...Ellipse From the general equation of all conic sections, [pic] and [pic] are not equal but of the same sign. Thus, the general equation of the ellipse is [pic] or [pic] Standard Equations of Ellipse Elements of the ellipse are shown in the figure above. 1. Center (h, k). At the origin, (h, k) is (0, 0). 2. Semi-major axis = a and semi-minor axis = b. 3. Location of foci c, with respect to the center of ellipse. [pic]. 4. Length latus rectum, LR 5. Consider the right triangle F1QF2: Based on the definition of ellipse: [pic] [pic] [pic] By Pythagorean Theorem [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] You can also find the same formula for the length of latus rectum of ellipse by using the definition of eccentricity. 6. Eccentricity, e DEFINITION: Eccentricity of Conic Eccentricity is a measure of how much a conic deviate from being circular, making the eccentricity of the circle obviously equal to zero. It is the ratio of focal distance to directrix distance of the conic section. [pic] From the figure...
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...MA1200 Basic Calculus and Linear Algebra I Lecture Note 1 Coordinate Geometry and Conic Sections υ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Topic Covered • Two representations of coordinate systems: Cartesian coordinates [ሺݕ ,ݔሻcoordinates] and Polar coordinates [ሺߠ ,ݎሻ-coordinates]. • Conic Sections: Circle, Ellipse, Parabola and Hyperbola. • Classify the conic section in 2-D plane General equation of conic section Identify the conic section in 2-D plane - Useful technique: Rotation of Axes - General results φ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Representations of coordinate systems in 2-D There are two different types of coordinate systems used in locating the position of a point in 2-D. First representation: Cartesian coordinates We describe the position of a given point by considering the (directed) distance between the point and -ݔaxis and the distance between the point and -ݕaxis. ݕ 0 ܽ ܲ ൌ ሺܽ, ܾሻ ܾ ݔ Here, ܽ is called “-ݔcoordinate” of ܲ and ܾ is called “-ݕcoordinate” of ܲ. χ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections ܲଶ ൌ ሺݔଶ , ݕଶ ሻ ܲଵ ൌ ሺݔଵ , ݕଵ ሻ Given two points ܲଵ ൌ ሺݔଵ , ݕଵ ሻ and ܲଶ ൌ ሺݔଶ , ݕଶ ሻ, we learned that • the distance between ܲଵ and ܲଶ : ܲଵ ܲଶ ൌ ඥሺݔଶ െ ݔଵ ሻଶ ሺݕଶ െ ݕଵ...
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...2a. General Equation of the Ellipse From the general equation of all conic sections, A and C are not equal but of the same sign. Thus,the general equation of the ellipse is Ax2 + Cy2 + Dx + Ey + F = 0 or Standard Equations of Ellipse From the figure above, and From the definition above, Square both sides Square again both sides From triangle OV3F2 (see figure above) Thus, Divide both sides by a2b2 The above equation is the standard equation of the ellipse with center at the origin and major axis on the x-axis as shown in the figure above. Below are the four standard equations of the ellipse. The first equation is the one we derived above. Ellipse with center at the origin Ellipse with center at the origin and major axis on the x-axis. Ellipse with center at the origin and major axis on the y-axis. Ellipse with center at (h, k) Ellipse with center at (h, k) and major axis parallel to the x-axis. Ellipse with center at (h, k) and major axis parallel to the y-axis. The Hyperbola Submitted by Romel Verterra on February 21, 2011 - 1:33pm Definition Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a. General Equation From the general equation of any conic (A and C have opposite sign, and can be A > C, A = C, or A < C.) ...
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...Term Paper Mathematics NAME: BIPIN SHARMA ROLL NO: B59 SECTION: C1903 Conics Conic sections are the curves which result from the intersection of a plane with a cone. These curves were studied and revered by the ancient Greeks, and were written about extensively by both Euclid and Appolonius. They remain important today, partly for their many and diverse applications. Although to most people the word “cone” conjures up an image of a solid figure with a round base and a pointed top, to a mathematician a cone is a surface, one which is obtained in a very precise way. Imagine a vertical line, and a second line intersecting it at some angle f (phi). We will call the vertical line the axis, and the second line the generator. The angle f between them is called the vertex angle. Now imagine grasping the axis between thumb and forefinger on either side of its point of intersection with the generator, and twirling it. The generator will sweep out a surface, as shown in the diagram. It is this surface which we call a cone. Notice that a cone has an upper half and a lower half (called the nappes), and that these are joined at a single point, called the vertex. Notice also that the nappes extend indefinitely far both upwards and downwards. A cone is thus completely determined by its vertex angle. Now, in intersecting a flat plane with a cone, we have three choices, depending on the angle the plane makes to the vertical axis of the cone. First, we may...
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...Lombos Avenue, San Isidro, Parañaque City www.patts.edu.ph ------------------------------------------------- ------------------------------------------------- VISION ------------------------------------------------- To become the Centre of Excellence in aviation education ------------------------------------------------- MISSION ------------------------------------------------- a. To provide quality aviation education and ------------------------------------------------- b. To help our graduates in the labor market BASIC STUDIES EDUCATIONAL OBJECTIVES | Mission | | a | b | 1. To provide students with a good and solid foundation in mathematics, basic engineering sciences, engineering drawing, physics, general chemistry and other branches of natural sciences and to apply knowledge to aviation and other related discipline. | √ | √ | 2. To develop communicative skills in listening, speaking, reading, writing and graphics communication pertaining to technical drawing interpretation. | √ | √ | 3. To teach and train students the importance of humanistic values and respect of cultural differences through humanities and social sciences. | √ | √ | 4. To impart high ethical standards to the students through assimilation and incorporation in the learning activities. | √ | √ | 5. To infuse students with enhanced computer concepts and expertise through incorporating competent applications and disciplines. | √ | √ | 6. To acquire the total human development...
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...Research on Butterfly Theorem Butterfly Theorem is one of the most appealing problems in the classic Euclidean plane geometry. The name of Butterfly Theorem is named very straightforward that the figure of Theorem just likes a butterfly. Over the last two hundreds, there are lots of research achievements about Butterfly Theorem that arouses many different mathematicians’ interests. Until now, there are more than sixty proofs of the Butterfly Theorem, including the synthetical proof, area proof, trigonometric proof, analytic proof and so on. And based on the extension and evolution of the Butterfly Theorem, people can get various interesting and beautiful results. The definition of the Butterfly Theorem is here below: “Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.” (Bogomolny) This is the most accurate definition currently. However, Butterfly Theorem has experienced some changes and developments. The first statement of the Butterfly Theorem appeared in the early 17th century. In 1803, a Scottish mathematician, William Wallace, posed the problem of the Butterfly Theorem in the magazine The Gentlemen’s Mathematical Companion. Here is the original problem below: “If from any two points B, E, in the circumference of a circle given in magnitude and position two right lines BCA, EDA, be drawn cutting the circle in C and D, and meeting in A; and...
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...Reflection Paper: Apollonius of Perga Background: Apollonius of Perga was known as 'The Great Geometer'. Little is known of his life but his works have had a very great influence on the development of mathematics, in particular his famous book Conics introduced terms which are familiar to us today such as parabola, ellipse and hyperbola. ‘The Great Geometer’ was born in Perga, Pamphylia which today is known as Murtina, and is now in Antalya, Turkey. Perga was a centre of culture at that time and it was the place of worship of Artemis, a nature goddess. When he was a young man, Apollonius went to Alexandria where he studied under the followers of Euclid and later he taught there. Reflection: In his writings on the conic sections, Apollonius deliberately alluded those who have worked on the realm of his studies through restating his view on their ideas. These have encompassed most of his early books which were elementary elaborations on Geometry. He then evaluated them in his own perspective and explored the uncharted possibilities which the early pioneers have overlooked. In his experience, one quote came to me, the one which I have pondered on earlier this year. In Apollonius’ feat he wasn’t set off to search the ambiguity of the field unarmed. He had with him the knowledge that comes from the known world as established by the avid foregoers of knowledge. He had stepped on the shoulders of giants to see the looming picture of what lies ahead. And it has done him good...
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...____________ Date: _______________ Section 1: Logarithms and Exponential Relations Definitions to Know: * Natural Logarithm * Common Logarithm * Mathematical * Exponential Growth * Exponential Decay Question 1) Change the following from exponential form to logarithmic form (1 mark each): a) b) Question 2) Change the following from logarithmic form to exponential form (1 mark each): a) b) Question 3) Solve for WITHOUT using a calculator. Show all of your work. (Hint: Use the definition of a logarithm.) (2 marks each) a) b) c) d) Question 4) Apply the Change of Base Formula to rewrite the logarithms with the common logarithm. (1 mark each) a) b) Question 5) Solve for the variable. Show all of your work and all of your steps. (Hint: Use the properties of logarithms.) (4 marks each) a) b) c) d) Question 6) Solve for the variable. Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the common logarithm.) (4 marks each) a) b) c) Question 7) Solve for . Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the natural logarithm and the definition of a logarithm.) (4 marks each) a) b) c) Question 8) Ms. Mary bought a condo for $145 000. Assuming that the value of the condo will appreciate at most 5% a year, how much will the condo be worth in 5 years? Section 2: Conic Sections Standard forms to Know: * Parabola ...
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...9.1 Circles: 1. Write the equation in standard form for the circle with center at and radius 2. Write the equation in standard form for the circle with center at and radius 3. Graph the circle given by 4. Graph the circle given by 5. Write the equation of the circle in standard form given: 9.1 Parabolas: 6. Find the focus of the parabola 7. Find the focus of the parabola 8. Write the equation of the parabola in standard form and find the focus and directrix. 9. Write the equation of the parabola in standard form and find the focus and directrix. 10. Write the equation for the parabola with vertex and focus 11. Write the equation for the parabola with vertex and directrix 9.2 Ellipses: 12. Identify the center, vertices, & foci of the ellipse given by and graph. 13. Identify the center, vertices, & foci of the ellipse given by and graph. 14. Write the equation in standard form: 9.5 Parametric Equations: 15. Write and in rectangular form. 16. Write each pair of parametric equations in rectangular form: 17. Write and in rectangular form. 18. Write and in rectangular form. 9.6 Polar Equations: 19. Graph the following polar coordinate: 20. Graph the following polar coordinate: 21. Graph the following polar coordinate: 22. Graph the following polar coordinate: 23. Find the polar coordinate of the point 24. Find the polar coordinate of the point 25...
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...Reading Pictographs (A) Answer the questions about the pictograph. Number of School Buses at Schools in Great Line School District vvvvvvv Euclid P.S. vvvvv Central P.S. vv North Central P.S. vvvvvv M.C. Escher P.S. vvv B. Pascal P.S. v = 2 buses 1. Which school has the most buses? How many? 2. Which school has the fewest buses? How many? 3. How many buses are there all together at all the schools? 4. How many more buses does North Central P.S. have than M.C. Escher P.S.? 5. If North Central P.S. added two buses, they would have the same number of buses as what other school? Math-Drills.Com Reading Pictographs (A) Answers Answer the questions about the pictograph. Number of School Buses at Schools in Great City, USA vvvvvvv Euclid P.S. vvvvv Central P.S. vv North Central P.S. vvvvvv M.C. Escher P.S. vvv B. Pascal P.S. v = 2 buses 1. Which school has the most buses? How many? B. Pascal P.S.; 14 buses 2. Which school has the fewest buses? How many? Central P.S.; 4 buses 3. How many buses are there all together at all the schools? 46 buses 4. How many more buses does North Central P.S. have than M.C. Escher P.S.? 6 more buses 5. If North Central P.S. added two buses, they would have the same number of buses as what other school? B. Pascal P.S. Math-Drills.Com...
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...* Link Mechanism A link mechanism can be defined as a system of connecting parts/rods that move or work together when a particular action is pre-determined. * Parabola The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and equal to one. This constant is also known as the eccentricity. * Ellipse The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and less than one. This constant is also known as the eccentricity. * Hyperbola: The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and greater than one. This constant is also known as the eccentricity. * Archimedean Spiral The path traced out by a point along a rod, as the rod pivots about a fixed end. The linear movement of the point along the rod is constant with the angular movement of the rod. * Involute The path traced out by a point when an end of a plane figure is wrapped or unwrapped when held firm. * Epicycloid Tracing the path of a point as a circular disc rolls on the outside...
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...1. Since we are looking for the time it takes for the student to be able to type 55 WPM N(t) will equal 55, when we plug this into the equation and solve, we see that the it would take approximately 8 weeks for the student to type 55 WPM 55 = 157/(1+5e^-0.12t) 1+5e^-0.12t = 157/55 ≈ 2.85454 5e^-0.12t = 1.85454 e^-0.12t = 1.85454/5 = 0.370908 -0.12t = ln 0.370908 t = ln 0.370908/-0.12 = 8.265 ≈ 8 weeks 2. According to the information (bacteria in culture after 3 hours is 100 and 5 hours is 400), the bacteria appears doubling every hour. If this is the case a) the growth rate is 200% every hour . b) Following this same trend to find the initial growth rate we would divide 100 by 2, 3 times. Thus the initial number of bacteria would be 12.5 c) Since the number of bacteria of 5 hours is 400, to find the number of bacteria at 6 hours is going to be double the amount at 5 hours, 800. 3. To solve we need to find the equation of the parabola in vertex form, since we know a point of the vertex, also that is vertically symmetrical from this point. Recall that the equation we’ll use is the quadratic y = a(x-h)² + k. Since we are looking for how wide the parabola is at 10 meters(8 meters) and its vertex’s x coordinate is at 0 and its vertically symmetrical the a point that is passed through at that height(10 meters) is 4 (4,10).Substitute this information into the equation (10=a(4-0)^2) + 12 simplify -2=16a 12/16=a a = -1/8 Now substitute the equation with a=-1/8...
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... | | |The introduction of your paper should acquaint the reader with the company being analyzed and demonstrate your ability to succinctly describe the company| | |from a historical perspective. Take this opportunity to highlight key factors and past strategies, which have led the company to its present position. | | |It is important for you to understand precisely why the company has been successful (or unsuccessful) in the past. Taking time to articulate this may | | |suggest distinctive competencies that you might otherwise overlook when writing the following sections. | | | | |Conclude the “Introduction” of your paper by...
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...with an ethical issue. Deliverable: 1. Introduction. It was something I witnessed, an ethical issue which I wish to point out. It is not really similar to that of the Lab’s video. It is about an investigation into a safety incident whereby a soldier, Solider A, fractured his hand during a botched attempt to lift an armored vehicle’s engine panel. 2. Background leading to incident. A background to the incident is necessary for a complete understanding of the ethical issue. Corporal A belonged to Platoon 1 Section 1, which was not assigned the task of lifting the engine panel. Platoon 2 Section 2 was assigned the task. Platoon 2 Section 2 Leader, Sergeant B, was not yet certified to operate the crane, hence they were in a fix as to how to proceed. However, Sergeant B is the senior ranking soldier and was supposed to be in charge of the operation. Corporal A came along and notice the situation, and decided to help. He and some of his buddies from Section 1 decided to operate the crane. The chains were not secured properly and gave way due to the weight, and crushed...
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...How To Read University Texts or Journal Articles Choose a section preferrably not longer than 25 or 30 pages - perhaps one chapter, or a section of a chapter - that you can handle at one sitting. Step 1. Read the title, the introduction, and the conclusion (5 minutes). Step 2. Read the title, the introduction again, all sub-headings, and the conclusion, again. (5-10 minutes). Step 3. Read the title, the introduction one more time, sub-headings, the Topic Sentence of each paragraph - usually the first or second sentence, (you may read the last sentence as well, if you have time), any italicized or boldfaced words, lists (you can skim these), and the conclusion (10 minutes). (Force yourself to do steps 1 to 3 in less than 25 minutes.) Step 4. Close your textbook. Step 5. Make a Mind-Map of all you can remember in the chapter. Do not stop until at least half an hour is up, even if you feel that you can't possibly remember any more--more will surface if you give yourself the time. DO NOT REFER TO THE TEXT WHILE YOU ARE DOING THIS. If you come to a dead end, try alternative memory techniques to the ones you have been using: associating ideas, either from within the section itself or from other related material; visualizing pages, pictures, graphs etc.; recalling personal associations that may have come to mind; staring out the window and blanking out your thoughts; and so on. This is strenuous, but it is rewarding. It will show you exactly how much you have learned...
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