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Unformatted text preview: MATH 147 Assignment #3 Due: Friday, October 30 1) A function is defined by f ( x ) = x + 5 a if x ≤ - 3- ax + b if- 3 < x < 3 2 x + 10 b if x ≥ 3 where a and b are constants. Find values of a and b that will ensure that f is continuous for all x . Sketch the graph of the resulting function. 2) Evaluate the following limits if they exist: i) lim x → sin | x | x ii) lim x → 1 | x |-| x- 2 | x- 1 iii) lim x → sin(4 x ) tan( πx ) iv)lim x → cos( 1 x ) 3a) Suppose that sin( πx ) ≤ f ( x ) ≤ 1 4 x (1- x ) for all x ∈ (0 , 1). What can you say about lim x → 1 2 f ( x )? Is f ( x ) continuous at x = 1 2 ? b) Use Maple to plot the graphs of g ( x ) = sin( πx ) and h ( x ) = 1 4 x (1- x ) on the same axes with x ∈ [0 , 1]. 4) Suppose that f ( a ) > 0 and that f ( x ) is continuous at x = a. Prove that there exists an open interval I containing a such that f ( x ) > 0 for all x ∈ I....
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This note was uploaded on 10/21/2010 for the course MATH 147 taught by Professor Wolzcuk during the Fall '09 term at Waterloo.
- Fall '09