... ------------------------------------------------- ------------------------------------------------- ------------------------------------------------- ------------------------------------------------- ------------------------------------------------- Submitted by: John Charlemagne Buan ------------------------------------------------- Submitted to: Ms. Harlene Santos ------------------------------------------------- ------------------------------------------------- ------------------------------------------------- Analytic geometry From Wikipedia, the free encyclopedia Analytic geometry, or analytical geometry, has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties. This article focuses on the classical and elementary meaning. In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the...
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...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 according to its physical, mental, emotional, social aspects in promoting a healthy lifestyle. | √ | √ | COURSE SYLLABUS 1. Course Code : MATH 121 2. Course Title : Analytic Geometry 3. Pre-Requisite : MATH 111, MATH 112 4. Co-Requisite : MATH 122, MATH 123 5. Credit/Class Schedule : 3 units 6. Course Description : Slope of a line; distance between two...
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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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...MC1704 Computer Graphics and Multimedia System 2 Marks Questions 1. What is scan conversion? A major task of the display processor is digitizing a picture definition given in an application program into a set of pixel-intensity values for storage in the frame buffer. This digitization process is called scan conversion. 2. Write the properties of video display devices? Properties of video display devices are persistence, resolution, and aspect ratio. 3. What is rasterization? The process of determining the appropriate pixels for representing picture or graphics object is known as rasterization. 4. Define Computer graphics. Computer graphics remains one of the most existing and rapidly growing computer fields. Computer graphics may be defined as a pictorial representation or graphical representation of objects in a computer. 5. Name any four input devices. Four input devices are keyboard, mouse, image scanners, and trackball. 6. Write the two techniques for producing color displays with a CRT? Beam penetration method, shadow mask method 7. What is vertical retrace of the electron beam? In raster scan display, at the end of one frame, the electron beam returns to the left top corner of the screen to start the next frame, is called vertical retrace of the electron beam. 8. Short notes on video controller? Video controller is used to control the operation of the display device. A fixed area of the system is reserved for the frame buffer, and the video controller...
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...First we to evaluate what was given: Evaluating f(x): Following are the values shown in tabular form: x | f(x) | -1 | -9 | 0 | -1 | 1 | 7 | Recall the to get slope (m) between P1 with coordinates (x1,y1) and P2 with coordinates (x2,y2) we use the formula: m = (y2 – y1) / (x2 – x1) say P1 = (x1,y1) = (-1,-9) and P2 = (x2,y2) = (0,-1) then m = (-1 - -9)/ (0 - -1) negative of negative is positive so m = (-1+9)/(0+1) m = slope = 8 Recall that the intercept of the line is point when x=0 so from the table x=0 when f(x) = -1 Thus our y-intercept is (0,-1). Evaluating g(x): g(x) = 3x -2 Recall that for an equation in the form y = mx+b the value of m is the slope of the line and b is the y intercept. Thus m=slope = 3 Thus our y-intercept is ( 0 , -2) COMPARISONS Part A : Write a sentence to compare the slope of the two functions and show the steps you used to determine the slope of f(x) and g(x). f(x) : slope = 8 g(x): slope = 3 From above we can deduce that f(x) has a bigger slope. Par B: Which function has a greater y-intercept? Justify your answer. f(x) : y-intercept is (0,-1). g(x) : y-intercept is ( 0 , -2) If our reference is the x-axis, then we can deduce that with g(x) y-intercept being further from the axis, g(x) has the greater...
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...Linear Functions Natasha D. Collins MTH/208 14 April 2011 John Rudin Linear Functions Question: Using the readings in Ch. 3 of the text, identify and explain at least one real-world application of algebraic concepts for one of the following areas: business, health and wellness, science, sports, and environmental sustainability. Do you think it is easier to relate this concept to one of these areas over any other? Explain why. I think the point slope method would be good in professional sports, because you can see the process of an athlete’s career. Question: Imagine that a line on a Cartesian graph is approximately the distance y in feet a person walks in x hours. What does the slope of this line represent? How is this graph useful? Provide another example for your colleagues to explain. Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4. Question: Can one line have two slopes? Explain how or why not. If the line is a straight line, meaning 180degrees, it can only have one slope. If it is a function (f(x) = or y=) then the line may have more than one, one, or an undefined slope. Find the first differential of the function and plug in your given x value to find the slope at any given point. Question: What is the difference between a scatter plot and a line graph? Provide...
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...Sergio Hernandez Jose Roque Paul Zuniga October 7th, 2012 Laboratory Report: Acceleration on an Incline Purpose: 0 Use a Motion Detector to measure the speed and acceleration of a cart rolling down an incline. 1 Determine the mathematical relationship between the angle of an incline and the acceleration of the cart. 2 Determine the value of free fall acceleration, g, by extrapolating the acceleration vs. sine of track angle graph. 3 Determine if an extrapolation of the acceleration vs. sine of track angle is valid. Materials: * Computer * Vernier computer interface. * Logger Pro. * Vernier Motion Detector. * Dynamics cart. * Meter stick. * Ramp. * Books. Procedure: 1. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface. 2. Place a single book under one end of a 1 – 3 m long board or track so that it forms a small angle with the horizontal. Adjust the points of contact of the two ends of the incline, so that the distance, x, in Figure 1 is between 1 and 3 m. 3. Place the Motion Detector at the top of an incline. Place it so the cart will never be closer than 0.4 m. 4. Open the file “04 g On An Incline” from the Physics with Vernier folder. 5. Hold the cart on the incline about 0.5 m from the Motion Detector. 6. Click to begin collecting data; release the cart after the Motion Detector starts to click. Get your hand out of the Motion Detector path quickly. You may have to adjust the position...
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...1. Juan and Romella are standing at the seashore 10 miles apart. The coastline is a straight line between them. Both can see the same ship in the water. The angle between the coastline and the line between the ship and Juan is 35 degrees. The angle between the coastline and the line between the ship and Romella is 45 degrees. How far is the ship from Juan? 2. Jack is on one side of a 200-foot-wide canyon and Jill is on the other. Jack and Jill can both see the trail guide at an angle of depression of 60 degrees. How far are they from the trail guide? 3. Tom, Dick, and Harry are camping in their tents. If the distance between Tom and Dick is 153 feet, the distance between Tom and Harry is 201 feet, and the distance between Dick and Harry is 175 feet, what is the angle between Dick, Harry, and Tom? 4. Three boats are at sea: Jenny one (J1), Jenny two (J2), and Jenny three (J3). The crew of J1 can see both J2 and J3. The angle between the line of sight to J2 and the line of sight to J3 is 45 degrees. If the distance between J1 and J2 is 2 miles and the distance between J1 and J3 is 4 miles, what is the distance between J2 and J3? 5. Airplane A is flying directly toward the airport which is 20 miles away. The pilot notices airplane B 45 degrees to her right. Airplane B is also flying directly toward the airport. The pilot of airplane B calculates that airplane A is 50 degrees to his left. Based on that information, how far is airplane B...
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...CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level ADDITIONAL MATHEMATICS 4037/01 Paper 1 May/June 2003 2 hours Additional Materials: Answer Booklet/Paper Graph paper Mathematical tables READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Write your answers on the separate answer booklet/paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. This document consists of 5 printed pages and 3 blank pages. MCS UCB246 S38225/2 © CIE 2003 http://www.xtremepapers.net [Turn over 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, –b...
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...Rotational Slumping A slump is a form of mass wasting that occurs when a large mass of loosely materials or rock layers moves a short distance down a slope. Movement is characterized by sliding along a curved surface. Causes of slumping include earthquake shocks, thorough wetting, freezing and thawing, undercutting, and loading of a slope. Rotational slumps occur when a slump block, composed of sediment or rock, slides along a concave-upward slip surface with rotation about an axis parallel to the slope. Rotational movement causes the original surface of the block to become less steep, and the top of the slump is rotated backward. This results in internal deformation of the moving mass consisting chiefly of overturned folds called sheath folds. Slumps have several characteristic features. The cut, which forms as the landmass breaks away from the slope, is called the scarp and is often cliff-like and concave. In rotational slumps, the main slump block often breaks into a series of secondary slumps and associated scarps to form stair step pattern of displaced blocks. The upper surfaces of the blocks are rotated backwards, forming depressions that may accumulate water to create ponds or swampy areas. The surface of the detached mass often remains relatively undisturbed, especially at the top. However, hummocky ridges may form near the toe of the slump. Addition of water and loss of sediment cohesion at the toe may transform slumping material into an earthflow. Thorough wetting...
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...COMP 274 Week 5 Homework Questions Answer the following questions: 1. Describe the graphical coordinate system in Java. Where is the origin? What units apply to the x,y coordinates? The horizontal axis is the x coordinate and the vertical axis is the y coordinate. The origin is the upper left corner of the display area. Moving to the right increases the x value, moving down increases the y value. The units are pixels. 2. How would you use the Graphics class to draw a line between 2 specific points? Give an example. public void paint(Graphics g) { Graphics2D g2 = (Graphics2D) g; Line2D lin = new Line2D.Float(5, 30, 380, 30); g2.draw(lin); } 3. How do you specify a particular color to be used as fill when using the Graphics class? g2.setPaint(Color.BLACK); 4. How would you create a SanSerif font of point size 14 that is bold and italic? Give an example. JButton b = new JButton("OK"); b.setFont(new Font("sansserif", Font.BOLD, 32)); 5. Given a graphics object g, write a few lines of code to have that graphics object draw a green circle (filled in) that has a diameter of 100 pixels. import java.awt.*; import java.applet.*; public class FirstApplet extends Applet { public void paint (Graphics g) { g.drawRect(40,20,200,100); g.drawOval(40,20,200,100); } 6. Given a graphics object g, write a few lines of code to draw a red rectangle...
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...4-5 Exact Values of Sine, Cosine, and Tangent RECALL: All Students Take Calculus | |Q1 |Q2 |Q3 |Q4 | |SINE (y) |+ |+ |- |- | |COSINE (x) |+ |- |- |+ | |TANGENT |+ |- |+ |- | *ALL positive *Sine positive *Tangent positive * Cosine pos ALL Students Take Calculus KEY POINTS (COS [pic], SIN [pic]) WE ALREADY KNOW: [pic]= 0/360 degrees [pic] = 90 degrees [pic] = 180 degrees [pic] = 270 degrees POINTS WE WILL MASTER TODAY: [pic] = 45 degrees [pic] = 30 degrees [pic] = 60 degrees Consider a rotation of 45 degrees ( or [pic] radians) What type of triangle do you get? (45-45-90 Right Triangle) BUT we need a radius/hypotenuse of 1!!! ((((( When [pic] = [pic], cos [pic] = [pic] sin [pic] = [pic] SYMMETRY AROUND THE UNIT CIRCLE WHEN [pic] = multiples of[pic] [pic] *Value in the first quadrant is at [pic] = [pic] *LOOK AT SECOND QUADRANT WHERE [pic] = [pic] *Numbers are...
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...PURE MATHEMATICS Mensuration Surface area of sphere = 4πr 2 Area of curved surface of cone = πr × slant height Arithmetic series S n a l n[ a n d] u a n d n n ( ) 2 ( 1) = ( 1) 2 1 2 = 1 + = + − + − Geometric series 1 for 1 1 (1 ) 1 = − < − = − = ∞ − r r S a r S a r u ar n n n n Summations ( )1 2 1 1 + = Σ= r n n n r ( 1)(2 1) 6 1 1 2 + + = Σ= r n n n n r 2 2 4 1 1 3 ) 1 ( + = Σ= r n n n r Trigonometry – the Cosine rule a2 = b2 + c2 − 2bc cos A Binomial Series ∈ + + ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + + ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + = + − − a − b b n r n a b n a b n a b n an n n n r r n ( 2 1 ( ) 1 2 2 … … ) where !( )! C ! r n r n r n r n − = = ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + = + + − + + − − + x + x < n∈ r x n nx n n x n n n r r ( 1, 1.2 ( 1) ( 1) 1.2 (1 ) 1 ( 1) 2 … … … … ) Logarithms and exponentials ax = exln a Complex numbers {r(cosθ + i sinθ )}n = r n (cos nθ + i sin nθ ) eiθ = cosθ + i sinθ The roots of z n = 1 are given by n k z 2π i = e , for k = 0, 1, 2, … , n −1 N R klj 5 Maclaurin’s series f( ) f(0) f (0) 2! f (0) … ! f ( ) (0) … 2 = + ′ + ′′ + + r + r r x x x x r x x x x x x r e exp( ) 1 2! ! for all 2 = = + + +…+ +… r x x x x x x r + = − + − + (−1)r+ + (−1 < ln(1 ) 2 3 2 3 1 … … 1) x r x x x x x r r for all (2 1)! ( 1) 3! 5! sin 3 5 2 1 … +… + = − + − + − + x r x x x x r r for all (2 )! ( 1) 2! 4! cos 1 2 4 2 = − + −…+ − +… Hyperbolic...
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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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...| | | Unit Test Slope and Proportional Thinking Unit Test, Part 2 Answer each question in the space provided. (7 points) |Score | | | 1. Refer to the equation 2x – 6y = 12. a) What is the x-intercept? Show your work. b) What is the y-intercept? Show your work. c) Use the x- and y-intercepts to graph the line. [pic] Answer: (8 points) |Score | | | 2. The graph shows the number of laps Kailee ran around a track over a given number of minutes. [pic] a) What is the slope of the line? Use the slope formula and show your work. b) How many laps did she run per minute? c) If she continued to run at the same rate, how many laps would she run in 24 min? Answer: (13 points) |Score | | | 3. The equation y = 12x describes the amount of money Louis earns, where x is the number of hours he works and y is the money he earns. The table shows the amount of money Carl earns for different numbers of hours worked. |Carl’s Earnings | |Time (h) |3 |5 |8 |10 ...
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