360 numerical differentiation

(7 + 0.1)2– 72

–––––––

–

0.1

1

–

6

2

–––

2√

–

36

h

h

h

h

their solutions. For example, the classification of the

P

YTHAGOREAN TRIPLES

would be considered a problem

in elementary number theory, as would the solution of

many D

IOPHANTINE EQUATION

s. (The use of the word

elementary here by no means implies that the level of

mathematical sophistication used is elementary.)

ANA

-

LYTIC NUMBER THEORY

incorporates the notion of

LIMIT

in the study of numbers, and algebraic number

theory extends the study of number theory to a general

study of

ALGEBRAIC NUMBER

s and new number systems

that include solutions to otherwise unsolvable algebraic

equations.

See also

ABSTRACT ALGEBRA

; C

ATALAN CONJEC

-

TURE

; C

OLLATZ

’

S CONJECTURE

; E

UCLID

’

S PROOF OF THE

INFINITUDE OF PRIMES

; E

UCLIDEAN ALGORITHM

;

FUN

-

DAMENTAL THEOREM OF ARITHMETIC

; P

EANO

’

S POSTU

-

LATES

;

PRIME

;

PRIME

-

NUMBER THEOREM

.

numerical differentiation The

DERIVATIVE

of a

function f(x) can be well approximated as a “Newton

quotient”:

at least for small values of h. Any use of this formula to

approximate the value of a derivative is called numeri-

cal differentiation. For example, we can approximate

the derivative of f(x) = x2at x= 7 simply as f′(7) ≈

= 14.1.

Rewriting the formula for the Newton quotient

gives:

f(x+ h) ≈f(x) + hf ′(x)

If the derivative of the function is known, then this for-

mula can be used to approximate values of f. For exam-

ple, to estimate square roots, set f(x) = √

–

xto obtain:

Thus √

–

38, for example, is approximately √

–

36 +

= 6 + ≈6.167.

The second derivative of a function is well

approximated by the quotient:

This follows using the approximation f″(x) ≈

, with and

f′(x) ≈.

See also N

EWTON

’

S METHOD

.

numerical integration According to the theory of

INTEGRAL CALCULUS

, the numerical value of a definite

integral ∫b

af(x)dx is determined by finding an antideriva-

tive F(x) to the integrand f(x) and then computing the

quantity F(b) – F(a).Although theoretically sound, it

is rare in real-world applications that such a proce-

dure can ever be completed. There are two possible

complications:

1. An antiderivative to the integrand cannot be found.

(Consider the integral ∫2

1dx, for instance.)

2. The function f(x) might not be completely specified.

(In performing an experiment, one can only ever

record a finite number of data values, in which case

the values of a function f(x) are known only at a

finite number of points.)

Nonetheless, despite these limitations, scientists

and engineers often still require a numerical value for

the area under the curve y= f(x),at least to some speci-

fied degree of accuracy. Numerical integration is any

technique that allows one to find an approximate value

for a definite integral ∫b

af(x)dx. There are two elemen-

tary methods currently in use:

1. Trapezoidal Rule (also known as the trapezium

rule):Divide the interval [a,b] into n+ 1 equally

spaced points a= x0, x1,…xn–1, xn= b. For conve-

nience denote f(xi) by fiand let Pidenote the point

(xi, fi) on the curve above x= xi. The straight-line

segment connecting Pito Pi+1 can be used as an

approximation for the curve y= f(x) between and xi

and xi+1. The area under this part of the curve is

thus approximately the area of a trapezoid of width

h= , left edge of height fiand right edge of

height fi+1. This area is given by: h(fi+ fi+1).

1

–

2

b– a

–––

n

ex

–

x