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Figure 12-12 Illustration for Problem 5
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Call the independent variable x and the dependent variable y Call the relation f Determine f (x) and f 1(x) mathematically State them both using relation notation 6 What is the real-number domain of the relation f (x) that you determined when you solved Problem 5 What is its real-number range 7 Sketch a graph of the inverse relation you found when you solved Problem 5 What is its real-number domain What is its real-number range 8 The relation described and graphed in Problem 5 can be modified by restricting its domain to the set of reals greater than or equal to 2 Show graphically, by means of the vertical-line test, that this restriction makes the inverse f 1 into a function 9 The relation described and graphed in Problem 5 can be modified by restricting its domain to the set of reals smaller than or equal to 2 Show graphically, by means of the vertical-line test, that this restriction makes the inverse f 1 into a function 10 The relation described and graphed in Problem 5 can be modified by restricting its range to the set of nonnegative reals Show graphically, and by means of verticalline tests, that this restriction makes f into a function, but does not make f 1 into a function
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CHAPTER
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In this chapter, we ll learn the fundamental properties of curves called conic sections These curves include the circle, the ellipse, the parabola, and the hyperbola The conic sections can always be represented in the Cartesian plane as equations that contain the squares of one or both variables
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Imagine a double right circular cone with a vertical axis that extends infinitely upward and downward Also imagine a flat, infinitely large plane that can be moved around so that it slices through the double cone in various ways, as shown in Fig 13-1 The intersection between the plane and the double cone is always a circle, an ellipse, a parabola, or a hyperbola, as long as the plane doesn t pass through the point where the apexes of the cones meet
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Geometry of a circle and an ellipse Figure 13-1A shows what happens when the plane is perpendicular to the axis of the double cone In that case, we get a circle In Fig 13-1B, the plane is not perpendicular to the axis of the cone, but it isn t tilted very much The curve is closed, but it isn t a perfect circle Instead, it s an elongated circle or ellipse Geometry of a parabola As the plane tilts farther away from a right angle with respect to the double-cone axis, the ellipse becomes increasingly elongated Eventually, we reach an angle of tilt where the curve is no longer closed At precisely this threshold angle, the intersection between the plane and the cone is a parabola (Fig 13-1C) Geometry of a hyperbola So far, the plane has only intersected one half of the double cone If we tilt the plane beyond the angle at which the intersection curve is a parabola, the plane intersects both halves of the cone In that case, we get a hyperbola If we tilt the plane as far as possible so that it becomes parallel to the cone s axis, we still get a hyperbola (Fig 13-1D)
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Conic Sections
Double circular cone
Flat plane
Double circular cone Flat plane
Flat plane
Flat plane
Double circular cone
Double circular cone
Figure 13-1
The conic sections can be defined by the intersection of a flat plane with a double right circular cone At A, a circle At B, an ellipse At C, a parabola At D, a hyperbola
Are you confused
You might ask, We haven t mentioned the flare angle of the double cone (the measure of the angle between the axis of the cone and its surface) Does the size of this angle make any difference Quantitatively, it does As the flare angle increases (the cones become fatter ), we get ellipses less often and hyperbolas more often As the flare angle decreases (the cones get slimmer ), we obtain ellipses more often and hyperbolas less often However, we can always get a circle, an ellipse, a parabola, or a hyperbola by manipulating the plane to the desired angle, regardless of the flare angle
Here s a challenge!
Imagine that you re standing on a frozen lake at night, holding a flashlight that throws a coneshaped beam with a flare angle of p /10; in other words, the outer face of the light cone subtends an angle of p /10 with respect to the beam center How can you aim the flashlight so that the edge of the light cone forms a circle on the ice An ellipse A parabola
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