NUMERICAL METHODS FOR ENGINEERING: MATHEMATICAL APPROXIMATION

EXAMPLES

1.ACCURACY AND PRECISION:

A student measures 1.73 meters in the class are asked to five students measure this fellow with a tape measure, these were the results of the 5:

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Student 1: 1.72 m

Student 2: 1.72 m

Student 3: 1.72 m

Student 4: 1.72 m

Student 5: 1.72 m

The measured values were accurate, but not exact, since the student's actual height is 1.73 m.

2.SIGNIFICANT FIGURES:

Suppose we want to measure a piece of aluminum, a balance capable of measuring up to 0.0001g. Then we could report the mass of aluminum 2.2405 + - 0. 0001 g, the notation plus / minus expresses the measurement uncertainty.

In scientific notation is omitted plus / minus as it is understood that there is an uncertainty of at least one unit on the last digit of the measured quantity. Then: the measured quantities are usually reported in such a way that only the last digit is uncertain.

What is the difference between 4.0 g and 4.00 g? Answer: Many people would say no difference, but a scientist would notice the difference in the number of significant figures in the two measurements. The value 4.0 has two significant figures while 4.00 has three, this means that the measurement with two figures is more uncertain than the second.

The mass of 4.0indica that the value is between 3.9 - 4.1 g.
In contrast, the mass of 4.00 indicates that the value is between 3.99 - 4.01g.
The more significant figures to have a truer measurement is measurement.

3. ROUNDING:

2.4 2.36105 Rounding up the tenth.
2.36 2.36105 rounding up the decimal.
2361 2.36105 Rounding up to the thousandths.

TAYLOR SERIES

In mathematics, the Taylor series of a function f (x) infinitely differentiable (real or complex) defined on an open interval (ar, a + r) is defined as the following sum:

sin (x) and Taylor approximations centered at 0, with polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
Here, n! NYF is the factorial (n) (a) indicates the nth derivative of f in a.

If this series converges for all x belonging to the interval (ar, a + r) and the sum is equal to f (x), then the function f (x) is called analytic. To check whether the series converges to f (x), is often used an estimate of the rest of Taylor's theorem. A function is analytic if and only if it can be represented by a power series, the coefficients of this series are necessarily determined in the formula for the Taylor series.

If a = 0, the series is called Maclaurin series.

This representation has three important advantages:

The derivation and integration of these series can be performed term by term, which are trivial operations.
It can be used to calculate approximate values of the function.
It can be shown that the feasibility of the transformation of a function to a Taylor series, is the best possible approximation.
Some functions can not be written as Taylor series because they have some uniqueness. In these cases you can usually get a series expansion using negative powers of x (see Laurent series. For example f (x) = exp (-1 / x ²) can be expanded as Laurent series.

The Taylor series of a function f of real numbers or complex that is infinitely differentiable in an environment of real or complex numbers, is the power series:

it can be written in a more compact as

where n! NYF is the factorial (n) (a) denotes the nth derivative of f at the point, the zero derivative of f is defined as the self-fy (x - a) 0 and 0! are both defined as one.

fuente:

Kline, M. (1990) Mathematical Thought from Ancient to Modern Times. Oxford University Press. pp. 35-37.
Boyer, C. and Merzbach, U. (1991) A History of Mathematics. John Wiley and Sons. pp. 202-203.

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