w04e - f has at least 8 roots in-2 2 or f has at most 6...

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Math 151, Workshop 4 1. Show that the following function is continuous at x 0 = 0: f ( x ) = x 2 sin 1 x if x 6 = 0 , 0 if x = 0 . Suggestion: Use the squeeze theorem and knowledge of the limit lim x 0 (sin x ) /x . 2. Find the following limit: lim x 0 (cot x )(1 - cos 2 x ) x . 3. Let f ( x ) = Ax + B x 2 where A and B are constants. a) Find values of A and B so that y = 2 x + 1 is tangent to y = f ( x ) when x = - 1. b) Graph the resulting f ( x ) and the tangent line together when - 4 x 2 and - 6 y 4. 4. A continuous function f is defined on the interval [ - 2 , 2]. The values of f at some of the points of the interval are given by the following table: x - 2 - 1 0 1 2 f ( x ) 2 - 1 2 - 1 2 a) Using only this information, what can be concluded about the roots of f , that is, the solutions of f ( x ) = 0, in the interval [ - 2 , 2]? Suggestion: The answer should be something like:
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Unformatted text preview: f has at least 8 roots in [-2 , 2], or f has at most 6 roots in [-2 , 2] . b) If f ( x ) = x 4-4 x 2 + 2, verify that the relevant values of f are given by the table above. Sketch the graph of y = f ( x ) in the viewing window [-2 . 5 , 2 . 5] × [-3 , 3]. How many roots does f have in the interval [-2 , 2]? Find the roots alge-braically. Suggestion: Substitute t = x 2 . c) If f ( x ) = x 4-4 x 2 +2+5(2 x-1) x ( x 2-1)( x 2-4), verify that the relevant values of f are given by the table above. Using the Intermediate Value Theorem find as many roots of f in the interval [-2 , 2] as possible. 1...
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