418 probability density function

d

––

dx

COUNT

; M

ONTE

C

ARLO METHOD

; M

ONTY

H

ALL PROB

-

LEM

;

ODDS

;

RANDOM WALK

;

STATISTICS

;

TWO

-

CARD

PUZZLE

.

probability density function See

DISTRIBUTION

.

Proclus See E

UCLID

’

S POSTULATES

.

product rule The

DERIVATIVE

of the product of two

functions fand gis 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 rectan-

gle, 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 hbecomes 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

COEFFICIENT

s:

This result is called Leibniz’s theorem.

See also

CHAIN RULE

;

HIGHER DERIVATIVE

;

QUO

-

TIENT RULE

.

projection Any mapping of a geometric figure onto a

PLANE

to produce a two-dimensional image of that fig-

ure 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 dis-

tance from the Earth, rays of sunlight are essentially

PARALLEL

, and shadows cast by it are parallel projec-

tions. 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 G

IRARD

D

ESARGUES

(1591–1661) observed that the central pro-

jection of any

CONIC SECTION

is another conic section.

(For instance, the shape cast on the ground by the cir-

cular 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 bis a vector parallel to bwhose length is

the length of the “shadow” cast by aif the two vectors

are placed at the same location in space and the “rays

of light” casting the shadow are parallel and

PERPEN

-

DICULAR

to b. Thus the projection of aonto bis a vec-

tor of the form xb, for some value x, with the property

that the vector connecting the tip of xbto the tip of ais

perpendicular to b. Using the

DOT PRODUCT

of vectors,

this yields the equation: