Unformatted text preview: Sivakumar Some definitions and theorems  Exam 3 M151H 1. Suppose that f is a function defined in an open interval containing the point c. We say that c is a critical number of f is f is not differentiable at c, or if f (c) = 0. 2. Suppose that f is defined in an open interval I. (a) We say that f has a local maximum at a point c in I if there is an interval I0 containing c such that f (x) f (c) for every x in I0 . (b) We say that f has a local minimum at a point d in I if there is an interval I1 containing d such that f (x) f (d) for every x in I1 . 3. Suppose that f is a function defined in an open interval I. We say that a function F is an antiderivative (or a primitive) of f in I if F (x) = f (x) for every x in I. 4. (Fermat's Theorem) Suppose that f is a function defined in an open interval, and let c be a point in the interval. If f has a local maximum or a local minimum at c, then c is a critical number of f . 5. (Extreme Value Theorem) Let a and b be real numbers with a < b. Suppose that f is continuous on the closed interval [a, b]. Then there exist points p and q in [a, b] such that f (p) f (x) f (q) for every a x b. 6. (Mean Value Theorem) Let a and b be real numbers with a < b. Suppose that f is continuous on the closed interval [a, b], and that it is differentiable on the open interval (a, b). Then there is some point c in (a, b) such that f (b)  f (a) = f (c). ba 7. (Fundamental Theorem of Calculus, Part I) Let a and b be real numbers with a < b. Suppose that f is continuous on the closed interval [a, b]. Define
x F (x) :=
a f (t) dt, a x b. Then F (x) = f (x) for every a < x < b, F+ (a) = f (a), and F (b) = f (b). 8. (Fundamental Theorem of Calculus, Part II) Let a and b be real numbers with a < b. Suppose that f is continuous on the closed interval [a, b]. Let G be any function satisfying each of the following conditions: (i) G is continuous on the closed interval [a, b], and (ii) G (x) = f (x) for every a < x < b. Then
b f (t) dt = G(b)  G(a).
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 Spring '08
 Skrypka
 Calculus, Continuous function, open interval

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