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Unformatted text preview: sides of (1). by b-a, transpose a term, and change the name of b td z. Now it's not a theorem about slopes; instead, it says that the value of f at some point z can be estimated, provided you know the value of f at some fixed point a, and have information about the size of f' on the interval [a, z]. In other words, from information about f', we can get information about f. (Such information can also be gotten by integration; one can think of the Mean-value Theorem as a poor-person's substitute for integration.) The special case of (1) in which f(a) = f(b) = 0 is. usually called Rolle's theorem; it says that if f is differentiable on [a, b], f(a) = f(b) = 0 Sf'(c) = 0 for some c, where a < c < b. Exercises: Section 2G Isee Simmons, p. 76...
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