121616949-math.56 - 42 Chapter 2 Instantaneous Rate Of...

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42 Chapter 2 Instantaneous Rate Of Change: The Derivative true for function values near x = - 1 on the graph in (a) which is not continuous at that location. DEFINITION 2.18 Continuity at a Point A function f is continuous at a point a if lim x a f ( x ) = f ( a ). DEFINITION 2.19 Continuous A function f is continuous if it is continuous at every point in its domain. Strangely, we can also say that (d) is continuous even though there is a vertical asymp- tote. A careful reading of the definition of continuous reveals the phrase “ at every point in its domain. Because the location of the asymptote, x = 0, is not in the domain of the function, and because the rest of the function is well-behaved , we can say that (d) is continuous. Differentiability. Now that we have introduced the derivative of a function at a point, we can begin to use the adjective differentiable . We can see that the tangent line is well- defined at every point on the graph in (c). Therefore, we can say that (c) is a differentiable function. DEFINITION 2.20 Differentiable at a Point
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Unformatted text preview: A function f is differentiable at point a (excluding endpoints and isolated points in the domain of f ) if f ± ( a ) exists. DEFINITION 2.21 Differentiable A function f is differentiable if is differentiable at every point (excluding endpoints and isolated points in the domain of f ) in the domain of f . Take note that, for technical reasons not discussed here, both of these definitions exclude endpoints and isolated points in the domain from consideration. We now have a collection of adjectives to describe the very rich and complex set of objects known as functions. We close with a useful theorem about continuous functions: THEOREM 2.22 Intermediate Value Theorem If f is continuous on the interval [ a, b ] and d is between f ( a ) and f ( b ), then there is a number c in [ a, b ] such that f ( c ) = d . This is most frequently used when d = 0. EXAMPLE 2.23 Explain why the function f = x 3 + 3 x 2 + x-2 has a root between 0 and 1....
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