Greenwood press
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book-20600
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- Exponential Decay
ANGLE
5 Horizontal distance (meters) trav- eled by a golf ball hit at 70 meters per second at an angle of α radians. The angle of light rays will change after hitting a different surface, such as water. Snell’s law can be used to determine the angle of refraction, or the angle in which light bends as it passes through a new surface. Gems such as diamonds have a high index of refraction, allowing them to trap light and reflect it internally, which consequently makes them sparkle. online sources for further exploration The best angle to view a baseball game: <http://forum.swarthmore.edu/pow/solutio65.html> Diamond design: Finding your way with map and compass: The mathematics of rainbows: Navigation problems: Photography angles: Projectile motion simulations: River crossing–swimming angles: Robot navigation angle: Sailing strategies: Snell’s law: Throwing a boomerang: ▲ ▼ ▲ 6 ANGLE ASYMPTOTE An asymptote is an imaginary line or curve that a function approaches as its independent variable approaches infinity or an undefined value. A vertical asymptote of x = c exists on a function f(x) if at a point of discontinuity, x = c, the limit of f(x) as x approaches c equals positive or negative infinity. A horizon- tal asymptote of y = k exists on a function f(x) if the limit of f(x) as x approaches positive or negative infinity equals k. For example, the function f (x) = x−2 x+3 has a vertical asymptote at x = −3 because lim x→−3 x−2 x+3 = ±∞ and a horizontal asymptote at y = 1 because lim x→±∞ x−2 x+3 = 1. In the real world, horizontal asymptotes typically represent a leveling-off effect, such as the radioactive decay of a particle diminishing until it is almost gone. (See Exponential Decay.) If the dependent variable y is the amount of the particle, then in this case there would be a horizontal asymptote of y = 0 on the graph, because the amount of the particle approaches zero. Most substances that have a decaying effect, such as the amount of power supply in a battery, will have an asymptote of y = 0 on a graph that describes its amount as a function of time. The cooling of hot liquids in a mug, such as coffee, illustrates asymptotic behavior because the liquid gradually approaches room temperature after sitting awhile in the cup. The warming of liquids, such as ice sitting in a cup, demon- strates a similar phenomenon, except that the temperature graph rises towards the Download 1.81 Mb. Do'stlaringiz bilan baham: |
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