Calculus: Difference between revisions

Latest revision as of 02:58, 11 December 2024

Mathematical study of continuous change

A visual representation of the secant line approaching the tangent line.

Calculus is a branch of mathematics that studies continuous change. It is a foundational part of modern mathematics education and is used in a variety of fields, including physics, engineering, economics, and biology. Calculus is divided into two main branches: differential calculus and integral calculus.

History[edit]

The development of calculus is attributed to Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. However, the ideas behind calculus can be traced back to ancient mathematicians such as Archimedes and Eudoxus of Cnidus.

Differential Calculus[edit]

Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. The derivative is a fundamental tool in calculus, used to find the slope of a curve at any given point. The process of finding a derivative is called differentiation.

Applications[edit]

Differential calculus is used to solve problems involving motion, such as finding the velocity and acceleration of an object. It is also used in optimization problems, where the goal is to find the maximum or minimum values of a function.

Integral Calculus[edit]

Integral calculus is concerned with the concept of the integral, which represents the accumulation of quantities. The integral is used to calculate areas under curves, volumes of solids, and other quantities that accumulate over a range.

Applications[edit]

Integral calculus is used in a variety of applications, including calculating the area under a curve, determining the total distance traveled by an object, and finding the center of mass of an object.

Fundamental Theorem of Calculus[edit]

The Fundamental Theorem of Calculus links differential and integral calculus. It states that differentiation and integration are inverse processes. This theorem provides a way to evaluate definite integrals without directly calculating the limit of a sum.

Notable Mathematicians[edit]

Also see[edit]

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-Calculus consists of mineralized bacterial plaque that forms on the surfaces of natural teeth and dental prostheses.
+{{Short description|Mathematical study of continuous change}}
+{{Use dmy dates|date=October 2023}}
+[[File:Parabolic segment and inscribed triangle.svg|thumb|A parabolic segment illustrating the concept of integration.]]
+[[File:Maria Gaetana Agnesi.jpg|thumb|Maria Gaetana Agnesi, an Italian mathematician known for her work in calculus.]]
+[[File:Tangent line to a curve.svg|thumb|A tangent line to a curve, illustrating the concept of derivatives.]]
+[[File: Sec2tan.gif|thumb|A visual representation of the secant line approaching the tangent line.]]
-=== Types of Calculus ===
+'''Calculus''' is a branch of [[mathematics]] that studies continuous change. It is a foundational part of modern mathematics education and is used in a variety of fields, including [[physics]], [[engineering]], [[economics]], and [[biology]]. Calculus is divided into two main branches: [[differential calculus]] and [[integral calculus]].
-==== Supragingival calculus ====
-* Supragingival calculus is located coronal to the gingival margin and therefore is visible in the oral cavity.
-* It is usually white or whitish yellow in color; hard, with a claylike consistency; and easily detached from the tooth surface. After removal, it may rapidly recur, especially in the lingual area of the mandibular incisors.
-* The color is influenced by contact with such substances as tobacco and food pigments.
-* It may localize on a single tooth or group of teeth, or it may be generalized throughout the mouth. The two most common locations for the development of supragingival calculus are the buccal surfaces of the maxillary molars and the lingual surfaces of the mandibular anterior teeth Saliva from the parotid gland flows over the facial surfaces of the upper molars via the parotid duct, whereas the submandibular duct and the lingual duct empty onto the lingual surfaces of the lower incisors from the submaxillary and sublingual glands, respectively.
-* In extreme cases, calculus may form a bridgelike structure over the interdental papilla of adjacent teeth or cover the occlusal surface of teeth that are lacking functional antagonists.
-<gallery widths="300" heights="250">
-File:MandibularAnteriorCalculus.JPG|MandibularAnteriorCalculus
-File:Calculus deposit on x-ray image.jpg|Calculus deposit on x-ray image
-</gallery>
-==== Subgingival calculus ====
-* Subgingival calculus is located below the crest of the marginal gingiva and therefore is not visible on routine clinical examination. The location and extent of subgingival calculus may be evaluated by careful tactile perception with a delicate dental instrument such as an explorer. Subgingival calculus is typically hard and dense; it frequently appears to be dark brown or greenish black in color, and it is firmly attached to the tooth surface.
-=== Composition ===
+==History==
-==== Inorganic Content ====
+The development of calculus is attributed to [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] in the late 17th century. However, the ideas behind calculus can be traced back to ancient mathematicians such as [[Archimedes]] and [[Eudoxus of Cnidus]].
-Supragingival calculus consists of inorganic(70% to 90%) and organic components. The major inorganic proportions of calculus have been reported as approximately 76% calcium phosphate, (Ca3[PO4]2); 3% calcium carbonate (CaCO3); and traces of magnesium phosphate (Mg3[PO4]2) and other metals. The percentage of inorganic constituents in calculus is similar to that of other calcified tissues of the body. The principal inorganic components have been reported as approximately 39% calcium, 19% phosphorus, 2% carbon dioxide, and 1% magnesium as well as trace amounts of sodium, zinc, strontium, bromine, copper, manganese, tungsten, gold, aluminum, silicon, iron, and fluorine.
-At least two thirds of the inorganic component is crystalline in structure. The four main crystal forms and their approximate
+==Differential Calculus==
-percentages are as follows: hydroxyapatite, 58%; magnesium whitlockite, 21%; octacalcium phosphate, 12%; and brushite, 9%.
+Differential calculus focuses on the concept of the [[derivative]], which represents the rate of change of a function. The derivative is a fundamental tool in calculus, used to find the slope of a curve at any given point. The process of finding a derivative is called [[differentiation]].
-Two or more crystal forms are typically found in a sample of calculus. Hydroxyapatite and octacalcium phosphate are detected
-most frequently (i.e., in 97% to 100% of all supragingival calculus) and constitute the bulk of the specimen. Brushite is more common in the mandibular anterior region, and magnesium whitlockite is found in the posterior areas. The incidence of the four crystal forms varies with the age of the deposit.
+===Applications===
-==== Organic Content ====
+Differential calculus is used to solve problems involving motion, such as finding the velocity and acceleration of an object. It is also used in [[optimization]] problems, where the goal is to find the maximum or minimum values of a function.
-The organic component of calculus consists of a mixture of protein–polysaccharide complexes, desquamated epithelial cells, leukocytes, and various types of microorganisms.
-Between 1.9% and 9.1% of the organic component is carbohydrate, which consists of galactose, glucose, rhamnose, mannose, glucuronic acid, galactosamine, and sometimes arabinose, galacturonic acid, and glucosamine. All of these organic components are present in salivary glycoprotein, with the exception of arabinose and rhamnose. Salivary proteins account for 5.9% to 8.2% of the organic component of calculus and include most amino acids. Lipids account for 0.2% of the organic content in the form of neutral fats, free fatty acids, cholesterol, cholesterol esters, and phospholipids.
+==Integral Calculus==
-The composition of subgingival calculus is similar to that of supragingival calculus, with some differences. It has the same
+Integral calculus is concerned with the concept of the [[integral]], which represents the accumulation of quantities. The integral is used to calculate areas under curves, volumes of solids, and other quantities that accumulate over a range.
-hydroxyapatite content, more magnesium whitlockite, and less brushite and octacalcium phosphate. The ratio of calcium to phosphate is higher subgingivally, and the sodium content increases with the depth of periodontal pockets.94 These altered compositions may be attributed to the origin of subgingival calculus being plasma, whereas supragingival calculus is partially composed of saliva constituents. Salivary proteins present in supragingival calculus are not found subgingivally.11 Dental calculus, salivary duct calculus, and calcified dental tissues are similar in inorganic composition.
-{{stub}}
+===Applications===
-{{dictionary-stub1}}
+Integral calculus is used in a variety of applications, including calculating the area under a curve, determining the total distance traveled by an object, and finding the center of mass of an object.
+==Fundamental Theorem of Calculus==
+The [[Fundamental Theorem of Calculus]] links differential and integral calculus. It states that differentiation and integration are inverse processes. This theorem provides a way to evaluate definite integrals without directly calculating the limit of a sum.
+==Notable Mathematicians==
+* [[Isaac Newton]]
+* [[Gottfried Wilhelm Leibniz]]
+* [[Maria Gaetana Agnesi]]
+==Also see==
+* [[Mathematical analysis]]
+* [[Limits (mathematics)]]
+* [[Series (mathematics)]]
+* [[Multivariable calculus]]
+* [[Vector calculus]]
+{{Calculus}}
+{{Mathematics}}
+[[Category:Calculus]]
+[[Category:Mathematical analysis]]