a5 242 - MATH 242 Analysis 1 (Fall, 2011) Assignment 5. Due...

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Unformatted text preview: MATH 242 Analysis 1 (Fall, 2011) Assignment 5. Due in class Thursday, November 24. Instructions: This assignment is out of 50 points. You may choose any questions whose points add up to 50. As in the previous assignment, provide all details in your work. Also as previously, stars (*) indicate possibly more challenging questions. 1. (10 pts.) (a) If f ( x ) = 1 / x, x > 0, use the definition to find f ( x ) for each x > 0. For the following functions on R , find all points where f ( x ) does not exist, and find f ( x ) when it does exist, proving your answers in each case: (b) f ( x ) = x sin(1 /x ) for x negationslash = 0, f (0) = 0, (c) f ( x ) = : x 2 if x is rational, x 3 if x is irrational. 2. (10 pts.) Use the Mean Value Theorem to prove that ( x 1) /x < ln x < x 1 when x > 1. [You may assume the facts that the function ln x has derivative 1 /x and ln x = ln(1 /x ), on x > 0.] 3. (10 pts.) Let f : [ a, b ] R be continuous on [ a, b ] and differentiable in...
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a5 242 - MATH 242 Analysis 1 (Fall, 2011) Assignment 5. Due...

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