent Essays
  • Improved Essays

    When discussing his discoveries that dealt with magnets, Newton incorporated his math, as he says, ‘A needle placed on the magnet (or rubbed) parallely [sic] to its equator will not acquire any verticity nor will two needles in y⁺ position to y ᵉ magnet attract….’ In his notes on the Principia, Newton also uses the y’s to prove his mathematical hypotheses. Overall, Newton made discoveries in various fields of science, and his contributions to the subject has been important for centuries, proving he…

    • 788 Words
    • 4 Pages
    Improved Essays
  • Improved Essays

    Donald Duck Analysis

    • 772 Words
    • 4 Pages

    The journey Donald Duck is taking in this short film tells the origins of mathematics and how mathematics can be found in everyday life. The film shows you how it all began by taking Donald back to ancient Greece. Pythagoras is the father of mathematics and he showed us how everything does in fact include math in it. Pythagoras learned that from taking one thread of string and dividing it in half then dividing that in half and so on, he learned they each have their own tones due to each length of the strings. He realized music can be produced by mathematics and developed the music scale.…

    • 772 Words
    • 4 Pages
    Improved Essays
  • Improved Essays

    For example one of Newton’s most famous publications the “Philosophie, Natrulis, Principa Mathematica” or the “Principa.” In which he highlights the concepts of universal gravitation and the laws of motion; both of which are still used in the forefront of science today. Furthermore Isaac closely followed the work of Isaac Barrow, learning his theories and methods. Newton furthered his work in binomial theorem, which he extended to include fractional and negative exponents. He succeeded in this enlarging the applicability of binomial theorem by applying the algebra of finite values in an analysis of infinite series. Also he was willing to view infinite series as approximate devices and as alternative forms of expressing a term.…

    • 464 Words
    • 2 Pages
    Improved Essays
  • Improved Essays

    He had made Nuclear power as in like working with math and the facts for it. Like about the mass of a nucleus and the amount of weight and everything to do with that and he had many inventions also this is how it is contributed to mathematics While Einstein was remembered for his contributions to physics he also made contributions in mathematics. He paid several equations to calculus and geometry ten of which are called the Einstein Field Equations. He first published these equations in 1915. One of these equations proves how stress energy causes turn of space time.…

    • 1228 Words
    • 5 Pages
    Improved Essays
  • Improved Essays

    Using Triangle Inequality

    • 1571 Words
    • 7 Pages

    Theorem 3.2. Suppose that u ∈ Cp+1(Ω); s = 1; 2;∞;Ω ⊂ R2 and that uh ∈ Wh on a quasi-uniform family {Qh} of meshes on Ω into quadrilaterals. Then a necessary condition for ∥u − uh∥ Ls(Ω) = O(hp+1) is for uh to be Wp+1 s smooth. In particular, for ∥u − uh∥ L∞(Ω) = O(hp+1) a necessary condition is that all the kth partial derivatives at xi ∈ T satisfy (41) @ (u − uh)(xi) = O(hp+1−k); | | = k; 0 ≤ k ≤ p: In other words, we have a simultaneous approximation result. Here all smoothness refers to interior smoothness and {xi} is any collection of points, one from each element.…

    • 1571 Words
    • 7 Pages
    Improved Essays
  • Improved Essays

    Quadratic Functions Essay

    • 1468 Words
    • 6 Pages

    Next, the constant number is to be moved to the other side of the equation. Another important rule in math is that whenever a number is moved, the sign changes. In this case the constant “c” is positive, and later changed as “-c” when moved to the right side of the equation. As a result, it should look like the…

    • 1468 Words
    • 6 Pages
    Improved Essays