lecture14 - 1 Procedure: 1. Find all number(s) where f ( x...

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Math 006 (Lecture 14) Continuity Properties Theorem 1. Continuity properties of some functions: 1. A polynomial is continuous for all x . 2. A rational function is continuous for all x except those values that make a denomi- nator zero. 3. For n an odd positive integer greater than 1, n p f ( x ) is continuous whenever f ( x ) is continuous. 4. For n an even positive integer, n p f ( x ) is continuous whenever is continuous and nonnegative. Example 1. Determine where each function is continuous (a) f ( x ) = x 2 - 2 x + 2 (b) f ( x ) = x x 2 - x - 2 (c) f ( x ) = 3 x 2 - 1 (d) f ( x ) = 4 x - 4 Solving inequalities using continuity properties Theorem 2. If f is continuous on a < x < b and f ( x ) 6 = 0 for all x in a < x < b , then either f ( x ) > 0 for all x in a < x < b or f ( x ) < 0 for all x in a < x < b . Example 2. Solve x + 2 x - 1 > 0.
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Unformatted text preview: 1 Procedure: 1. Find all number(s) where f ( x ) is discontinuous and all zero(s) of f ( x ). 2. Plot the number(s) found in step 1 on a real number line, which divide the number line into intervals. 3. Select a test number in each interval determine in step 2 and evaluate f ( x ) at each test number to determine whether f ( x ) is positive or negative in each interval. 4. Construct a sign chart using the real number line in step 2. This will show the sign of f ( x ) on each interval. Example 3. Solve each of the following inequalities: (a) ( x-1)( x + 2)( x + 3) &gt; (b) x 2-1 x-3 ≤ 2...
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This note was uploaded on 09/04/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

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lecture14 - 1 Procedure: 1. Find all number(s) where f ( x...

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