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Diaz, Jeane Paul T. BSIT – III Math 6: Calculus Significant Rules of Calculus in Information Technology Calculus is us

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Diaz, Jeane Paul T. BSIT – III Math 6: Calculus

Significant Rules of Calculus in Information Technology

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. 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. 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's used to derive dosing laws. In nuclear medicine, it's 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. Calculus is also used to find approximate solutions to equations; in practice it's 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. Basic calculus comes up in machine learning and signal processing, I've never had a need for any of the differential equations and fancy integrals that come up in later calculus classes. Sometimes programmers are forced to deal with physics problems, in writing simulations, which involves a lot of math, but doing that properly effectively requires a second degree with most of the math attributable to the other one. For games programming, you need to be very comfortable with vectors, because math with vectors makes up the underpinnings of graphics and computational geometry, which involve fairly hardcore math. Linear algebra comes up more often than you'd think. A lot of common problems are solved by reducing them to sparse matrix operations, because matrix operations are the most heavily optimized algorithms in existence. It's simple enough that you can mostly pick it up as-needed, but learning linear algebra

early will save you some pain later. Everyone should learn statistics, for the same reasons you should learn writing and civics. Discrete math is usually used as a catch-all for the entire math which someone decided CS majors ought to have, but which didn't fit anywhere else; some of it is useful, some isn't. Prime numbers matter to cryptography, but nowhere else. Pay attention to the graph theory and graph algorithms, because you will need them. Pick up the concepts of automata theory, but don't ever try to build an automaton as part of a real project. If you haven't tried writing a game yet, do it, with a group that contains at least one other programmer and at least one dedicated artist. Games programming tends to lead off into areas you don't expect, some of which are theory-heavy. You might not find it as fun as you expected; and remember, the job market for game developers is already oversaturated.

Following are some areas of computer science where calculus/analysis is applicable. 1. 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. 2. Design and analysis of algorithms. The behavior of a combinatorial algorithm on very large instances is often most easily analyzed using calculus. This is especially true for randomized algorithms; modern probability theory is heavily analytic. In the other direction, sometimes one can design an algorithm for a discrete problem by considering a continuous analogue, using calculus to solve the continuous problem, and then discrediting to obtain an algorithm for the original problem. The simplest example of this might be finding an approximate root of a polynomial equation; using calculus, one can formulate Newton's method, and then discrete it. 3. 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.