Find the perimeter and area of the shaded sections of these shapes. (i) [7] (ii) [7] Total 60 marks Geometry and trigonometry Solutions to Topic assessment 1. The curve is a circle, centre O and radius 2. [2] 2. Substituting [pic] into [pic] gives [pic] Since the equation has a repeated root, the line meets the circle just once, and so the line is a tangent to the circle.
For Newton the calculus was geometrical while Leibniz took it towards analysis. According to "Http://mathforum.org/dr.math/faq/analysis_hyperreals.html" (1994-2015), the two founders of infinitesimal calculus results were the same, their motivations and interpretations were different. Leibniz established his calculus based on differential quantities, ratios (derivatives), and infinite sums (integrals). Newton, used notions of speed and motion, very apparent by his terminology of "fluent" for functions and "fluxions" for derivatives. Both established an excellent perception of the infinitesimals and found regular rules, lemmas, and theorems, by thinking of notions such as dy/dx in the context of a change in y less than any assignable quantity divided by a change in x less than any other assignable quantity.
Simon Stevin was an influential mathematician and engineer with a broad range of interests. He offered new insights and discoveries in the development of decimal numbers and the laws of inclines, gravity, hydrostatics, and fortification. Although Stevin never earned the same lasting reputation as Galileo or Isaac Newton, his contributions to the advancement of mathematical theory are noteworthy. Stevin began publishing his writings on mathematics while still a student. In 1582 he employed a printing shop in Antwerp to produce Tables of Interest, which outlined the rules for computing interest and provided tables for understanding discounts and annuities.
These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons
His theorem makes it possible to change the order of integration with double integrals, which is used to find area under a plane. Fubini assumed that both integrals are less then infinite spaces. Therefore Fubini’s theorem states that if the absolute value of the function is finite, the order or integration doesn’t matter and can be reversed. If we integrate first in respect to x, then in respect to y, we get the same answer as if we integrated in terms of y first, then x. Putting dydx means that we need to re-arrange the limits of integration so the inside integral has limits for y, and the outside integral has the limits for x, and vice versa.
Why? Because it seems like math is one of the sciences having the most authoritativeness. 2+2 = 4 is not an absolute statement, but rather coherent, as an argument would be true if it was logically put together. An example of this in mathematics could be geometry, as this area uses the assets of space. In geometry the famous Greek mathematician Euclid has made a model of reasoning, with its three key elements: axioms, deductive reasoning and theorems.
As a result of this complicated formula, quadratic equation has made significant breakthroughs in many areas of mathematics and science (Quadratic Equation, 2012). The quadratic equation is a formula that is derived from solving equations that are in the form quadratic (Origins, Derivation, & Applications, 2012). According to Origins, Derivation, & Applications (2012), “A quadratic is an equation in which the degree, or highest exponent, is a square. The degree also describes the number of possible solutions to the equation; therefore, the number of possible solutions for a quadratic is two” (p. 1). An easy way to remember the quadratic equation formula is to sing the formula to the tune of song Pop Goes the Weasel.
The History of the Calculus The Calculus, being a difficult subject requires much more than the intuition and genius of one man. It took the work and ideas of many great men to establish the advanced concepts now known as calculus. The history of the Calculus can be traced back to c. 1820 BC to the Egyptian Moscow papyrus, in which an Egyptian successfully calculated the volume of a pyramidal frustum. [1][2] Calculating volumes and areas, the basic function of integral calculus, can be traced back from the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus. [3] The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle.
Other examples that they recorded included $13^2=5^2+12^2$ and even $8161^2=4961^2+6480^2$. One of the great intellectual masterpieces of the ancient Greek world was Diophantus' {\sl Arithmetic}. This work, available in Latin translation in the seveteenth century, was an important inspiration for the scientific renaissance of that period, read by Fermat, Descartes, Newton and others. Fermat, a jurist from Toulouse, studied mathematics as a hobby. He didn't formally publish his work but rather disseminated his ideas in letters, challenging
The velocity can be obtained by finding the slope of the graph of position as a function of time. The acceleration can be obtained by finding the slope of the graph of velocity as a function of time. The critical concepts are contained in the equations for motion with constant acceleration in one dimension, as follows: x=x0+vxot+1/2axt2 Equation 1 vx=vx0+axt Equation 2 In these equations, x is the position at time t andx0 is the position at time t=0 of the object; vxis the velocity of the object along the direction of motion, x, at time t, and is the velocity of the object along the direction of motion, x, at time t=0 ; and ax is the acceleration of the object along the direction of motion, x. Uniformly accelerated linear motion is all around us. Architects often consider the safety of the slides by simulating and calculating the acceleration of a child slides down.