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CAL-24

Taylor order 2

Local parabola.

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CalculusTaylor

Governing equation

f(a+h)approxf+f′h+f′′h2/2f(a+h)\\approx f+f'h+f''h^2/2

h∈[−2,2]h \in \left[-2, 2\right]

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

CAL-24 · curve
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①

Governing law — red

f(a+h)approxf+f′h+f′′h2/2f(a+h)\\approx f+f'h+f''h^2/2

h∈[−2,2]h \in \left[-2, 2\right]

h=−2h = -2y=−0.4y = -0.4

②

Domain of the variable — blue

hmin⁡=−2,hmax⁡=2h_{\min} = -2,\quad h_{\max} = 2

h∈[−2,2]h \in \left[-2, 2\right]

h=−2h = -2h=−2h = -2

③

Response curve — green

f(a+h)approxf+f′h+f′′h2/2f(a+h)\\approx f+f'h+f''h^2/2

h∈[−2,2]h \in \left[-2, 2\right]

h=−2h = -2y=−0.4y = -0.4

④

Rate of change — amber

dydh\frac{\mathrm{d}y}{\mathrm{d}h}

h∈[−2,2]h \in \left[-2, 2\right]

h=−1.975h = -1.975y′=0.895y^{\prime} = 0.895

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: ff, f′f', f′′f'', hh →\rightarrow yy. Local parabola.

Purpose

Purpose: compute yy from ff, f′f', f′′f'', hh in Calculus via f(a+h)approxf+f′h+f′′h2/2f(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 —f = 1.000\,\mathrm{—}, f′=0.500 —f' = 0.500\,\mathrm{—}, f′′=−0.200 —f'' = -0.200\,\mathrm{—}, h=0.300 —h = 0.300\,\mathrm{—}, the governing relation f(a+h)approxf+f′h+f′′h2/2f(a+h)\\approx f+f'h+f''h^2/2 yields y=1.1410 —y = 1.1410\,\mathrm{—}. Local parabola. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • y y1.1410 —
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Narration of this film

Local parabola.

Local parabola.

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