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Application of Calculus

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Application of Calculus

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Application of Calculus

  1. 1. • MADE BY- NIKUNJ, SATISH, RAJ, HARDIK • BRANCH-MECHANICAL • SEMESTER-1st • ENROLLMENT NO. -130150119069 130150119109 130150119002 130150119057
  2. 2. • Calculus is the mathematical study of change ,in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally considered to have been founded in the 17th century by Isaac Newton and Gottfried Leibniz, today calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.
  3. 3. • Calculus is a major part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or " infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus.
  4. 4. The logarithmic spiral of the Nautilus shellis a classical image used to depict the growth and change related to calculus
  5. 5. SIR ISSAC NEWTON developed the use of calculus in laws of motion and gravitation
  6. 6. • The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term.[9] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1670.
  7. 7. • The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton.He is now regarded as an independent inventor of and contributor to calculus.
  8. 8. Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
  9. 9. • The fundamental theorem of calculus states that differentiation and integration are inverse operations. 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.
  10. 10. • The fundamental theorem of calculus states: If a function f is continuous on the interval [a, b ] and if F is a function whose derivative is f on the interval (a, b ), then • Furthermore, for every x in the interval (a,b) •
  11. 11. • This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
  12. 12. • Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators. • The indefinite integral is the anti derivative, the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) • The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum.
  13. 13. A motivating example is the distances traveled in a given time. DISTANCE =SPEED.TIME
  14. 14. Finding the Average of a Function A function can represent many things. One example is the path of an airplane. Using calculus you can calculate its average cruising altitude, velocity and acceleration. Same goes for a car, bus, or anything else that moves along a path. Now what would you do without a speedometer on your car?
  15. 15. • 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.
  16. 16. • Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are interrelated 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:
  17. 17. • historically stated it expressly uses the term "rate of change" which refers to the derivative saying The rate of 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 involves 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.
  18. 18. • 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.
  19. 19. • In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. From the decay laws for a particular drug's elimination from the body, it is used to derive dosing laws. In nuclear medicine, it is used to build models of radiation transport in targeted tumor therapies. • In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.
  20. 20. CALCULUS is that part of mathematics which has unique importance in almost every field of education. The discovery of calculus is considered as one of the major success in the field of mathematics. RR

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