Rolle’s Theorem

%%visits: 5 Let f ∈ C[a, b] be differentiable on (a, b) and assume f(a) = f(b). Then there exists a point c ∈ (a, b) such that f′(c) = 0

This is really just combining extreme_value_theorem with Fermats_Theorem. Can be proven case by case.

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Rolle’s thm

f(a) = f(b)

f is cts on $$

differentiable on (a, b)

 ⟹ ∃c ∈ (a, b) so f′(c)0

Link mean_value_theorem to rolles_theorem: Think of mean_value_theorem as a generalisation

$$ f'(p) = \frac{f(b) -f(a)}{b-a} $$

speed at some point is the same as the average speed.

If you travelled 60mph over an hour, at some point you travelled 60mph instantaneously

exam_clinic

I model a stock S(t) = Main(t) + Error(t). If I want to sell this stock, the error term needs to be low.

The behaviour of the stock over a year, modelled at every interval in a linear fashion.

sps I culd partition $$ into small intervals. I1, …, IN so sup|Error(t)| < 1p

  1. S(t) = M(t) + e(t) Where the absolute value of E(t) is never more athan 10 in the interval [0, 2]. suppose I want to subdivide [0, 2] into N intervals of even length so that the $max{|E(t) - E(s)| \le \frac{1}{1000}$ on each I.

How big must I choose N?

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