Taylor Series Approximation

1. Taylor Series Approximation Taylor series, Order of approximation, Newton�s method.

2. Orders of approximation In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order approximation, a second order approximation, and so forth. Formally, an nth order approximation is one where the order of magnitude of the error is at most n; in terms of big O notation, with error bounded by O(xn). In suitable circumstances, approximating a function by a Taylor polynomial of degree n yields an nth order approximation, by Taylor's theorem: a first order approximation is a linear approximation, and so forth.

3. Usage in science and engineering Zeroth-order approximation (also 0th order) is the term scientists use for a first educated guess at an answer. Many simplifying assumptions are made, and when a number is needed, an order of magnitude answer (or zero significant figures) is often given. For example, you might say "the town has a few thousand residents", when it has 3,914 people in actuality. This is also sometimes referred to as an order of magnitude approximation. A zeroth-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be constant, or a flat line with no slope: a polynomial of degree 0. First-order approximation (also 1st order) is the term scientists use for a further educated guess at an answer. Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4,000 residents"). A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1.

4. Second-order approximation (also 2nd order) is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has 3,900 residents") is generally given. A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a quadratic polynomial, geometrically, a parabola: a polynomial of degree 2. While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number. A third-order approximation would be required to fit four data points, and so on. These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (eg., "Of course the rotation of the earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect.

5. Taylor series As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sinx (in black) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).