Table of Contents Numerical Integration 2 Trapezoidal Rule 2 Simpson’s Rule 3 Roots of Equation: 4 Fixed‐Point Iteration 4 Newton‐Raphson Method 4 Systems of Linear Equations 4 LU Decomposition 4 Gauss‐Seidel 4 References: 4 Numerical Integration Numerical integration consist of a wide variety of different method for calculating the area under the curve. Some of the ones that I will cover in this portfolio are the Trapezoidal Rule and the Simpson 1/3 Rule. I will explain how some of these algorithms are used in relation to the numerical integration. Trapezoidal Rule The trapezoidal rule is used to approximate the area under a curve. The approximation uses the area of the trapezoid,h/2(p+q), to find area where the …show more content…
The fact that an infinitesimal number of trapezoids can give you the area under any curve it makes it an important concept. Understanding the Trapezoidal Rule will make understanding integration much easier. The Trapezoidal Rule has many applications in real word problem solving. It is used to weather forecasting and determining how much rainfall is going to fall in a certain part of town. It is a rough estimation of the amount of rainfall. Simpson’s Rule The Simpson Rule is used to approximate the area under a curve. This time instead of using trapezoids to approximate the area under the curve, we use parabolas. The parabolas will help us get a more precise approximation of the area under the curve because we will not have the void space that was caused by trapezoid. The Trapezoidal Rule was the approximated using a first order polynomial were as the Simpson Rule is approximated using a second order polynomial. The formal for the Simpson’s Rule would go as …show more content…
The Newton-Raphson method uses the same principles of the Fixed-Point Iteration with respects that it starts with an initial approximation and then the sequence is generated through iterations from the initial base point. The general form of the function looks as follows: g(x) =x-f(x)/(f^' (x) ) , where the sequence is generated by p_n =g(p_(n-1)) This function will replace the f by the tangent that was estimate by pn value. The function finds the upper and the lower limit and interpolates where the point in between will fall. The root of the equation will be given by this number. Figure 6 shows how the interpolation of the middle value is found by means of the tangent line. The tangent line, in red, is fixed at a point and uses the upper and lower limits to find a single iteration. The Newton-Raphson Method has many applications in approximating the root. When comparing the Newton-Raphson Method with the Fixed-Point Iteration, the best to use would be the Newton-Raphson Method because it will be able to iterate through any function a lot faster. Using the same function, e-x, the Newton-Raphson Method would be able to find the root within 8 iterations while the Fixed-Point Iteration would take 20 iterations to get to same
4. An engine performs 5000 Joules of work in 20 seconds. What is its power output in kilowatts and in
the length of the slope can be used to calculate the speed of the car
Areas of the following shapes were investigated: square, rectangle, kite, parallelogram, equilateral triangle, scalene triangle, isosceles triangle, right-angled triangle, rhombus, pentagon, hexagon, heptagon and octagon. Results The results of the analysis are shown in Table 1 and Fig 1. Table 1 showing the areas for the different shapes formed by using the
If I am to use a square of side length 10cm, then I can calculate the
The argument in this paper that even though the onus of the discovery of calculus lies with Isaac Newton, the credit goes to Leibniz for the simple fact that he was the one who published his works first. Appending to this is the fact that the calculus wars that ensue was merely and egotistic battle between humans succumbing to their bare primal instincts. To commence, a brief historical explanation must be given about both individuals prior to stating their cases.
4. Compute successive value of recursively using the computed values of (from step 2), the given initial estimate , and the input data .
As it is an extension of another concept, this paper aims first to introduce Pascal’s Triangle and then to apply its properties in deriving each layer of Pascal’s Pyramid. From there, a trinomial principle or theorem is to be formed that will govern the coefficients. For its use in trinomial expansion, the author also aims to compare using any formulas or sho...
Newton-Raphson method is of use when it comes to approximating the root or roots of an equation.
Many years ago humans discovered that with the use of mathematical calculations many things can be calculated in the world and even the universe. Mathematics consists of many different operations. The most important that is used by mathematicians, scientists and engineers is the derivative. Derivatives can help make calculations of anything with respect to another event or thing. Derivatives are mostly common when used with respect to time. This is a very important tool in this revolutionary world. With derivatives we can calculate the rate of change of anything with respect to time. This way we can have a sort of knowledge of upcoming events, and the different behaviors events can present. For example the population growth can be estimated applying derivatives. Not only population growth, but for example when dealing with plagues there can be certain control. An other example can be with diseases, taking all this events together a conclusion can be made.
Differential calculus is a subfield of Calculus that focuses on derivates, which are used to describe rates of change that are not constants. The term ‘differential’ comes from the process known as differentiation, which is the process of finding the derivative of a curve. Differential calculus is a major topic covered in calculus. According to Interactive Mathematics, “We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).” Not only are derivatives used to determine how to maximize or minimize functions, but they are also used in determining how two related variables are changing over time in relation to each other. Eight different differential rules were established in order to assist with finding the derivative of a function. Those rules include chain rule, the differentiation of the sum and difference of equations, the constant rule, the product rule, the quotient rule, and more. In addition to these differential rules, optimization is an application of differential calculus used today to effectively help with efficiency. Also, partial differentiation and implicit differentiation are subgroups of differential calculus that allow derivatives to be taken to more challenging and difficult formulas. The mean value theorem is applied in differential calculus. This rule basically states that there is at least one tangent line that produces the same slope as the slope made by the endpoints found on a closed interval. Differential calculus began to develop due to Sir Isaac Newton’s biggest problem: navigation at sea. Shipwrecks were frequent all due to the captain being unaware of how the Earth, planets, and stars mov...
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
Historically speaking, ancient inventors of Greek origin, mathematicians such as Archimedes of Syracuse, and Antiphon the Sophist, were the first to discover the basic elements that translated into what we now understand and have formed into the mathematical branch called calculus. Archimedes used infinite sequences of triangular areas to calculate the area of a parabolic segment, as an example of summation of an infinite series. He also used the Method of Exhaustion, invented by Antiphon, to approximate the area of a circle, as an example of early integration.
...hat it could also be used to show measurements, be used in science, and that it could be used to ride in things such as hot air balloons. It could be used to find measurements because you could be trying to fill up the balloon just to see how tall or how wide it can get, and then measure it. It could also be used in science for that reason, and if you were testing the acidity in something, and having it fill up with something other than air or helium.
Mathematics, the language of the universe, is one of the largest fields of study in the world today. With the roots of the math tree beginning in simple mathematics such as, one digit plus one digit, and one digit minus one digit, the tree of mathematics comes together in the more complex field of algebra to form the true base of calculations as the trunk. As we get higher, branches begin to form creating more specialized forms of numerical comprehension and schools of mathematical thought. Some examples of these are the applications into chemistry, economics and computers. Further up the tree we see the crown beginning to form with the introduction of calculus based organization. Calculus, a theoretical school of mathematical thought, had its creation in the middle ages with Newton. The main use of calculus is its application in advanced physics. Mathematics is everywhere because that is where we put them, everywhere. We, humans, represent everything with numbers, which therefore means that we impose mathematics on to the universe.
In my previous studies, I have covered all the four branches of mathematics syllabus and this has made me to develop a strong interest in pure mathematics and most importantly, a very strong interest in calculus.