1 Assignment of Math - What the uses of calculus in software engineering? PDF

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What the uses of calculus in software engineering?...

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Subject:

Calculus and Analytical Geometry

Topic:

The uses of Calculus in Software Engineering

Submitted to:

Ms. Humaira Muslim

Submitted By:

Mohsan Yaseen

Roll No: Dated:

351151 24-9-2019

Section:

W

Calculus:Calculus is a branch of mathematics which helps us understand changes between values that are related by a function. For example, if you had one formula telling how much money you got every day, calculus would help you understand related formulas like how much money you have in total, and whether you are getting more money or less than you used to. All these formulas are functions of time, and so that is one way to think of calculus — studying functions of time. There are two different types of calculus. Differential calculus divides things into small (different) pieces and tells us how they change from one moment to the next, while integral calculus joins (integrates) the small pieces together and tells us how much of something is made, overall, by a series of changes. Calculus is used in many different areas such as physics, astronomy, biology, engineering, economics, medicine and sociology.

History:Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India.

 Ancient:The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (13th dynasty, c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components. From the age of Greek mathematics, Eudoxus (c. 408–355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287– 212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle.[7] In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method that would later be called Cavalieri's principle to find the volume of a sphere.

 Medieval:In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 CE) 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. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of

Astronomy and Mathematics thereby stated components of calculus. A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations. However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".

Uses of calculus in Software Engineering field Calculus is really only ever used in:-

I.

Computer graphics:-

Knowledge of calculus is an important part of advanced computer graphics. If you plan to do research in graphics, I strongly recommend getting a basic grounding in calculus. This is true not just because it is a collection of tools that are often used in the field, but also because many researchers describe their problems and solutions in the language of calculus. In addition, a number of important mathematical areas require calculus as a prerequisite. This is the one area in mathematics in addition to basic algebra that can open the most doors for you in computer graphics in terms of your future mathematical understanding. For example: integral calculus is used to simulate the bouncing of light, subdivision and geometry are used in creating smooth surface, and harmonic coordinates are used in making characters move realistically. Calculus generally plays a role in contributing to two main functions that are used in animation: rendering objects and powering physics engines.

Rendering Objects:Rendering is the process of turning a 3D model into a 2D image that is seen in movies. In order to make the 2D images look 3D, animation requires the simulation of light bouncing throughout a scene until it reaches the camera or viewpoint.

II.

Language design:-

I n programming language and software engineering, the main mathematical tool is de facto some form of predicate logic. Yet, as elsewhere in applied mathematics, it is used mostly far below its potential, due to its traditional formulation as just a topic in logic instead of a calculus for everyday practical use. The proposed alternative combines a language of utmost simplicity (four constructs only) that is devoid of the defects of common mathematical conventions, with a set of convenient calculation rules that is sufficiently comprehensive to make it practical for everyday use in most (if not all) domains of interest. Its main elements are a functional predicate calculus and concrete generic functional. The first supports formal calculation with quantifiers with the same fluency as with derivatives and integrals in classical applied mathematics and engineering. The second achieves the same for calculating with functional, including smooth transition between point wise and point-free expression. The extensive collection of examples pertains mainly to software specification, language semantics and its mathematical basis, program calculation etc., but occasionally shows wider applicability throughout applied mathematics and

engineering. Often it illustrates how formal reasoning guided by the shape of the expressions is an instrument for discovery and expanding intuition, or highlights design opportunities in declarative and (functional) programming languages.

III.

Automation:-

Dealing with fuzzy logic so you smoothly transition states in increments, rather than trying to fully stop/fully start a conveyor belt full of milk bottles; look up “PID Controller”.

IV.

Data scientist:-

Although at high levels there are some data scientists who need deep mathematical skill, at a beginning level. You a need solid understanding of data analysis.

V.

regular expressions:-

Calculus is the language of engineers, scientists, and economists. The work of these professionals has a huge impact on our daily life – from your microwaves, cell phones, TV, and car to medicine, economy, and national defense. It is used to create mathematical models in order to arrive into an optimal solution. For example, in physics, calculus is used in a lot of its concepts. Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics.

VI.

AI and Machine learning:-

The first part, as Jegor van Opdorp said, just more generally machine learning uses differential calculus in order to find optimal solutions to problems. Some examples are finding the maximal margin in SVM and finding the maximum in the EM algorithm. Machine learning uses integral calculus in determining the probability of events, for example in finding the posterior in a Bayesian model, or bounding the error in a sequential decision according to the Neyman-Pearson lemma.

VII.

Signal analysis:-

Fourier transforms are used to examine oscillating functions, such as those which show up in communicating between cluster nodes in large clusters, or site monitoring (or anywhere there is network latency between communicating computers, actually)

VIII.

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 integral

IX.

Financial modeling:-

Calculus is essentially a way of identifying rates of change and allow optimization. In advanced finance (financial engineering) and complex optimization/boundary problems variants of calculus are used - stochastic calculus, ordinary differential equations and partial differential equations.

X.

Software testing:Prediction of running time vs. actual running time

XI.

Economy:-

Calculus, by determining marginal revenues and costs, can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. As long as marginal revenue exceeds marginal cost, the firm increases its profits.

XII.

Robotics:Partial differentials and gradients for interacting with the real world.

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Motion: mobile robots drive around, with a velocity (the derivative of position) and an acceleration (the derivative of velocity). Robotic arms move, each joint has an angular velocity and acceleration. This is probably your most basic application of calculus to robotics. Controls: after calculus, many people will take differential equations. A differential equal is often used as a description of how some type of system will change over time. It says "if the system is in some state right now, then this is how much the state will change by." For example, if you are on a roller coaster going down a hill. Your current "state" might be described by how fast you are moving (velocity) and how far down the hill you are. At that point in time, you can use a differential equation to describe how much your speed and position will change over the next instant of time. Let's say now that this roller coaster has brakes, and you don't want to move too fast, you need to control your speed. So you can look at your current state and use that to calculate how much to apply the brakes to make sure you stay at a constant speed. Perhaps you want to move as some speed back up the hill? Well you can calculate how much power to apply to make it happen. You can actually do this with a ton of different systems: how do you think a Segway stays upright? How does your cruise control make sure you keep going 60 on the highway? The answer is controls, and it's all based on calculus (and a ton of linear/matrix algebra). This is also how you make a robot move where you want it to, or pick up something where you want it to. Computer Vision: This one may seem a bit odd at first, applying calculus to images, but hear me out. A digital image is a giant array of pixels, and each pixel has some numbers representing the color for that pixel. To keep things simple, let's assume you have a black and white image, so each pixel value is just the brightness of that cell. Now, what if you want to find edges of objects in the image. Well, what defines an edge? A change in color, or in our case intensity (since we aren't using color). How do you calculate this change? Calculus! By subtracting the value of each pixel by the value of the pixel to the right, you can get a sense of how much change occurred between those two pixels. If there was enough change, (i.e. the derivative is high enough), then you predict that there is an edge located at that pixel. This isn't quite the same derivative that you would perform in a calculus class, but it is a derivative....