CAL-04

Taylor polynomial of degree 2

T2(x) = f + f' h + ½ f'' h², h = x − a.

Reading speed

SeriesTaylor

Governing equation

T_2(a+h)=f+f'h+\tfrac12 f'' h^2

where

f: f(a) (—)
f': f'(a) (—)
f'': f''(a) (—)
h: Step h (—)
T_2: Taylor value (—)

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-04 — Taylor polynomial of degree 2) is the form associated with Taylor. Working symbols: $f$ , $f'$ , $f''$ , $h$ $\rightarrow$ $T_2$ . The unique quadratic that matches f, f', f'' at a. Remainder ~ h³ f'''(ξ)/6.

Purpose

Purpose: compute

T_2

from

f

f'

f''

h

in Calculus via

T_2(a+h)=f+f'h+\tfrac12 f'' h^2

T2(x) = f + f' h + ½ f'' h², h = x − a. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f = 1.000\,\mathrm{—}

f' = 0.500\,\mathrm{—}

f'' = -2.000\,\mathrm{—}

h = 0.200\,\mathrm{—}

, the governing relation

T_2(a+h)=f+f'h+\tfrac12 f'' h^2

yields

T_2 = 1.0600\,\mathrm{—}

. Enter jets at a and a step h. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

f(a) f1 —f'(a) f'0.5 —f''(a) f''-2 —Step h h0.2 —

Outputs

Taylor value T_21.0600 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Enter jets at a and a step h.

The unique quadratic that matches f, f', f'' at a. Remainder ~ h³ f'''(ξ)/6.

Reading speed

Watch on YouTube