Calculus | Definition, Formulas & Applications

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Calculus is a branch of mathematics that studies continuous change. It encompasses two fundamental concepts: differential calculus and integral calculus. Calculus deals with rate of change in a specific time frame.

History of Calculus

Ancient Egytians and Ancient Greeks uses calculus for calculating volume and area with the help of integration in 1820 BC.

After the ancient age the medieval age came where two major persons gave beautiful theories about calculus who were

  1. Ibn-al-Haytham in 11th century, He was Arab Mathematician and physicist
  2. Bhaskara II was an indian mathematician and astronomer who also provided some theories about calculus

From 965 – c to 1040 AD, Hasan Ibn al Haytham, derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.

The history of calculus can be traced back to ancient civilizations, but it was formalized in the 17th century. Key milestones include:

  • Ancient Contributions: The method of exhaustion used by Greek mathematicians like Archimedes laid foundations for integral calculus.
  • Isaac Newton and Gottfried Wilhelm Leibniz: Both independently developed the fundamental concepts of calculus in the late 17th century. Newton focused on rates of change, while Leibniz introduced integral notation.
  • Expansion: In the 18th and 19th centuries, mathematicians such as Euler and Cauchy expanded calculus further, introducing more rigorous definitions and concepts.
  • Modern Developments: Calculus has evolved with the introduction of limit definitions, epsilon-delta definitions, and the development of real analysis.

Calculus remains an essential part of mathematics, influencing various fields such as physics, engineering, economics, and statistics.

Differential Calculus

Differential calculus focuses on the concept of the derivative, which represents the rate of change of a quantity. It involves the study of how functions change as their inputs change. Key concepts include:

  • Derivatives: A derivative measures how a function’s output changes as its input changes.
  • Applications: Used in finding slopes of curves, optimization problems, and in physics to determine velocity and acceleration.
{\displaystyle m={\frac {\text{rise}}{\text{run}}}={\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {\Delta y}{\Delta x}}.}
{\displaystyle m={\frac {f(a+h)-f(a)}{(a+h)-a}}={\frac {f(a+h)-f(a)}{h}}.}
{\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.}

Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x2 be the squaring function.

{\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-3^{2} \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6\end{aligned}}}

Leibniz notation

A common notation introduced by Leibniz.

{\displaystyle {\begin{aligned}y&=x^{2}\\{\frac {dy}{dx}}&=2x.\end{aligned}}}

In an approach based on limits, the symbol ⁠dy/dx⁠ is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above .  Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of ⁠d/dx⁠ as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:

{\displaystyle {\frac {d}{dx}}(x^{2})=2x.}
{\displaystyle {\begin{aligned}g(t)&=t^{2}+2t+4\\{d \over dt}g(t)&=2t+2\end{aligned}}}

Integral Calculus

Integral calculus is concerned with the concept of the integral, which represents the accumulation of quantities. It involves calculating areas under curves and the total accumulation of change. Key concepts include:

  • Integrals: An integral computes the accumulation of a quantity, often represented as the area under a curve.
  • Fundamental Theorem of Calculus: This theorem links differential and integral calculus, stating that differentiation and integration are inverse processes.
{\displaystyle \int _{a}^{b}f(x)\,dx.}
{\displaystyle \int f(x)\,dx.}
{\displaystyle \int 2x\,dx=x^{2}+C.}

Fundamental Theorem

The states that differentiation and integration are inverse operations.[49]: 290  More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.

Reference from wikipedia for application of calculus

Calculus is used in every branch of the physical sciences,[53]: 1  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.[54] 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.[55] 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.[56]: 37  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. 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 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 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.[57]

Maxwell’s theory of electromagnetism and Einstein‘s theory of general relativity are also expressed in the language of differential calculus.[58][59]: 52–55  Chemistry also uses calculus in determining reaction rates[60]: 599  and in studying radioactive decay.[60]: 814  In biology, population dynamics starts with reproduction and death rates to model population changes.[61][62]: 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.[63] 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.[64] Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a cancerous tumor grows.[65]

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.[66]: 387 

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