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CURVE SKETCHING. EXAM QUESTIONS. Question 1 (**) f ( x ) = 2 x + 6 x + 10 , x ∈ . ( a) Express f. where. b) Describe. x ) in the form . ( ) = ( x + a )2 + b , and b are integers. geometrically the transformations which map the graph of x the graph of. ( x ) . = , 3 b. translation by. 2 onto. − 3 1 . 2 (**+)
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this recipe will be justified. In order to make the recipe plausible, we shall begin by looking at the problem of how to approximate a given curve by polygons. The first question we must answer, however, is more fundamental: How are curves to be described in the first place? In this course the answer will usually be in terms of a ...
2.1 What is a curve? Recall the examples of curves you have seen in \Methods of Applied Maths": y= 2x+ 1 x2 + y2 = 1 y= x2 These examples are curves given by a Cartesian equation: f(x;y) = c where f: R2!R is a function in 2 variables: Put di erently, the curve is given as a level set C= f(x;y) 2R2 jf(x;y) = cg of the di erentiable function f.
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1. Curvature measures how quickly a curve turns, or more precisely how quickly the unit tangent vector turns. 1.1. Curvature for arc length parametrized curves. Consider a curve (s): ( ; ) 7!R3. Then the unit tangent vector of (s) is given by T (s) := _(s). Consequently, how quickly T (s) turns can be characterized by the number.
2.3 Geometry of curves: arclength, curvature, torsion Overview: The geometry of curves in space is described independently of how the curve is parameterized. The key notion of curvature measures how rapidly the curve is bending in space. In 3-D, an additional quantity, tor-sion, describes how much the curve is wobbling out of a plane. Alternative
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For each problem, find the area of the region enclosed by the curves. You may use the provided graph to sketch the curves and shade the enclosed region. 5) y = −2x2 − 1 y = −x + 3 x = 0 x = 1 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 6) y = 2 3 x2 y = x x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 7) y ...
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The rough idea is to imagine a curve as a roller-coaster along which you travel at a constant speed; the curvature is then the force necessary to keep you travelling along the curve. Curvature is a more difficult concept for surfaces.
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