Editing Calculus (section)

== Applications ==
[[File: NautilusCutawayLogarithmicSpiral.jpg|thumb|right|The [[logarithmic spiral]] of the [[Nautilus|nautilus shell]] is a classical image used to depict the growth and change related to calculus.]]
Calculus is used in every branch of the physical sciences,<ref>{{Cite book |last=Baron |first=Margaret E. |title=The origins of the infinitesimal calculus |date=1969 |isbn=978-1-483-28092-9 |location=Oxford |publisher=Pergamon Press |oclc=892067655 |author-link=Margaret Baron}}</ref>{{Rp|page=1}} [[actuarial science]], [[computer science]], [[statistics]], [[engineering]], [[economics]], [[business]], [[medicine]], [[demography]], and in other fields wherever a problem can be [[mathematical model|mathematically modeled]] and an [[optimization (mathematics)|optimal]] solution is desired.<ref>{{cite news |last1=Kayaspor |first1=Ali |date=28 August 2022 |title=The Beautiful Applications of Calculus in Real Life |url=https://ali.medium.com/the-beautiful-applications-of-calculus-in-real-life-81331dc1bc5a |access-date=26 September 2022 |work=Medium |archive-date=26 September 2022 |archive-url=https://web.archive.org/web/20220926011601/https://ali.medium.com/the-beautiful-applications-of-calculus-in-real-life-81331dc1bc5a |url-status=live }}</ref> 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.<ref>{{Cite book |last=Hu |first=Zhiying |title=2021 2nd Asia-Pacific Conference on Image Processing, Electronics, and Computers |chapter=The Application and Value of Calculus in Daily Life |date=14 April 2021 |series=Ipec2021 |location=Dalian China |publisher=ACM |pages=562–564 |isbn=978-1-4503-8981-5 |s2cid=233384462 |doi=10.1145/3452446.3452583}}</ref> 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 [[expectation value]] of a continuous random variable given a [[probability density function]].<ref>{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=[[Cambridge University Press]] |isbn=978-0-521-87342-0 |oclc=860391091}}</ref>{{Rp|37}} In [[analytic geometry]], the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, [[Concave function|concavity]] and [[inflection points]]. 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.

[[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, and the [[potential energy|potential energies]] due to gravitational and electromagnetic forces can all be found by the use of calculus. An example of the use of calculus in mechanics is [[Newton's laws of motion|Newton's second law of motion]], which states that the derivative of an object's [[momentum]] concerning time equals the net [[force]] upon it. Alternatively, Newton's second law can be expressed by saying that the net force equals the object's mass times its [[acceleration]], which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.<ref>{{Cite book|first=Elizabeth|last=Garber|title=The language of physics: the calculus and the development of theoretical physics in Europe, 1750–1914|date=2001|publisher=Springer Science+Business Media|isbn=978-1-4612-7272-4 |oclc=921230825}}</ref>

Maxwell's theory of [[electromagnetism]] and [[Albert Einstein|Einstein]]'s theory of [[general relativity]] are also expressed in the language of differential calculus.<ref>{{Cite journal|last=Hall|first=Graham|date=2008|title=Maxwell's Electromagnetic Theory and Special Relativity|journal=Philosophical Transactions: Mathematical, Physical and Engineering Sciences|volume=366|issue=1871 |pages=1849–1860|doi=10.1098/rsta.2007.2192|jstor=25190792|pmid=18218598 |bibcode=2008RSPTA.366.1849H|s2cid=502776|issn=1364-503X}}</ref><ref>{{Cite book |last=Gbur|first=Greg|title=Mathematical Methods for Optical Physics and Engineering|date=2011 |publisher=Cambridge University Press |isbn=978-0-511-91510-9|location=Cambridge|oclc=704518582|author-link=Greg Gbur}}</ref>{{Rp|pages=52–55}} Chemistry also uses calculus in determining reaction rates<ref name=":3">{{Cite book|last1=Atkins|first1=Peter W. |title=Chemical principles: the quest for insight|last2=Jones|first2=Loretta|date=2010|publisher=W.H. Freeman|isbn=978-1-4292-1955-6|edition=5th|location=New York |oclc=501943698}}</ref>{{Rp|page=599}} and in studying radioactive decay.<ref name=":3" />{{Rp|page=814}} In biology, population dynamics starts with reproduction and death rates to model population changes.<ref>{{Cite book|last=Murray|first=J. D. |title=Mathematical biology. I, Introduction|date=2002 |publisher=Springer|isbn=0-387-22437-8 |edition=3rd|location=New York |oclc=53165394}}</ref><ref>{{Cite book|last=Neuhauser|first=Claudia|title=Calculus for biology and medicine|date=2011 |publisher=Prentice Hall|isbn=978-0-321-64468-8|edition=3rd|location=Boston|oclc=426065941|author-link=Claudia Neuhauser}}</ref>{{Rp|page=631}}

[[Green's theorem]], which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a [[planimeter]], which is used to calculate the area of a flat surface on a drawing.<ref>{{Cite journal |first=R. W. |last=Gatterdam |title=The planimeter as an example of Green's theorem |journal=[[The American Mathematical Monthly]] |volume=88 |year=1981 |issue=9 |pages=701–704 |doi= 10.2307/2320679|jstor=2320679 }}</ref> For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.

In the realm of medicine, calculus can be used to find the optimal branching angle of a [[blood vessel]] to maximize flow.<ref>{{Cite journal|last=Adam|first=John A.|date=June 2011|title=Blood Vessel Branching: Beyond the Standard Calculus Problem |journal=[[Mathematics Magazine]] |volume=84|issue=3|pages=196–207 |doi=10.4169/math.mag.84.3.196|s2cid=8259705|issn=0025-570X}}</ref> Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a [[cancer]]ous tumor grows.<ref>{{cite journal |url=https://archive.siam.org/pdf/news/203.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://archive.siam.org/pdf/news/203.pdf |archive-date=9 October 2022 |url-status=live |title=Mathematical Modeling and Cancer |journal=[[SIAM News]] |date=2004 |volume=37 |number=1 |first=Dana |last=Mackenzie}}</ref>

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both [[marginal cost]] and [[marginal revenue]].<ref>{{Cite book|last=Perloff|first=Jeffrey M.|title=Microeconomics: Theory and Applications with Calculus |date=2018|isbn=978-1-292-15446-6|edition=4th global|location=Harlow |publisher=Pearson|oclc=1064041906}}</ref>{{Rp|page=387}}