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Continuity Curves without gaps? Animation (infinite length) Continuity Definition A function f is continuous at a number a if lim f(x) = f(a). x -> a 1. f (a) is defined. 2. lim f(x) exists. x -> a 3.lim f(x) = f(a). x -> a Continuity 1. f (a) is defined. 2. lim f(x) exists. x -> a 3.lim f(x) = f(a). x -> a Animation sin (1/x) If f is not continuous at a , we say f is discontinuous at a, or f has a discontinuity at a . Example Where is the function f(x)=(x 2 – x – 2)/(x – 2) discontinuous? Definition A function f is continuous from the right at a number a if lim f(x) = f(a), x -> a + and f is continuous from the left a if lim f(x) = f(a). x -> a - Definition A function f is continuous on an interval if it is continuous at every number in the interval. At an endpoint of the interval we understand continuous to mean continuous from the right or continuous from the left. Example Use the definition of continuity and the properties of limits to show that the function _____ f (x) = x 16 –x2 is continuous on the interval [-4, 4]. Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1. f + g 2. f – g 4. f g 5. ( f / g) if g(a) is not 3. c f equal to 0. Theorem (a)Polynomials are continuous everywhere; that is continuous on R = (-, ). (b)Any rational function is continuous wherever it is defined; that is, it is continuous on its domain. Theorem The following types of functions are continuous at every number in their domains: -polynomials -rational functions -root functions -exponential functions -trigonometric functions -inverse trigonometric functions -logarithmic functions. Example Evaluate lim arctan x -> 2 2 ((x - 4) / (3x 2– 6x)). Theorem If f is continuous at b and lim g(x) = b, then, lim f(g(x)) = f(b). x -> a x -> a In other words, lim f(g(x)) = f(lim g(x)). x -> a x -> a Theorem If g is continuous at a and f is continuous at g(a), then (f o g)(x) = f(g(x)) is continuous at a. The Intermediate Value Theorem Suppose that f is continuous on the closed interval [a, b] and let N be any number strictly between f (a) and f (b). Then there exists a number c in (a,b) such that f (c)= N . Example Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ln x = e –x , (1,2).