CAL-24

Taylor order 2

Local parabola.

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CalculusTaylor

Governing equation

f(a+h)\\approx f+f'h+f''h^2/2

where

f: f (—)
f': f' (—)
f'': f'' (—)
h: h (—)
y: y (—)

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-24 — Taylor order 2) is the form associated with Taylor. Working symbols: $f$ , $f'$ , $f''$ , $h$ $\rightarrow$ $y$ . Local parabola.

Purpose

Purpose: compute

y

from

f

f'

f''

h

in Calculus via

f(a+h)\\approx f+f'h+f''h^2/2

Local parabola. 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'' = -0.200\,\mathrm{—}

h = 0.300\,\mathrm{—}

, the governing relation

f(a+h)\\approx f+f'h+f''h^2/2

yields

y = 1.1410\,\mathrm{—}

. Local parabola. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

f f1 —f' f'0.5 —f'' f''-0.2 —h h0.3 —

Outputs

y y1.1410 —

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Narration of this film

Local parabola.

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