The document discusses calculating the acute angle between two lines with slopes m1 and m2. It provides an example problem that asks to find the possible values of m for the acute angle between the lines y=3x+5 and y=mx+4.
The document summarizes key aspects of hydrographic surveys. It discusses controlling horizontal and vertical positions, measuring depths through sounding, and producing charts. Soundings are taken using various instruments and referenced to tidal datums. Depths are plotted on charts along with shorelines, depth contours, and navigational features. Hydrographic surveys provide critical data for safe navigation and engineering projects.
The document discusses finding the angle between two lines given by their gradients (slopes). It provides the formula for calculating the tangent of the acute angle between the lines as (m1 - m2) / (1 + m1m2), where m1 and m2 are the gradients of the two lines. It then works through three examples of applying the formula to find the acute angle between pairs of lines.
11 x1 t08 03 angle between two lines (2013)Nigel Simmons
This document describes how to find the acute angle between two lines given their slopes m1 and m2. The method is to take the tangent of both sides of the equation m1/
This document discusses slope and how to calculate it from the coordinates of two points on a line. It defines slope as the tangent of the angle of inclination of the line with respect to the x-axis. It provides the formula for calculating slope given two points and discusses how to determine if two lines are parallel, perpendicular, or the angle between them based on their slopes.
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.
A really simple presentation about Stereo Vision, especially the stereo vision in real time applied to mobile robotics. In the talk I explained how the stereo vision works, I presented a simplified mathematical model to achieve it and I proposed what for me is the best hardware to use to achieve real-time stereo vision, that is, the sensor ZED produced by Stereolabs used together with the board Jetson TX1 by Nvidia.
The presentation is really simple and has been made for an audience with limited knowledge about computer vision and the underhood mathematics.
The talk was held during the event "Officine Robotiche 2016" in Rome, 21-22 May 2016
Two cameras displaced horizontally from one another can obtain two differing views of a scene, similar to human binocular vision. By comparing these two images and examining the relative positions of objects, 3D information like depth can be extracted from the 2D images. This process is called stereo vision. The depth information is contained in pixel displacements between the two images called disparities, which are inversely proportional to distance. Knowing camera intrinsics and their relative pose allows reconstructing 3D point positions through triangulating corresponding image points. Reconstruction accuracy depends on factors like disparity, baseline distance, and focal length.
The document discusses dividing a line segment into parts at given ratios. It defines the point of division as the point that divides a line segment into a given ratio. It then presents the formula to find the point of division using similar triangles. The document provides examples of finding the point of division for line segments divided into equal parts or divided at specific ratios. It gives examples of finding the midpoint and trisection points of line segments.
The document summarizes key aspects of hydrographic surveys. It discusses controlling horizontal and vertical positions, measuring depths through sounding, and producing charts. Soundings are taken using various instruments and referenced to tidal datums. Depths are plotted on charts along with shorelines, depth contours, and navigational features. Hydrographic surveys provide critical data for safe navigation and engineering projects.
The document discusses finding the angle between two lines given by their gradients (slopes). It provides the formula for calculating the tangent of the acute angle between the lines as (m1 - m2) / (1 + m1m2), where m1 and m2 are the gradients of the two lines. It then works through three examples of applying the formula to find the acute angle between pairs of lines.
11 x1 t08 03 angle between two lines (2013)Nigel Simmons
This document describes how to find the acute angle between two lines given their slopes m1 and m2. The method is to take the tangent of both sides of the equation m1/
This document discusses slope and how to calculate it from the coordinates of two points on a line. It defines slope as the tangent of the angle of inclination of the line with respect to the x-axis. It provides the formula for calculating slope given two points and discusses how to determine if two lines are parallel, perpendicular, or the angle between them based on their slopes.
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.
A really simple presentation about Stereo Vision, especially the stereo vision in real time applied to mobile robotics. In the talk I explained how the stereo vision works, I presented a simplified mathematical model to achieve it and I proposed what for me is the best hardware to use to achieve real-time stereo vision, that is, the sensor ZED produced by Stereolabs used together with the board Jetson TX1 by Nvidia.
The presentation is really simple and has been made for an audience with limited knowledge about computer vision and the underhood mathematics.
The talk was held during the event "Officine Robotiche 2016" in Rome, 21-22 May 2016
Two cameras displaced horizontally from one another can obtain two differing views of a scene, similar to human binocular vision. By comparing these two images and examining the relative positions of objects, 3D information like depth can be extracted from the 2D images. This process is called stereo vision. The depth information is contained in pixel displacements between the two images called disparities, which are inversely proportional to distance. Knowing camera intrinsics and their relative pose allows reconstructing 3D point positions through triangulating corresponding image points. Reconstruction accuracy depends on factors like disparity, baseline distance, and focal length.
The document discusses dividing a line segment into parts at given ratios. It defines the point of division as the point that divides a line segment into a given ratio. It then presents the formula to find the point of division using similar triangles. The document provides examples of finding the point of division for line segments divided into equal parts or divided at specific ratios. It gives examples of finding the midpoint and trisection points of line segments.
This document provides an overview of stereo vision algorithms and applications. It begins with an introduction to stereo vision and the correspondence problem. Key steps in a stereo vision system are discussed, including calibration, rectification, stereo matching algorithms, and triangulation. Both local and global stereo matching approaches are described. Several challenges in stereo correspondence are highlighted. The document also outlines datasets, architectures, and commercial stereo cameras for evaluation and implementation.
The document discusses key concepts in Euclidean geometry including definitions, notation, terminology, naming conventions, angle theorems, and constructing proofs. It defines parallel lines, perpendicular lines, congruent lines, and similar lines using notation symbols. It provides examples of naming angles, polygons, and parallel lines in a consistent cyclic order. It also outlines the steps and logical structure required to properly construct geometric proofs.
The document discusses the slope (gradient) of a line and how to calculate it. It provides four methods to calculate slope:
(1) The rise over the run between two points (vertical change over horizontal change)
(2) The change in y-values over the change in x-values between two points using a formula
(3) The slope of a line is equal to the tangent of the angle of inclination
(4) The relationship between slopes of parallel and perpendicular lines. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1. An example problem demonstrates finding the value of a that results in two lines being parallel or perpendicular.
11 x1 t07 04 quadrilateral family (2012)Nigel Simmons
The document presents a diagram showing the hierarchical classification of quadrilaterals. It begins with quadrilateral as the parent shape and branches down to more specific shapes such as squares, rectangles, rhombi, parallelograms, trapezoids, and kites. The text explains that quadrilaterals can be classified based on their properties, and moving down the hierarchy, the shapes become more specialized. It also notes that any property of the parent shapes also applies to the child shapes.
The document discusses tests for determining if triangles are similar. There are three tests: (1) corresponding sides are in proportion (SSS), (2) two pairs of corresponding sides are in proportion and included angles are equal (SAS), (3) all three angles are equal (AA). An example problem finds the length of side AD using the properties of similar triangles. The side lengths are in the same ratio as the corresponding sides between the two triangles.
The document discusses several triangle and polygon theorems:
- The sum of the interior angles of any triangle is 180 degrees.
- The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- The sum of the interior angles of any quadrilateral is 360 degrees.
- The sum of the interior angles of any pentagon is 540 degrees.
Proofs of these theorems are presented using angle properties, parallel lines, and adding individual triangle angle sums.
11 x1 t05 04 point slope formula (2013)Nigel Simmons
The document discusses the point slope formula and provides examples of its use. Specifically, it contains:
1) The point slope formula: y - y1 = m(x - x1)
2) An example that finds the equation of the line passing through (-3,4) and (2,-6).
3) A second example that finds the equation (3x + 4y + 6 = 0) of the line passing through (2,-3) and parallel to the given line (3x + 4y - 5 = 0).
11 x1 t05 06 line through pt of intersection (2013)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
The document discusses properties of transversals and ratios of intercepts formed when a transversal cuts parallel lines. It includes examples of calculating lengths of intercepts using given ratios and applying the ratio property that the ratio of corresponding intercepts is equal to the ratio of distances from the transversal to the parallel lines.
The document discusses congruent triangles and the different tests that can be used to prove triangles are congruent. There are four main tests: side-side-side (SSS), side-angle-side (SAS), angle-angle-side (AAS), and right angle-hypotenuse-side (RHS). It also provides examples of applying the SAS test to prove corresponding sides and angles of two triangles are congruent. Additionally, it defines different types of triangles such as isosceles and equilateral triangles based on their side lengths and angle measures.
The document discusses finding the greatest term in polynomial expansions. It provides an example of finding the greatest coefficient in the expansion of (2 + 3x)^20. Through algebraic steps, it is shown that the greatest coefficient is T13 = 20C12 * 2831^2. It then gives an example of finding the greatest term in the expansion of (3x - 4)^15 when x = 1/2. After setting up expressions for the terms Tk+1 and Tk, it derives an inequality to determine the value of k that makes Tk+1 the greatest term.
11 x1 t05 01 division of an interval (2013)Nigel Simmons
The document discusses division of intervals in coordinate geometry. It states that for a 2 unit math exam, interval division questions are restricted to finding the midpoint, which divides an interval in a 1:1 ratio. For Extension 1 exams, intervals can be divided in any ratio internally or externally. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint, while the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
The document discusses several key concepts about triangles:
- The sum of the angles of any triangle is always 180 degrees.
- It defines different types of triangles based on angle measures, such as acute, obtuse, right, isosceles, equilateral, and scalene triangles.
- Parallel lines and auxiliary lines are used to help prove theorems about triangles. The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the remote interior angles.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
This document provides an overview of stereo vision algorithms and applications. It begins with an introduction to stereo vision and the correspondence problem. Key steps in a stereo vision system are discussed, including calibration, rectification, stereo matching algorithms, and triangulation. Both local and global stereo matching approaches are described. Several challenges in stereo correspondence are highlighted. The document also outlines datasets, architectures, and commercial stereo cameras for evaluation and implementation.
The document discusses key concepts in Euclidean geometry including definitions, notation, terminology, naming conventions, angle theorems, and constructing proofs. It defines parallel lines, perpendicular lines, congruent lines, and similar lines using notation symbols. It provides examples of naming angles, polygons, and parallel lines in a consistent cyclic order. It also outlines the steps and logical structure required to properly construct geometric proofs.
The document discusses the slope (gradient) of a line and how to calculate it. It provides four methods to calculate slope:
(1) The rise over the run between two points (vertical change over horizontal change)
(2) The change in y-values over the change in x-values between two points using a formula
(3) The slope of a line is equal to the tangent of the angle of inclination
(4) The relationship between slopes of parallel and perpendicular lines. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1. An example problem demonstrates finding the value of a that results in two lines being parallel or perpendicular.
11 x1 t07 04 quadrilateral family (2012)Nigel Simmons
The document presents a diagram showing the hierarchical classification of quadrilaterals. It begins with quadrilateral as the parent shape and branches down to more specific shapes such as squares, rectangles, rhombi, parallelograms, trapezoids, and kites. The text explains that quadrilaterals can be classified based on their properties, and moving down the hierarchy, the shapes become more specialized. It also notes that any property of the parent shapes also applies to the child shapes.
The document discusses tests for determining if triangles are similar. There are three tests: (1) corresponding sides are in proportion (SSS), (2) two pairs of corresponding sides are in proportion and included angles are equal (SAS), (3) all three angles are equal (AA). An example problem finds the length of side AD using the properties of similar triangles. The side lengths are in the same ratio as the corresponding sides between the two triangles.
The document discusses several triangle and polygon theorems:
- The sum of the interior angles of any triangle is 180 degrees.
- The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- The sum of the interior angles of any quadrilateral is 360 degrees.
- The sum of the interior angles of any pentagon is 540 degrees.
Proofs of these theorems are presented using angle properties, parallel lines, and adding individual triangle angle sums.
11 x1 t05 04 point slope formula (2013)Nigel Simmons
The document discusses the point slope formula and provides examples of its use. Specifically, it contains:
1) The point slope formula: y - y1 = m(x - x1)
2) An example that finds the equation of the line passing through (-3,4) and (2,-6).
3) A second example that finds the equation (3x + 4y + 6 = 0) of the line passing through (2,-3) and parallel to the given line (3x + 4y - 5 = 0).
11 x1 t05 06 line through pt of intersection (2013)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
The document discusses properties of transversals and ratios of intercepts formed when a transversal cuts parallel lines. It includes examples of calculating lengths of intercepts using given ratios and applying the ratio property that the ratio of corresponding intercepts is equal to the ratio of distances from the transversal to the parallel lines.
The document discusses congruent triangles and the different tests that can be used to prove triangles are congruent. There are four main tests: side-side-side (SSS), side-angle-side (SAS), angle-angle-side (AAS), and right angle-hypotenuse-side (RHS). It also provides examples of applying the SAS test to prove corresponding sides and angles of two triangles are congruent. Additionally, it defines different types of triangles such as isosceles and equilateral triangles based on their side lengths and angle measures.
The document discusses finding the greatest term in polynomial expansions. It provides an example of finding the greatest coefficient in the expansion of (2 + 3x)^20. Through algebraic steps, it is shown that the greatest coefficient is T13 = 20C12 * 2831^2. It then gives an example of finding the greatest term in the expansion of (3x - 4)^15 when x = 1/2. After setting up expressions for the terms Tk+1 and Tk, it derives an inequality to determine the value of k that makes Tk+1 the greatest term.
11 x1 t05 01 division of an interval (2013)Nigel Simmons
The document discusses division of intervals in coordinate geometry. It states that for a 2 unit math exam, interval division questions are restricted to finding the midpoint, which divides an interval in a 1:1 ratio. For Extension 1 exams, intervals can be divided in any ratio internally or externally. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint, while the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
The document discusses several key concepts about triangles:
- The sum of the angles of any triangle is always 180 degrees.
- It defines different types of triangles based on angle measures, such as acute, obtuse, right, isosceles, equilateral, and scalene triangles.
- Parallel lines and auxiliary lines are used to help prove theorems about triangles. The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the remote interior angles.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses key features that should be noticed when analyzing graphs of functions. It covers basic curve shapes like straight lines, parabolas, cubics, and more that can be identified from their equations. It also discusses concepts like odd and even functions, symmetry, dominance of terms as x increases, and types of asymptotes. The document provides examples and notes for identifying these various graphical features from the equation of a function.
The document discusses the primitive function and how it can be used to find the original curve when the equation of its tangent is known. It provides examples of calculating primitive functions from equations of tangents. It also gives an example of finding the equation of a curve given its point of intersection and gradient function.
This document contains notes from a calculus class covering topics including: implicit differentiation, related rates, linear approximations, maximum and minimum values, the mean value theorem, limits at infinity, and curve sketching. Example problems are provided for each topic to demonstrate key concepts and techniques.
The document provides information about linear equations and their graphs. It defines linear equations and discusses how to write equations in slope-intercept form, point-slope form, and standard form. It also describes how to graph linear equations by plotting intercepts and using slope. Key topics covered include finding the slope between two points, determining if lines are parallel or perpendicular based on their slopes, and recognizing the intercepts on a graph of a linear equation in two variables.
The document provides a review of calculating area and volume. It lists 13 practice problems involving finding the area between curves or the volume of solids obtained by rotating areas about axes. The problems involve skills like identifying the bounds of integration, setting up integrals of common functions like polynomials, and evaluating the integrals to find the final numeric or symbolic answer.
1. The document discusses various properties of graphs including symmetry, even and odd functions, translations, reflections, and stretching or compressing graphs.
2. It provides examples of applying these properties to determine if a graph is symmetric, even or odd, or to sketch graphs based on translations, reflections, or vertical/horizontal stretching and compressing of a original graph.
3. Several exercises are provided to apply these graph properties and transformations to specific functions and points on graphs.
This document contains a table with geometry vocabulary terms from Chapter 3 including definitions and examples. It lists terms such as alternate angles, parallel lines, perpendicular lines, slope, and the point-slope form of a linear equation. The table is intended to be filled in as readers work through Chapter 3. There are also review questions testing understanding of lines, angles, parallel and perpendicular lines, slopes of lines, and representing lines in the coordinate plane.
This document discusses linear equations in two variables. It defines linear equations and explains how to find the slope of a line using the slope formula. It presents the slope-intercept form and point-slope forms of linear equations and how to graph lines using these forms. It also discusses parallel and perpendicular lines based on the slopes of the lines.
The document discusses transformations of graphs including:
1) Symmetry with respect to the x-axis, y-axis, and origin.
2) Translations that shift graphs horizontally or vertically by adding/subtracting constants.
3) Reflections that flip graphs across the x-axis or y-axis.
4) Stretching/compressing graphs vertically or horizontally by multiplying the function by constants greater than or less than 1.
This document provides instruction on writing linear equations in slope-intercept form, point-slope form, and finding equations of lines parallel or perpendicular to given lines. It defines key vocabulary like slope, parallel, and perpendicular. Examples are worked through, like writing the equation of a line given its slope and a point, or finding the equation of a line perpendicular to another line passing through a given point. The problem set provided practices writing various linear equations.
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
11 X1 T05 06 Line Through Pt Of IntersectionNigel Simmons
The document discusses finding the equation of a line that passes through the intersection of two other lines, 2x + y + 1 = 0 and 3x + 5y - 9 = 0, and through the point (1,2).
The document defines locus as the collection of all points whose location is determined by some stated law. It then provides examples of finding the locus of points that satisfy specific conditions: (1) always being 4 units from the origin, forming a circle; (2) always being 5 units from the y-axis, forming two lines; (3) always being 3 units from the line y=x+1, forming two parabolas. Finally, it gives an example of a point whose distance from the x-axis is always 5 times its distance from the y-axis.
A summation is the sum of terms in a sequence. There are four special summations that were presented: the sum of the first n natural numbers, the sum of the first n integers, the sum of the squares of the first n integers, and the sum of the cubes of the first n integers. Riemann sums are used to approximate the area under a curve between two values by dividing the interval into subintervals and creating rectangles to estimate the area, with the true area found as the number of subintervals approaches infinity. An example of using a Riemann sum to calculate the distance traveled given the velocity function over time between two points was provided.
This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
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
The document discusses key features to notice when analyzing graphs of functions. It identifies the basic shapes of straight lines, parabolas, cubics, polynomials, hyperbolas, exponentials, circles, ellipses, logarithmics, and trigonometric/inverse trigonometric functions based on their equations. It also covers concepts of odd and even functions, symmetry across the line y=x, and dominance of terms for large values of x.
Similar to 11 X1 T05 07 Angle Between Two Lines (20)
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
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3
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The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
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ملزمة تشريح الجهاز الهيكلي (نظري 3)
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#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).