handout_09_18

handout_09_18 - f ( x ) by f ( x ) = xe cos(1 /x ) for x 6...

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Math 1a: Continuity September 18, 2009 1. At which values of x are the following functions continuous? (a) a ( x ) = b x c , the greatest integer less than or equal to x . (b) b ( x ) = the taxi fare (in dollars) for distance x (in miles). Assume that the meter ticks every 0 . 1 miles. (c) c ( x ) = sin( x ) + cos( x ) . (d) d ( x ) = | x | , the absolute value of x . 2. A phone plan states that “Charges are calculated on a per minute basis, and rounded up to the next whole minute”. Is the price of a phone call a continuous function of its length? 1
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3. Define the function
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Unformatted text preview: f ( x ) by f ( x ) = xe cos(1 /x ) for x 6 = 0 . How should I define f (0) to make f continuous? 4. Define the function f ( x ) by f ( x ) = cos( x ) if x < 0, ax + b if 0 ≤ x ≤ 2, x 3 if 2 < x . What should the constants a and b be so that f ( x ) is continuous for all x ? 5. (Challenge problem) Define f ( x ) by f ( x ) = ( 1 if x is rational, if x is irrational. At which points is f ( x ) continuous? 2...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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handout_09_18 - f ( x ) by f ( x ) = xe cos(1 /x ) for x 6...

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