CAPS LS 3

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A brief animated introduction to Curve Sketching highlighting fascination of curves from early days to modern times plus basic techniques of curve sketching

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11/18/2011 1 NA MA S TE 2011 On Behalf of Centre for Academic and Profession al Services And Nepal Mathematics Centre is pleased to welcome you all at CAPS Lecture Series --3

Where to Begin? and How to Begin?:

Where to Begin? and How to Begin? Curve Sketching ?

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SOMETHING NON-MATHEMATICAL AND SOMETHING MATHEMATICAL We begin with

Where We Come From?:

Where We Come From?

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The Milky Way or The spiral galaxy

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Hundreds of Thousands of Stars 200000000000 Our Galaxy STARS Age 13,600 ± 800 million years MILKY WAY WE COME FROM

Curves in Solar System:

Curves in Solar System

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8 T H E S O L A R S Y S T E M Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto (?) Age 4.560 million years Distance between the Earth and the Sun 149598000 km

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SOLAR SYSTEM Distance between the Earth and the Sun 149598000 km Born 4,560 million years ago

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LAURASIA TECHTONIC MOVEMENT GONDAWANALAND

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TECHTONIC MOVEMENT Formation of Continents

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12 EARTH Rotating Earth Born 4.5 billion years ago

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13 THE WORLD

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14 ANCIENT CIVILIZATIONS

WORLD’S MAIN CIVILIZATIONS:

15 WORLD’S MAIN CIVILIZATIONS

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16 INDUS CIVILIZATION

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17 INDUS CIVILIZATION

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INDUS CIVILIZATION

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Range x-axis y-axis

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20 WE COME FROM Nepal The land where a well developed number system existed as early as the beginning of the first millennium CE. KATHMANDU VALLEY :jlguM

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21 WE COME FROM NEPAL Nepal The land where a well developed number system existed as early as the beginning of the first millennium CE . 107 AD Maligaon Inscription

ALL TIME EXPERIENCES:

ALL TIME EXPERIENCES

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All Time Experiences

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All Time Experiences

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All Time Experiences

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All Time Experiences

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All Time Experiences

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All Time Experiences

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All Time Experiences

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Early Interests in Curves

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Bhimbetka rock painting (India) Stone Age Painting

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The Fighting Cats, which is about 10,000 years old, is in serious danger of crumbling to pieces (LibyanSahara-Prehistoric Art)

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Megalithic Art from Newgrange showing an early interest in curves (New Stone age)

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Paleolithic Bulls and Other Animals Crowd Calcite Walls (Lascaux, France )

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Primitive Abstract Rock Art Forms Most common cave and rock arts found in widely separated locations on different continents Small Circles, Specks, Clusters Large Circles and Lines Zig Zags , Wayv Lines, Concentric Circles, Broken Lines Honeycombs, Grids, Networks, Amorphous areas Wavy Lines, Stars, Lines, Rotations

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Three Headed Seal , Indus Civilization

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Harrapan Plate , Indus Civilization

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A plate from Nepal, the decoration show a saucer - like shape and a large headed humanoid . (ca.3000 B . C . )

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Kalachakra Mandala

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The Silk Road extending from southern and eastern Europe through Egypt , Somalia , the Arabian Peninsula , Iran , Central Asia , Ancient India , Java-Indonesia , and Vietnam until it reaches China . The land routes are red, and the water routes are blue .

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Great Wall of China

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Historic painting of Swayambhu Nath Temple and its vicinity (Artist unknown) Image by Daniel Wright

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Swayambhu Temple Today

Something Common:

Something Common

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Range x-axis y-axis Domain

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Slide 4 Intercepts on the Axes y-intercept x-intercept  

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No x-intercept

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Symmetry about an Axis

Is a butterfly symmetrical?:

Is a butterfly symmetrical ?

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O O - 

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Three zeros of a cubic function R a n g e y  - Domain x - axis y - axis O -3 2 -1 x  -  (0, -6) y-intercept -6 Domain, Range and Intercepts on the Axes

Poles and Holes:

Poles and Holes f ( a ) → ∞ Hole at x = a Asymptote : x = a Pole at x = a f(a) undfined a ( a, 2 a ) f ( x ) = 1/( x – a )

Poles and Holes:

Poles and Holes f ( a ) → ∞ Hole at x = a Asymptote : x = a Pole at x = a f (3) undfined

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x y O (0, 1) .7 2.7 - .7 -2.7 -2 2 -3 0 -  +  Symmetry f ( x ) = - f (- x )

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x y O (0, 1) x- intercepts: y = 0  x   .7 and  2.7 .7 2.7 - .7 -2.7 + + + + + + + + + + + + + + -2 2 -3 Slope > or = or < 0; Intervals of Increase/Decrease 0 y-intercept x = 0  y = 1 -  +  Slope = 0 at x   2, 0

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Maxima, minima and stationary points y x Maximum Stationary points dy dx =0 B D B is a local maximum D is a local minimum Terminology Minimum Points on the curve, where dy/dx=0, are called stationary points y = f ( x )

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Introduction to Curve sketching In geometry , curve sketching ( or curve tracing ) includes techniques that can used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot . It is an application of the theory of curves to find their main features .

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Introduction to Curve sketching Ancient peoples of Babylonia, China, Egypt and Indian subcontinent are known to have a very good knowledge of various geometric shapes (and hence those of various kinds of curves) and their characteristics. The knowledge of various geometric shapes and their drawing can be considered as the precursors of what we now call the graph of a function and drawing or sketching of a curve.

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Introduction to Curve sketching Graphs are commonly used for many purposes. Usually, drawing graphs or sketching or tracing of curves virtually involves plotting of lot of points in the Cartesian plane. Such a graph or curve provides a very good pictorial view of the nature or various characteristics of what we call function. Simple algebraic techniques and methods of calculus are available for sketching graphs of functions. Although using such techniques seem to be somewhat tedious in an era of ST, students of mathematics are expected to know and use the traditional practice of sketching the curve or graphing a given function

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Different disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context . However many of these meanings are special instances of the definition which follows .

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A curve is a topological space which is locally homeomorphic to a line . In every day language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation . A simple example of a curve is the parabola , shown to the right . A large number of other curves have been studied in multiple mathematical fields

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In Mathematics , a curve is, generally speaking, an object similar to a Line (geometry) but which is not required to be straight . Often curves in two - dimensional ( Plane curves ) or three - dimensional ( Space curves ) Euclidean space are of interest .

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Different disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context . A simple example of a curve is the Parabola , shown to the right . A large number of other curves have been studied in multiple mathematical fields .

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The term curve has several meanings in non - mathematical language as well . For example, it can be almost synonymous with Mathematical function ( as in Learning curve ) , or Graph of a function ( as in Phillips curve ).

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Fascination with curves began long before they were the subject of mathematical study . This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times . [1] Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach . Historically, the term 'line' was used in place of the more modern term 'curve '

Fascination of Curves in Ancient Times:

Fascination of Curves in Ancient Times Man got fascinated with curves long before they became the subject of mathematical study . This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times . [1] Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach . Path of snake on a desert

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Curve Sketching Region of existence ( domain and range) Passage through certain points on the x-axis and y-axis ( x- and y- intercepts) Symmetry with respect to certain lines ( lines parallel to the axes, slanting line, and origin) and as decided by odd or even or periodic nature of the function Sketching a curve is just having a qualitatively good looking representation that will help visualize certain behaviour such as

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Curve Sketching 4 Closeness or nearness to certain line or lines but not meeting the line (Asymptote), Rise and fall, (increasing or decreasing nature of the function - monotonocity) Extremum value (maximum or minimum value, point where the slope of the tangent is zero and switches from increasing to decreasing or vice versa),

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Curve Sketching 7. Changes in concavity (Changes form CU to CD or vice versa – inflection) 8. Sketch (free-hand drawing or putting together some or all of the above information).

D I S A I M I S:

O M A I N N T E R C E P T S Y M M E T R Y S Y M P T O T E S N T E R V A L S A X M I N N F L E C T I O N K E T C H D I S A I M I S Basic elements of Curve sketching (In A Nutshell )

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x y O (0, 1) Step Two x- intercepts: y = 0  x   .7 and  2.7 Range: x = ±   y =  .7 2.7 - .7 -2.7 CURVE SKETCHING ? + + + + + + + + + + + + + + -2 2 -3 Step One Domain: x : -  to +  0 y-intercept x = 0  y = 1 -  + 

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Thanks Bye

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