121616949-math.57

# 121616949-math.57 - c continuous at x = 0 not continuous at...

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2.5 Adjectives For Functions 43 Since f (0) = - 2 and f (1) = 3, and 0 is between - 2 and 3, there is a c [0 , 1] such that f ( c ) = 0. This example also points the way to a simple method for approximating roots. EXAMPLE 2.24 Approximate the root of the previous example to one decimal place. If we compute f (0 . 1), f (0 . 2), and so on, we find that f (0 . 6) < 0 and f (0 . 7) > 0, so by the Intermediate Value Theorem, f has a root between 0 . 6 and 0 . 7. Repeating the process with f (0 . 61), f (0 . 62), and so on, we find that f (0 . 61) < 0 and f (0 . 62) > 0, so f has a root between 0 . 61 and 0 . 62, and the root is 0 . 6 rounded to one decimal place. Exercises 1. Along the lines of Figure 2.3 , for each part below sketch the graph of a function that is: a. bounded, but not continuous. b. differentiable and unbounded.
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Unformatted text preview: c. continuous at x = 0, not continuous at x = 1, and bounded. d. diﬀerentiable everywhere except at x =-1, continuous, and unbounded. 2. Is f ( x ) = sin( x ) a bounded function? If so, ﬁnd the smallest M . 3. Is s ( t ) = 1 / (1 + t 2 ) a bounded function? If so, ﬁnd the smallest M . 4. Is v ( u ) = 2 ln | u | a bounded function? If so, ﬁnd the smallest M . 5. Consider the function h ( x ) = 2 x-3 , if x < 1 , if x ≥ 1. Show that it is continuous at the point x = 0. Is h a continuous function? 6. Approximate a root of f = x 3-4 x 2 + 2 x + 2 to one decimal place. 7. Approximate a root of f = x 4 + x 3-5 x + 1 to one decimal place....
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