Pc 10.1 notes_tangents
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This document discusses finding the angle of inclination and slope of a line, as well as the angle between two lines. It defines inclination as the angle measured counterclockwise from the x-axis to the line. Slope is related to inclination by the formula that if a non-vertical line has an inclination of θ and a slope of m, then tanθ = m. It also provides the formula for finding the angle between two non-perpendicular lines based on their slopes. Several examples are worked through to demonstrate finding inclination, slope, and the angle between lines.
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1. The assignment can be completed in groups of 4 students and must be typed using Microsoft Word and Equation Editor. References from at least 2 books must be used in addition to internet sources.
2. L'Hospital's Rule can be used to evaluate limits involving indeterminate forms such as 0/0, ∞/∞, and other cases. It uses derivatives to help find the limit.
3. Conic sections are curves formed by the intersection of a plane with a double cone. The five types are a pair of intersecting lines, a circle, a parabola, an ellipse, and a hyperbola. Each has a specific formation method and properties that can be used to graph
Minimum enclosingdisk
The document discusses methods for solving the smallest k-enclosing circle problem, which is finding the smallest circle that encloses at least k points from a set of n points. It presents two simple algorithms: 1) generating circles that pass through every pair of points, running in O(n^3 log(n)) time; and 2) generating circles hinged at each point, running in O(n^2 log(n) log(d)) time where d is the geometric span of the points. The problem has applications in facility location and military targeting.
Dimension Reduction Introduction & PCA.pptx
PCA is a dimensionality reduction technique that uses linear transformations to project high-dimensional data onto a lower-dimensional space while retaining as much information as possible. It works by identifying patterns in data and expressing the data in such a way as to highlight their similarities and differences. Specifically, PCA uses linear combinations of the original variables to extract the most important patterns from the data in the form of principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible.
ANALYTIC-GEOMETRY(4).pptx
This document provides information about various concepts in analytic geometry including:
- Points, lines, planes, line segments, circles, angles, polygons, and their definitions.
- Finding midpoints, distances between points, slopes and inclinations of lines.
- Calculating slopes, tangents of angles, and the angle between two lines.
- Examples are provided to demonstrate finding midpoints, distances, slopes, angles, and solving geometric problems using coordinate systems.
Geometry 201 unit 5.4
1) A median of a triangle is a segment from a vertex to the midpoint of the opposite side. The medians of any triangle are concurrent at the centroid.
2) An altitude of a triangle is a perpendicular segment from a vertex to the opposite side. The three altitudes of a triangle are concurrent at the orthocenter.
3) This document discusses properties of medians and altitudes of triangles, including using the centroid and orthocenter to solve problems involving lengths of segments and finding coordinates of special points like the centroid and orthocenter. Examples are provided to illustrate these concepts.
Analytic geometry lecture1
Lecture on Analytic Geometry on the topic Rectangular coordinate system,line segment, distance formula,parallel lines and equation of the line.
1516 contouring
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Chapter 9 plane figures
- A triangle is a three-sided polygon with three angles that sum to 180 degrees. Triangles can be classified based on side length (scalene, isosceles, equilateral) or angle type (acute, right, obtuse).
- The triangle inequality theorem states that any side of a triangle must be less than the sum of the other two sides and greater than the difference of the other two sides.
- A quadrilateral is a four-sided polygon. Quadrilaterals can be simple or complex, and simple ones can be convex or concave. The interior angles of any simple quadrilateral sum to 360 degrees.
- A circle is the set of all points in a plane equid
Assignment of straight lines and conic section
1. The document contains 35 problems related to straight lines and conic sections like circles, ellipses, parabolas and hyperbolas. The problems involve finding equations of lines and conic sections given certain properties, finding properties like foci, vertices, axes, etc. from given equations.
2. Specific problems include finding slopes of lines, angle between lines, perpendicular lines, finding equations of lines passing through points or perpendicular/parallel to other lines.
3. For conic sections, problems include finding equations given foci, directrix, passing points, intersecting lines, centers and radii of circles, lengths of axes, foci, latus rectum and eccentricities of ellipses and hyperbol
1539 graphs linear equations and functions
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Math - analytic geometry
1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
Presentación1
1) The document defines various geometric terms including points, lines, planes, collinear and coplanar points, segments, rays, intersections, angles, and transformations.
2) It provides examples of how to use formulas to find distances between points, add segments, use the Pythagorean theorem, find midpoints and angle bisectors, calculate perimeter and area of squares, rectangles, and triangles.
3) The five step problem solving process is outlined as read the problem, highlight important information, draw a picture, solve the equation, and answer the question.
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Pc 8.5 notes_appsof_det
This document discusses using determinants and Cramer's rule to solve systems of equations, find the area of triangles formed by three points, and determine if three points are collinear or lie on the same line. It provides examples of using Cramer's rule to solve a system of two equations with two unknowns, finding the area of triangles formed by two example point sets, and checking if an example point set defines collinear points. It also mentions finding the equation of a line through two given points.
Pc 8.3 notes
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Pc 8.4 notes
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Pc 10.1 notes_tangents
- 1. 10.1 Tangent Lines Objective: 1) Find the angle of inclination and slope of a line. 2) Find the angle between 2 lines. Standard: Trigonometry 7.0
- 2. Warm-up: 1) Find the slope of the line containing the two points (–4, 1) and (3, 6). 2) What is the angle this line makes with the x-axis?
- 3. Inclination and Slope Inclination: · The ___________ _________ less than ___ measured _________________ from the x- axis to the line. Slope: · If a non-vertical line has an inclination of θ and a slope of m, then…
- 4. Ex. 1) Find the inclination. Find the angle of inclination of the line given by 5x–7y+3=0.
- 5. Ex. 1b) Find the inclination of 3x+4y=7.
- 6. What about the angle between 2 lines?
- 7. Angles between lines: (Where m1 and m2 are the slopes of two non-perpendicular lines)
- 8. Ex. 2) Find the angle between the two following lines. 2x + y = 4 x–y=2
- 9. Ex. 3) Application: A straight road rises with the inclination of 0.20 radians from the horizontal. Find the slope of the road and the change in elevation of a 2-mile stretch of the road.
- 11. Ex. 4) Find the equation of the line with an inclination of 120° and through the point (-3, 5).
- 12. Assignment: #5 p. 732 #9-31 (odd), 51, 53, 56