HISTORY AND UTILITY VALUE OF CALCULUS

1. ❖MATHS
INNOVATIVE TEST
❑ NAME:-VICKY KUMAR YADAV
❑ ROLL NO:-4175
❑ STD:-XII SCI-K
❑ SUBJECT:-MATHEMATICS
❑ SCHOOL:-ST. XAVIER’S HSS
❑ ACADEMIC YEAR:-2021-2022

2. HISTORY
AND
UTILITY
VALUE OF
CALCULUS

3. INDEX
SR
NO
TITLE PAGE
NO
1 INTRODUCTION 1
2 HISTORY OF CALCULUS 2-3
3 UTILITY VALUE OF CALCULUS 4-5
4 REFERENCES 6

4.  Modern calculus was developed in 17th-century
Europe by Isaac Newton and Gottfried Wilhelm
Leibniz (independently of each other, first
publishing around the same time) but elements of
it appeared in ancient Greece, then in China and
the Middle East, and still later again in medieval
Europe and in India.
❑INTRODUCTION

5.  Calculus, known in its early history as infinitesimal calculus, is a mathematical
discipline focused on limits, continuity, derivatives, integrals, and infinite series.
Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory
of infinitesimal calculus in the later 17th century. By the end of the 17th century,
both Leibniz and Newton claimed that the other had stolen his work, and the
Leibniz–Newton calculus controversy continued until the death of Leibniz in 1716.
 Ancient
The ancient period introduced some of the ideas that led to integral calculus, but
does not seem to have developed these ideas in a rigorous and systematic way.
Calculations of volumes and areas, one goal of integral calculus, can be found in
the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for
concrete numbers, some are only approximately true, and they are not derived by
deductive reasoning. Babylonians may have discovered the trapezoidal rule while
doing astronomical observations of Jupiter.
HISTORY OF CALCULUS

6.  Medieval
 In the Islamic Middle East, the 11th-century Arab mathematician Ibn al-Haytham
(Alhazen) derived a formula for the sum of fourth powers. He used the results to
carry out what would now be called an integration, where the formulas for the sums
of integral squares and fourth powers allowed him to calculate the volume of a
paraboloid. In the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī
discovered the derivative of cubic polynomials. His Treatise on Equations
developed concepts related to differential calculus, such as the derivative function
and the maxima and minima of curves, in order to solve cubic equations which may
not have positive solutions.
 Some ideas on calculus later appeared in Indian mathematics, at the Kerala school
of astronomy and mathematics Madhava of Sangamagrama in the 14th century, and
later mathematicians of the Kerala school, stated components of calculus such as the
Taylor series and infinite series approximations. However, they were not able to
combine many differing ideas under the two unifying themes of the derivative and
the integral, show the connection between the two, and turn calculus into the
powerful problem-solving tool we have today.
 Early Modern
 In the 17th century, European mathematicians Isaac Barrow, René Descartes, Pierre
de Fermat, Blaise Pascal, John Wallis and others discussed the idea of a derivative.
In particular, in Methodus ad disquirendam maximam et minima and in De
tangentibus linearum curvarum, Fermat developed an adequality method for
determining maxima, minima, and tangents to various curves that was closely
related to differentiation. Isaac Newton would later write that his own early ideas
about calculus came directly from "Fermat's way of drawing tangents."

7. Utility value is how the task relates to future goals. While students may not enjoy an
activity, they may value a later reward or outcome it produces (Wigfield, 1994). The
activity must be integral to their vision of their future, or it must be instrumental to
their pursuit of other goals. Because goals can play a key role in attaining later
outcomes, educators and parents should help students see beyond the immediate
activity to the long-term benefits it produces.
 Calculus is used in every branch of the physical sciences, actuarial science, computer
science, statistics, engineering, economics, business, medicine, demography, and in
other fields wherever a problem can be mathematically modeled and
an optimal solution is desired. It allows one to go from (non-constant) rates of
change to the total change or vice versa, and many times in studying a problem we
know one and are trying to find the other.
 Physics makes particular use of calculus; all concepts in classical
mechanics and electromagnetism are related through calculus. The mass of an object
of known density, the moment of inertia of objects, as well as the total energy of an
object within a conservative field can be found by the use of calculus. An example of
the use of calculus in mechanics is Newton's second law of motion: historically
stated it expressly uses the term "change of motion" which implies the derivative
saying The change of momentum of a body is equal to the resultant force acting on
the body and is in the same direction. Commonly expressed today as
Force = Mass × acceleration, it implies differential calculus because acceleration is
the time derivative of velocity or second time derivative of trajectory or spatial
position. Starting from knowing how an object is accelerating, we use calculus to
derive its path
UTILITY VALUE OF CALCULUS

8.  Calculus can be used in conjunction with other mathematical disciplines. For
example, it can be used with linear algebra to find the "best fit" linear
approximation for a set of points in a domain. Or it can be used in probability
theory to determine the probability of a continuous random variable from an
assumed density function. In analytic geometry, the study of graphs of
functions, calculus is used to find high points and low points (maxima and
minima), slope, concavity and inflection points.
 In economics, calculus allows for the determination of maximal profit by
providing a way to easily calculate both marginal cost and marginal revenue
 Calculus is also used to find approximate solutions to equations; in practice it
is the standard way to solve differential equations and do root finding in most
applications. Examples are methods such as Newton's method, fixed point
iteration, and linear approximation. For instance, spacecraft use a variation of
the Euler method to approximate curved courses within zero gravity
environments

9. https://en.wikipedia.org/wiki/Utility
https://amandakovacs.wordpress.com/2008/11/22/newton-
vs-leibniz/
https://en.wikipedia.org/wiki/Calculus
https://en.wikipedia.org/wiki/Calculus#History
REFERENCES

10. THANK YOU