Greenwood press

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.

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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:


jarapplet.html>
River crossing–swimming angles:

Robot navigation angle:

Sailing strategies:

Snell’s law:




Throwing a boomerang:


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6
ANGLE

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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