CAL-08

Mean-value snapshot

f'(c) = (f(b)−f(a))/(b−a). A slope that matches the chord.

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TheoremsMean value theorem

Governing equation

f'(c)=\dfrac{f(b)-f(a)}{b-a}

$f(b) \in \left[-20, 20\right]$

where

f(a): f(a) (—)
f(b): f(b) (—)
a: a (—)
b: b (—)
f'(c): MVT slope (—)

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.

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①

Governing law — red

f'(c)=\dfrac{f(b)-f(a)}{b-a}

$f(b) \in \left[-20, 20\right]$

$f(b) = -20$ $f'(c) = -10.5$

Along the curve

②

Domain of the variable — blue

f(b)_{\min} = -20,\quad f(b)_{\max} = 20

$f(b) \in \left[-20, 20\right]$

$f(b) = -20$ $f(b) = -20$

Along the curve

③

Response curve — green

f'(c)=\dfrac{f(b)-f(a)}{b-a}

$f(b) \in \left[-20, 20\right]$

$f(b) = -20$ $f'(c) = -10.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f'(c)}{\mathrm{d}f(b)}

$f(b) \in \left[-20, 20\right]$

$f(b) = -19.75$ $f'(c)^{\prime} = 0.5$

Along the curve

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-08 — Mean-value snapshot) is the form associated with Mean value theorem. Working symbols: $f(a)$ , $f(b)$ , $a$ , $b$ $\rightarrow$ $f'(c)$ . Rolle's theorem after subtracting the chord. Existence of c, not a construction.

Purpose

Purpose: compute

f'(c)

from

f(a)

f(b)

a

b

in Calculus via

f'(c)=\dfrac{f(b)-f(a)}{b-a}

f'(c) = (f(b)−f(a))/(b−a). A slope that matches the chord. 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(a) = 1.000\,\mathrm{—}

f(b) = 5.000\,\mathrm{—}

a = 0.000\,\mathrm{—}

b = 2.000\,\mathrm{—}

, the governing relation

f'(c)=\dfrac{f(b)-f(a)}{b-a}

yields

f'(c) = 2.0000\,\mathrm{—}

. Enter endpoint values; the sheet reports the guaranteed slope. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

f(a) f(a)1 —f(b) f(b)5 —a a0 —b b2 —

Outputs

MVT slope f'(c)2.0000 —

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Narration of this film

Enter endpoint values; the sheet reports the guaranteed slope.

Rolle's theorem after subtracting the chord. Existence of c, not a construction.

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