Application of calculus in cse
- 1. Presented by Md. Janibul Hoque ID: 143-15-4595 Dept. Computer Science and Engineering Daffodin Internationa University
- 2. What is Calculus: • 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. About Calculus: • 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. Calculus in CSE • Computer Graphics/Image Processing, and here will also be needed Analytic Geometry and Linear Algebra. In this sector students need to study some Differential Geometry (which has multivariate Calculus as a minimum prerequisite). But he'll need Calculus here even for very basic things like "Fourier Transform" or "Wavelets", for example -- these are two very fundamental tools for people working with images.
- 5. Calculus in CSE • Optimization, non-linear mostly, where multivariate Calculus is the fundamental language used to develop everything. But even linear optimization benefits from Calculus (the derivative of the objective function is absolutely important) • Probability/Statistics. These cannot be seriously studied without multivariate Calculus. • Machine Learning, which makes heavy use of Statistics (and consequently, multivariate Calculus)
- 6. Calculus in CSE • Data Mining and related subjects, which also use lots of Statistics; • Robotics, where a programmer will need to model physical movements of a robot, so he will need to know partial derivatives and gradients. • Analysis of Algorithms, where an analyzer uses the notion of limit right from the start
- 7. Calculus in CSE • Discrete Math and Combinatorics if anyone get serious enough about generating functions, he'll need to know how to integrate and derivate certain formulas. And that is useful for Analysis of Algorithms Similarly, Taylor Series and calculus can be useful in solving certain kinds of recurrence relations, which are used in algorithm analysis.
- 8. Calculus in CSE • Scientific computing. Computer algebra systems that compute integrals and derivatives directly, either symbolically or numerically, are the most blatant examples here, but in addition, any software that simulates a physical system that is based on continuous differential equations (e.g., computational fluid dynamics) necessarily involves computing derivatives and integrals. • Asymptotic enumeration. Sometimes the only way to get a handle on an enumeration problem is to form a generating function and use analytic methods to estimate its asymptotic behavior. See the book Analytic Combinatorics by Flajolet and Sedgewick
- 9. Calculus in CSE • In stochastic simulation, we are often interested in estimating the expected value of a random variable. The expected value of a continuous random variable is an integral over the real numbers. To estimate this quantity, we use the Monte Carlo method which consists of generating instances of this random variable from pseudorandom uniform variables. From these uniform variables, we can generate random variables from a given distribution by inverting the cumulative distribution function which is defined itself as an integral.
- 10. CALCULUS is that part of mathematics which has unique importance in almost every field of education Specially in CSE. The discovery of calculus is considered as one of the major success in the field of mathematics.