COUNT; MONTE CARLO METHOD; MONTY HALL PROBLEM; ODDS; RANDOM WALK; STATISTICS; TWO-CARD
PUZZLE.
probability density function See DISTRIBUTION.
Proclus See EUCLID’S POSTULATES.
product rule The DERIVATIVE of the product of two
functions f and g is given by the product rule:
For example, we have
(xsinx) = 1 . sinx + x . cosx =
sinx + xcosx. The rule can be proved using the limit
definition of the derivative as follows:
Alternatively, one can recognize that the quantity
f(x + h) · g(x + h) is the formula for the area of a rectangle, one that contains the smaller rectangle of area f(x) ·
g(x). Writing a formula for the area of the L-shaped
region between the two rectangles, dividing by h, and
taking the limit as h becomes small leads to the same
formula for the product rule.
The product rule can be generalized to apply to
any finite product of functions. For example, for the
product of three functions we have:
(f(x) · g(x) · h(x))′ = f′(x) · g(x) · h(x) + f(x) · g′(x) · h(x)
+ f(x) · g(x) · h′(x)
If one of the functions in a product is a constant k,
then the product rule shows:
(kf(x))′ = 0 · f(x) + k · f′(x) = kf′(x)
Two applications of the product rule give the second
derivative of a product of two functions:
(f(x) · g(x))′′ = f(x) · g′′(x) + 2f′(x) · g′(x) + f′′(x) · g(x)
In general, the nth derivative of a product of two
functions is a sum of products containing BINOMIAL
COEFFICIENTs:
This result is called Leibniz’s theorem.
See also CHAIN RULE; HIGHER DERIVATIVE; QUOTIENT RULE.
projection Any mapping of a geometric figure onto a
PLANE to produce a two-dimensional image of that figure is called a projection. For instance, the daytime
shadow cast by an outdoor object is an example of a
projection onto the ground. Since the Sun is a great distance from the Earth, rays of sunlight are essentially
PARALLEL, and shadows cast by it are parallel projections. Shadows cast by a single point of light, however,
such as the flame of a candle, have different shapes
than those cast by the sun. Such projections are called
central projections.
French mathematician and engineer GIRARD
DESARGUES (1591–1661) observed that the central projection of any CONIC SECTION is another conic section.
(For instance, the shape cast on the ground by the circular rim of a flashlight is an ELLIPSE.) This led him to
study those properties of geometric figures that remain
unchanged under central projections, thereby founding
the field of PROJECTIVE GEOMETRY.
In VECTOR analysis, the projection of a vector a
onto a vector b is a vector parallel to b whose length is
the length of the “shadow” cast by a if the two vectors
are placed at the same location in space and the “rays
of light” casting the shadow are parallel and PERPENDICULAR to b. Thus the projection of a onto b is a vector of the form xb, for some value x, with the property
that the vector connecting the tip of xb to the tip of a is
perpendicular to b. Using the DOT PRODUCT of vectors,
this yields the equation:
f x g x
n
k
f x g
x
n
k
nk
k
n
( ) ( )
( )
( )
( )
( )
(
)
⋅
(
) =
⎛
⎝
⎜
⎞
⎠
⎟
⋅
−
=
∑
0
d
dx
f x g x
f x h g x h f x g x
h
f x h g x h f x g x h f x g x h f x g x
h
f x h f x
h
g x
h
h
h
( ) ( ) lim
(
) (
) ( ) ( )
lim
(
) (
) ( ) (
) ( ) (
) ( ) ( )
lim
(
) ( ) (
⋅
=
+ ⋅
+ −
⋅
=
+ ⋅ + −
⋅ + +
⋅ + −
⋅
=
+ −
⋅
+
(
)
→
→
→
0
0
0
h h f x
g x h g x
h
f x g x f x g x
) ( )
(
) ( )
( ) ( ) ( ) ( )
+
⋅
+ −
= ′ ⋅
+
⋅ ′
⎛
⎝
⎜
⎞
⎠
⎟
d
––
dx
d
dx
f g
df
dx
g f
dg
dx
⋅
( ) = ⋅ + ⋅
418 probability density function
PUZZLE.
probability density function See DISTRIBUTION.
Proclus See EUCLID’S POSTULATES.
product rule The DERIVATIVE of the product of two
functions f and g is given by the product rule:
For example, we have
(xsinx) = 1 . sinx + x . cosx =
sinx + xcosx. The rule can be proved using the limit
definition of the derivative as follows:
Alternatively, one can recognize that the quantity
f(x + h) · g(x + h) is the formula for the area of a rectangle, one that contains the smaller rectangle of area f(x) ·
g(x). Writing a formula for the area of the L-shaped
region between the two rectangles, dividing by h, and
taking the limit as h becomes small leads to the same
formula for the product rule.
The product rule can be generalized to apply to
any finite product of functions. For example, for the
product of three functions we have:
(f(x) · g(x) · h(x))′ = f′(x) · g(x) · h(x) + f(x) · g′(x) · h(x)
+ f(x) · g(x) · h′(x)
If one of the functions in a product is a constant k,
then the product rule shows:
(kf(x))′ = 0 · f(x) + k · f′(x) = kf′(x)
Two applications of the product rule give the second
derivative of a product of two functions:
(f(x) · g(x))′′ = f(x) · g′′(x) + 2f′(x) · g′(x) + f′′(x) · g(x)
In general, the nth derivative of a product of two
functions is a sum of products containing BINOMIAL
COEFFICIENTs:
This result is called Leibniz’s theorem.
See also CHAIN RULE; HIGHER DERIVATIVE; QUOTIENT RULE.
projection Any mapping of a geometric figure onto a
PLANE to produce a two-dimensional image of that figure is called a projection. For instance, the daytime
shadow cast by an outdoor object is an example of a
projection onto the ground. Since the Sun is a great distance from the Earth, rays of sunlight are essentially
PARALLEL, and shadows cast by it are parallel projections. Shadows cast by a single point of light, however,
such as the flame of a candle, have different shapes
than those cast by the sun. Such projections are called
central projections.
French mathematician and engineer GIRARD
DESARGUES (1591–1661) observed that the central projection of any CONIC SECTION is another conic section.
(For instance, the shape cast on the ground by the circular rim of a flashlight is an ELLIPSE.) This led him to
study those properties of geometric figures that remain
unchanged under central projections, thereby founding
the field of PROJECTIVE GEOMETRY.
In VECTOR analysis, the projection of a vector a
onto a vector b is a vector parallel to b whose length is
the length of the “shadow” cast by a if the two vectors
are placed at the same location in space and the “rays
of light” casting the shadow are parallel and PERPENDICULAR to b. Thus the projection of a onto b is a vector of the form xb, for some value x, with the property
that the vector connecting the tip of xb to the tip of a is
perpendicular to b. Using the DOT PRODUCT of vectors,
this yields the equation:
f x g x
n
k
f x g
x
n
k
nk
k
n
( ) ( )
( )
( )
( )
( )
(
)
⋅
(
) =
⎛
⎝
⎜
⎞
⎠
⎟
⋅
−
=
∑
0
d
dx
f x g x
f x h g x h f x g x
h
f x h g x h f x g x h f x g x h f x g x
h
f x h f x
h
g x
h
h
h
( ) ( ) lim
(
) (
) ( ) ( )
lim
(
) (
) ( ) (
) ( ) (
) ( ) ( )
lim
(
) ( ) (
⋅
=
+ ⋅
+ −
⋅
=
+ ⋅ + −
⋅ + +
⋅ + −
⋅
=
+ −
⋅
+
(
)
→
→
→
0
0
0
h h f x
g x h g x
h
f x g x f x g x
) ( )
(
) ( )
( ) ( ) ( ) ( )
+
⋅
+ −
= ′ ⋅
+
⋅ ′
⎛
⎝
⎜
⎞
⎠
⎟
d
––
dx
d
dx
f g
df
dx
g f
dg
dx
⋅
( ) = ⋅ + ⋅
418 probability density function