The document discusses geometrical theorems about parabolas, including two main topics:
1) Focal chords - It proves that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix.
2) Reflection property - It discusses how any line parallel to the axis of the parabola is reflected towards the focus, and proves that the angle of incidence equals the angle of reflection.
The document discusses geometrical theorems about parabolas, including:
1) Focal chords and proving that tangents from a focal chord's endpoints intersect the directrix at right angles.
2) The reflection property - any line parallel to the parabola's axis reflects toward the focus, and any line from the focus reflects parallel to the axis. It proves that the angle of incidence equals the angle of reflection.
3) Additional exercises are listed at the end.
The document discusses two geometrical theorems about parabolas:
1. It proves that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix.
2. It proves that the angle of incidence of a line parallel to the axis of a parabola is equal to the angle of reflection. This establishes that a line and its reflection are equally inclined to the normal and tangent of the parabola.
The document discusses electronic excited states and challenges in simulating their dynamics. It describes how excited states are described using different representations like adiabatic and diabatic. It highlights how conical intersections are better described than avoided crossings and how nonadiabatic transitions occur. It also summarizes challenges with excited state electronic structure methods like TDDFT in accurately describing conical intersections.
1) The posterior chamber of the eye is filled with aqueous humor. It flows from the ciliary body into the posterior chamber and then into the anterior chamber through the pupil.
2) Governing equations for fluid motion in the posterior chamber are developed using the assumptions of incompressible, Newtonian flow with negligible motion in the φ direction.
3) The equations are non-dimensionalized and simplified using scaling arguments to obtain equations for the velocity profiles and pressure distribution.
Linear vs. semidefinite extended formulationsspokutta
The document discusses linear programs (LPs) and extended formulations for solving problems like the traveling salesman problem (TSP). It presents the following key points:
1) A previous claim that the TSP could be solved in polynomial time via an LP was disproven, showing LPs require super-polynomial size.
2) The document shows that every extended formulation (EF) of the TSP polytope requires size 2Ω(n1/4), via reductions from EFs of quantum stabilizer problems and cut polytopes.
3) This establishes a new connection between semidefinite programs and quantum information, and generalizes prior work linking classical communication complexity to linear EFs. The
The document discusses Bhaskara II's treatise on Jyotpatti, the science of trigonometry in ancient India. It begins by providing context for the treatise as the last section of Bhaskara's comprehensive work Siddhanta Siromani. It then describes the graphical method used in ancient India to define and obtain values of trigonometric functions like jya (sine). Subsequently, it presents Bhaskara's mathematical method for calculating jya and other functions through formulas involving the radius of the reference circle. Key contributions of the treatise included the first accurate values for sines of particular angles derived through inscribing regular polygons in a circle.
A Multicover Nerve for Geometric InferenceDon Sheehy
We show that filtering the barycentric decomposition of a Cech complex by the cardinality of the vertices captures precisely the topology of k-covered regions among a collection of balls for all values of k.
Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.
Algebraic geometry and commutative algebraSpringer
This document provides an overview of the theory of Noetherian rings. It begins by defining Noetherian rings as rings whose ideals satisfy the ascending chain condition. It then discusses examples of Noetherian and non-Noetherian rings. A key tool in studying the structure of ideals in Noetherian rings is primary decomposition. The document explores primary decomposition and its properties, including uniqueness of associated prime ideals. It introduces Krull dimension and discusses its applications, such as proving that the dimension of a Noetherian local ring is finite. The document concludes by discussing related concepts like systems of parameters and regular local rings.
The document discusses geometrical theorems about parabolas, including:
1) Focal chords and proving that tangents from a focal chord's endpoints intersect the directrix at right angles.
2) The reflection property - any line parallel to the parabola's axis reflects toward the focus, and any line from the focus reflects parallel to the axis. It proves that the angle of incidence equals the angle of reflection.
3) Additional exercises are listed at the end.
The document discusses two geometrical theorems about parabolas:
1. It proves that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix.
2. It proves that the angle of incidence of a line parallel to the axis of a parabola is equal to the angle of reflection. This establishes that a line and its reflection are equally inclined to the normal and tangent of the parabola.
The document discusses electronic excited states and challenges in simulating their dynamics. It describes how excited states are described using different representations like adiabatic and diabatic. It highlights how conical intersections are better described than avoided crossings and how nonadiabatic transitions occur. It also summarizes challenges with excited state electronic structure methods like TDDFT in accurately describing conical intersections.
1) The posterior chamber of the eye is filled with aqueous humor. It flows from the ciliary body into the posterior chamber and then into the anterior chamber through the pupil.
2) Governing equations for fluid motion in the posterior chamber are developed using the assumptions of incompressible, Newtonian flow with negligible motion in the φ direction.
3) The equations are non-dimensionalized and simplified using scaling arguments to obtain equations for the velocity profiles and pressure distribution.
Linear vs. semidefinite extended formulationsspokutta
The document discusses linear programs (LPs) and extended formulations for solving problems like the traveling salesman problem (TSP). It presents the following key points:
1) A previous claim that the TSP could be solved in polynomial time via an LP was disproven, showing LPs require super-polynomial size.
2) The document shows that every extended formulation (EF) of the TSP polytope requires size 2Ω(n1/4), via reductions from EFs of quantum stabilizer problems and cut polytopes.
3) This establishes a new connection between semidefinite programs and quantum information, and generalizes prior work linking classical communication complexity to linear EFs. The
The document discusses Bhaskara II's treatise on Jyotpatti, the science of trigonometry in ancient India. It begins by providing context for the treatise as the last section of Bhaskara's comprehensive work Siddhanta Siromani. It then describes the graphical method used in ancient India to define and obtain values of trigonometric functions like jya (sine). Subsequently, it presents Bhaskara's mathematical method for calculating jya and other functions through formulas involving the radius of the reference circle. Key contributions of the treatise included the first accurate values for sines of particular angles derived through inscribing regular polygons in a circle.
A Multicover Nerve for Geometric InferenceDon Sheehy
We show that filtering the barycentric decomposition of a Cech complex by the cardinality of the vertices captures precisely the topology of k-covered regions among a collection of balls for all values of k.
Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.
Algebraic geometry and commutative algebraSpringer
This document provides an overview of the theory of Noetherian rings. It begins by defining Noetherian rings as rings whose ideals satisfy the ascending chain condition. It then discusses examples of Noetherian and non-Noetherian rings. A key tool in studying the structure of ideals in Noetherian rings is primary decomposition. The document explores primary decomposition and its properties, including uniqueness of associated prime ideals. It introduces Krull dimension and discusses its applications, such as proving that the dimension of a Noetherian local ring is finite. The document concludes by discussing related concepts like systems of parameters and regular local rings.
The document discusses two geometrical theorems about parabolas:
1) Focal chords theorem - It is proven that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix.
2) Reflection property theorem - Any line parallel to the axis of the parabola is reflected towards the focus, and the angle of incidence equals the angle of reflection. This is proven through several steps showing properties of the parabola and its tangents.
This document discusses solving locus problems by eliminating parameters from coordinate expressions. It outlines three types of locus problems based on the relationship between the x and y coordinates: 1) No parameters in x or y, 2) An obvious single-parameter relationship, 3) A non-obvious relationship requiring use of another proven relationship. Examples are provided for each type. The document also discusses finding the locus of the point where two tangents or normals to a parabola intersect.
This document presents two variations on the periscope theorem in geometry optics. It summarizes:
1) For a "spherical periscope" system of two mirrors that reflects rays emanating from a point back to that point, the vector field relating the incoming and outgoing ray directions is projectively gradient.
2) For a "reversed periscope" system reflecting upward rays downward, the local diffeomorphism relating the ray directions and the function describing the second mirror can be expressed in terms of the function for the first mirror.
3) In both cases, the document provides theorems characterizing the relationships between the mirror surfaces and ray mappings in terms of gradients and differential equations.
Curve generation %a1 v involute and evoluteTanuj Parikh
The document discusses evolutes and involutes of curves. It defines an evolute as the locus of the centers of curvature of a curve, and an involute as a curve that is traced by a point on a taut string unwinding from the evolute. Specifically:
- The evolute of a curve is the envelope of its normals, and the original curve is the involute of its evolute.
- Examples of evolutes include the semi-cubic parabola as the evolute of a parabola, and an equal spiral as the evolute of an equiangular spiral.
- Involutes can be used to generate gear teeth profiles, with the teeth profiles drawn as invol
This document provides information about conic sections, including definitions, types, important terms, properties, equations, and examples of parabolas and ellipses. Key points include:
- Conic sections are curves formed by the intersection of a plane and a right circular cone. The main types are parabolas, ellipses, and hyperbolas.
- Parabolas have an eccentricity of 1 and satisfy the property that the distance from the focus to any point is equal to the distance from that point to the directrix. Ellipses have an eccentricity less than 1.
- Important terms defined include focus, directrix, eccentricity, axes, and vertex. Standard and general equations are
The shortest distance between skew linesTarun Gehlot
The document discusses finding the angle and distance between two skew lines. It provides solutions for when a point and direction are given on each line, and for when the edges of a tetrahedron are known. The angle can be found using the dot product of the lines' direction vectors. The distance is the projection of the vector between points onto the cross product of the direction vectors. For a tetrahedron, the angle and distance are related to the lengths of opposite edges and the tetrahedron's volume.
- A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
- Important terms related to circles include chord, diameter, arc, sector, minor/major segments.
- A tangent touches the circle at one point, a secant intersects at two points, and there can be at most two parallel tangents for a given secant.
- The tangent radius theorem states that the tangent is perpendicular to the radius at the point of contact. The equal tangent lengths theorem says tangents from an external point are equal in length.
This document provides instructions for the 28th Indian National Mathematical Olympiad exam to be held on February 03, 2013. It states that calculators and protractors are not allowed, but rulers and compasses are. It includes 6 multi-part math problems to be solved on separate pages with clear numbering. The problems cover topics like properties of circles touching externally, positive integer solutions to equations, properties of polynomial equations, subsets with integer mean averages, relationships between areas of triangles formed by triangle centers, and inequalities relating positive real numbers.
1) The document discusses phonons, which are quantized lattice vibrations in crystals that carry thermal energy. It describes modeling crystal vibrations using a harmonic lattice approach.
2) Normal modes of the lattice vibrations can be described as a set of independent harmonic oscillators. Quantum mechanically, these normal modes are quantized as phonons with discrete energy levels.
3) Phonons can be thought of as quasiparticles that carry momentum and energy in the crystal lattice. Their propagation is described using a phonon field approach rather than independent normal modes.
1) The document discusses phonons, which are quantized lattice vibrations in crystals that carry thermal energy. It describes modeling crystal vibrations using a harmonic lattice approach.
2) Normal modes of the lattice vibrations can be described as a set of independent harmonic oscillators. Quantum mechanically, these normal modes are quantized as phonons with discrete energy levels proportional to their frequency.
3) Phonons can be thought of as quasiparticles that carry momentum and thermal energy within the crystal lattice. Their propagation is described using a phonon field model rather than independent normal modes.
Correspondence analysis is a technique for approximating a contingency table with lower rank tables to analyze the relationship between two categorical variables. It works by finding pairs of correspondence factors that have unit variance with respect to the marginal distributions and are maximally correlated. The correspondence factors and their correlations are obtained from the singular value decomposition of a normalized contingency table. Hypothesis tests can then be conducted to test the independence of the categorical variables and how well a lower rank approximation fits the data. The analysis also provides a spatial representation of the row and column categories in lower dimensions.
This document discusses different coordinate systems used to describe points in two-dimensional and three-dimensional spaces, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems, and gives examples of performing these conversions as well as writing equations of basic geometric shapes in different coordinate systems.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
This document discusses various types of engineering curves defined by the motion of a point or object along a path. It provides definitions and examples of involutes, cycloids, trochoids, spirals and helices. Methods for drawing tangents and normals to these curves are also mentioned. Specific problems are given to illustrate how to draw different types of curves step-by-step, including involutes with various string lengths, composite pole shapes, loci of rod ends rolling on a semicircular pole, standard and superior/inferior cycloids and trochoids, and epicycloids and hypocycloids defined by a smaller circle rolling on the outside or inside of a larger curved path, respectively.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
The document discusses different types of polygons including simple, convex, and star-shaped polygons. It then presents the convexification algorithm, which takes a star-shaped polygon and transforms it into a convex polygon through a series of edge swaps. The algorithm works by traversing the polygon and swapping any edges that form a left-hand turn. A proof is provided that this algorithm will always terminate with a convex polygon after at most n(n-1)/2 swaps, where n is the number of sides in the original polygon.
1) The document discusses concepts in planar projective geometry including planar twists, wrenches, and their addition. It shows that planar twists and wrenches can be represented by points and lines in a plane.
2) The addition of planar twists and wrenches follows graphical rules. The addition of two rotations results in a rotation located at a point determined proportionally to the original rotations. The addition of forces follows the parallelogram rule.
3) Invariants like the pole of a twist or the direction and distance of a wrench polar completely describe the motion or loading of a rigid body in the plane.
The document discusses orbital hybridization and bonding in methane, ethane, ethylene, and acetylene. It explains how promoting electrons from the carbon 2s orbital to the 2p orbitals allows for the formation of sp3, sp2, and sp hybrid orbitals. These hybrid orbitals overlap with hydrogen 1s orbitals to form sigma bonds. It also describes how unhybridized p orbitals overlap to form pi bonds in ethylene and acetylene, giving them double and triple bonds. The hybridization model provides insight into bonding and justifies the structures and bonding properties of these compounds.
The document discusses parabolas and their key properties:
- A parabola is the set of all points equidistant from a fixed line called the directrix and a fixed point called the focus.
- The standard equation of a parabola depends on the orientation of its axis and vertex.
- Key properties include the axis of symmetry, direction of opening, and the length of the latus rectum.
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 two geometrical theorems about parabolas:
1) Focal chords theorem - It is proven that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix.
2) Reflection property theorem - Any line parallel to the axis of the parabola is reflected towards the focus, and the angle of incidence equals the angle of reflection. This is proven through several steps showing properties of the parabola and its tangents.
This document discusses solving locus problems by eliminating parameters from coordinate expressions. It outlines three types of locus problems based on the relationship between the x and y coordinates: 1) No parameters in x or y, 2) An obvious single-parameter relationship, 3) A non-obvious relationship requiring use of another proven relationship. Examples are provided for each type. The document also discusses finding the locus of the point where two tangents or normals to a parabola intersect.
This document presents two variations on the periscope theorem in geometry optics. It summarizes:
1) For a "spherical periscope" system of two mirrors that reflects rays emanating from a point back to that point, the vector field relating the incoming and outgoing ray directions is projectively gradient.
2) For a "reversed periscope" system reflecting upward rays downward, the local diffeomorphism relating the ray directions and the function describing the second mirror can be expressed in terms of the function for the first mirror.
3) In both cases, the document provides theorems characterizing the relationships between the mirror surfaces and ray mappings in terms of gradients and differential equations.
Curve generation %a1 v involute and evoluteTanuj Parikh
The document discusses evolutes and involutes of curves. It defines an evolute as the locus of the centers of curvature of a curve, and an involute as a curve that is traced by a point on a taut string unwinding from the evolute. Specifically:
- The evolute of a curve is the envelope of its normals, and the original curve is the involute of its evolute.
- Examples of evolutes include the semi-cubic parabola as the evolute of a parabola, and an equal spiral as the evolute of an equiangular spiral.
- Involutes can be used to generate gear teeth profiles, with the teeth profiles drawn as invol
This document provides information about conic sections, including definitions, types, important terms, properties, equations, and examples of parabolas and ellipses. Key points include:
- Conic sections are curves formed by the intersection of a plane and a right circular cone. The main types are parabolas, ellipses, and hyperbolas.
- Parabolas have an eccentricity of 1 and satisfy the property that the distance from the focus to any point is equal to the distance from that point to the directrix. Ellipses have an eccentricity less than 1.
- Important terms defined include focus, directrix, eccentricity, axes, and vertex. Standard and general equations are
The shortest distance between skew linesTarun Gehlot
The document discusses finding the angle and distance between two skew lines. It provides solutions for when a point and direction are given on each line, and for when the edges of a tetrahedron are known. The angle can be found using the dot product of the lines' direction vectors. The distance is the projection of the vector between points onto the cross product of the direction vectors. For a tetrahedron, the angle and distance are related to the lengths of opposite edges and the tetrahedron's volume.
- A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
- Important terms related to circles include chord, diameter, arc, sector, minor/major segments.
- A tangent touches the circle at one point, a secant intersects at two points, and there can be at most two parallel tangents for a given secant.
- The tangent radius theorem states that the tangent is perpendicular to the radius at the point of contact. The equal tangent lengths theorem says tangents from an external point are equal in length.
This document provides instructions for the 28th Indian National Mathematical Olympiad exam to be held on February 03, 2013. It states that calculators and protractors are not allowed, but rulers and compasses are. It includes 6 multi-part math problems to be solved on separate pages with clear numbering. The problems cover topics like properties of circles touching externally, positive integer solutions to equations, properties of polynomial equations, subsets with integer mean averages, relationships between areas of triangles formed by triangle centers, and inequalities relating positive real numbers.
1) The document discusses phonons, which are quantized lattice vibrations in crystals that carry thermal energy. It describes modeling crystal vibrations using a harmonic lattice approach.
2) Normal modes of the lattice vibrations can be described as a set of independent harmonic oscillators. Quantum mechanically, these normal modes are quantized as phonons with discrete energy levels.
3) Phonons can be thought of as quasiparticles that carry momentum and energy in the crystal lattice. Their propagation is described using a phonon field approach rather than independent normal modes.
1) The document discusses phonons, which are quantized lattice vibrations in crystals that carry thermal energy. It describes modeling crystal vibrations using a harmonic lattice approach.
2) Normal modes of the lattice vibrations can be described as a set of independent harmonic oscillators. Quantum mechanically, these normal modes are quantized as phonons with discrete energy levels proportional to their frequency.
3) Phonons can be thought of as quasiparticles that carry momentum and thermal energy within the crystal lattice. Their propagation is described using a phonon field model rather than independent normal modes.
Correspondence analysis is a technique for approximating a contingency table with lower rank tables to analyze the relationship between two categorical variables. It works by finding pairs of correspondence factors that have unit variance with respect to the marginal distributions and are maximally correlated. The correspondence factors and their correlations are obtained from the singular value decomposition of a normalized contingency table. Hypothesis tests can then be conducted to test the independence of the categorical variables and how well a lower rank approximation fits the data. The analysis also provides a spatial representation of the row and column categories in lower dimensions.
This document discusses different coordinate systems used to describe points in two-dimensional and three-dimensional spaces, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems, and gives examples of performing these conversions as well as writing equations of basic geometric shapes in different coordinate systems.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
This document discusses various types of engineering curves defined by the motion of a point or object along a path. It provides definitions and examples of involutes, cycloids, trochoids, spirals and helices. Methods for drawing tangents and normals to these curves are also mentioned. Specific problems are given to illustrate how to draw different types of curves step-by-step, including involutes with various string lengths, composite pole shapes, loci of rod ends rolling on a semicircular pole, standard and superior/inferior cycloids and trochoids, and epicycloids and hypocycloids defined by a smaller circle rolling on the outside or inside of a larger curved path, respectively.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
The document discusses different types of polygons including simple, convex, and star-shaped polygons. It then presents the convexification algorithm, which takes a star-shaped polygon and transforms it into a convex polygon through a series of edge swaps. The algorithm works by traversing the polygon and swapping any edges that form a left-hand turn. A proof is provided that this algorithm will always terminate with a convex polygon after at most n(n-1)/2 swaps, where n is the number of sides in the original polygon.
1) The document discusses concepts in planar projective geometry including planar twists, wrenches, and their addition. It shows that planar twists and wrenches can be represented by points and lines in a plane.
2) The addition of planar twists and wrenches follows graphical rules. The addition of two rotations results in a rotation located at a point determined proportionally to the original rotations. The addition of forces follows the parallelogram rule.
3) Invariants like the pole of a twist or the direction and distance of a wrench polar completely describe the motion or loading of a rigid body in the plane.
The document discusses orbital hybridization and bonding in methane, ethane, ethylene, and acetylene. It explains how promoting electrons from the carbon 2s orbital to the 2p orbitals allows for the formation of sp3, sp2, and sp hybrid orbitals. These hybrid orbitals overlap with hydrogen 1s orbitals to form sigma bonds. It also describes how unhybridized p orbitals overlap to form pi bonds in ethylene and acetylene, giving them double and triple bonds. The hybridization model provides insight into bonding and justifies the structures and bonding properties of these compounds.
The document discusses parabolas and their key properties:
- A parabola is the set of all points equidistant from a fixed line called the directrix and a fixed point called the focus.
- The standard equation of a parabola depends on the orientation of its axis and vertex.
- Key properties include the axis of symmetry, direction of opening, and the length of the latus rectum.
Similar to 11 x1 t11 08 geometrical theorems (2012) (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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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 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 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
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
3. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
4. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
5. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
6. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
pq 1
7. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
pq 1
Tangents are perpendicular to each other
8. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
pq 1
Tangents are perpendicular to each other
3 Show that the point of intersection,T , of the tangents is
a p q , apq
9. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
pq 1
Tangents are perpendicular to each other
3 Show that the point of intersection,T , of the tangents is
a p q , apq y apq
10. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
pq 1
Tangents are perpendicular to each other
3 Show that the point of intersection,T , of the tangents is
a p q , apq y apq
y a pq 1
11. Geometrical Theorems about
(1) Focal Chords
Parabola
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove pq 1
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
pq 1
Tangents are perpendicular to each other
3 Show that the point of intersection,T , of the tangents is
a p q , apq y apq
y a pq 1
Tangents meet on the directrix
13. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
14. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
15. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
Prove: SPK CPB
16. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
Prove: SPK CPB
(angle of incidence = angle of reflection)
17. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
Prove: SPK CPB
(angle of incidence = angle of reflection)
Data: CP || y axis
18. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
Prove: SPK CPB
(angle of incidence = angle of reflection)
Data: CP || y axis
1 Show tangent at P is y px ap 2
19. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
Prove: SPK CPB
(angle of incidence = angle of reflection)
Data: CP || y axis
1 Show tangent at P is y px ap 2
2 tangent meets y axis when x = 0
20. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
Prove: SPK CPB
(angle of incidence = angle of reflection)
Data: CP || y axis
1 Show tangent at P is y px ap 2
2 tangent meets y axis when x = 0
K is 0,ap 2
21. (2) Reflection Property
Any line parallel to the axis of the parabola is reflected towards the
focus.
Any line from the focus parallel to the axis of the parabola is reflected
parallel to the axis.
Thus a line and its reflection are equally inclined to the normal, as well
as to the tangent.
Prove: SPK CPB
(angle of incidence = angle of reflection)
Data: CP || y axis
1 Show tangent at P is y px ap 2
2 tangent meets y axis when x = 0
K is 0,ap 2
d SK a ap 2
23. 2ap 0 ap a
2
d PS
2 2
4a 2 p 2 a 2 p 4 2a 2 p 2 a 2
a p4 2 p2 1
p 1
2
a 2
a p 2 1
24. 2ap 0 ap a
2
d PS
2 2
4a 2 p 2 a 2 p 4 2a 2 p 2 a 2
a p4 2 p2 1
p 1
2
a 2
a p 2 1 d SK
25. 2ap 0 ap a
2
d PS
2 2
4a 2 p 2 a 2 p 4 2a 2 p 2 a 2
a p4 2 p2 1
p 1
2
a 2
a p 2 1 d SK
SPK is isosceles two = sides
26. 2ap 0 ap a
2
d PS
2 2
4a 2 p 2 a 2 p 4 2a 2 p 2 a 2
a p4 2 p2 1
p 1
2
a 2
a p 2 1 d SK
SPK is isosceles two = sides
SPK SKP (base 's isosceles )
27. 2ap 0 ap a
2
d PS
2 2
4a 2 p 2 a 2 p 4 2a 2 p 2 a 2
a p4 2 p2 1
p 1
2
a 2
a p 2 1 d SK
SPK is isosceles two = sides
SPK SKP (base 's isosceles )
SKP CPB (corresponding 's , SK || CP)
28. 2ap 0 ap a
2
d PS
2 2
4a 2 p 2 a 2 p 4 2a 2 p 2 a 2
a p4 2 p2 1
p 1
2
a 2
a p 2 1 d SK
SPK is isosceles two = sides
SPK SKP (base 's isosceles )
SKP CPB (corresponding 's , SK || CP)
SPK CPB
29. 2ap 0 ap a
2
d PS
2 2
4a 2 p 2 a 2 p 4 2a 2 p 2 a 2
a p4 2 p2 1
p 1
2
a 2
a p 2 1 d SK
SPK is isosceles two = sides
SPK SKP (base 's isosceles )
SKP CPB (corresponding 's , SK || CP)
SPK CPB
Exercise 9I; 1, 2, 4, 7, 11, 12, 17, 18, 21