Generalized Intermediate Value Theorem Intermediate Value Theorem - - PowerPoint PPT Presentation

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Generalized Intermediate Value Theorem Intermediate Value Theorem Theorem Intermediate Value Theorem Suppose f is continuous on [ a , b ] and let N be any number between f ( a ) and f ( b ) , where f ( a ) = f ( b ) . Then there exists a

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Generalized Intermediate Value Theorem

Intermediate Value Theorem Theorem Intermediate Value Theorem Suppose f is continuous on [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N.

September 17, 2019 1 / 3

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Generalized Intermediate Value Theorem

Intermediate Value Theorem Theorem Intermediate Value Theorem Suppose f is continuous on [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N. Generalized Intermediate Value Theorem Theorem Let f be continuous on [a, b]. Let x0, x1, . . . , xn be points in [a, b] and a1, a2, . . . , an > 0. There exists a number c between a and b such that (a1 + · · · + an)f (c) = a1f (x1) + · · · + anf (xn)

September 17, 2019 1 / 3

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Generalized IVT applied to error estimates

Recall f ′(x) = f (x + h) − f (x − h) 2h − h2 12

f ′′′(c1) + f ′′′(c2)
for c1 ∈ (x, x + h) and c2 ∈ (x − h, x).

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Generalized IVT applied to error estimates

Theorem Let f be continuous on [a, b]. Let x0, x1, . . . , xn be points in [a, b] and a1, a2, . . . , an > 0. There exists a number c between a and b such that (a1 + · · · + an)f (c) = a1f (x1) + · · · + anf (xn) We can combine the error terms of the central difference formula as

( 1 12 + 1 12)f (c) = h2 12

f ′′′(c1) + f ′′′(c2)
for c ∈ (x − h, x + h) to obtain a nicer looking estimate:

f ′(x) = f (x + h) − f (x − h) 2h − h2 6 f ′′′(c) c ∈ (x − h, x + h)

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