m147A3sol - MATH 147 Assignment #3 Solutions 1) A function...

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Unformatted text preview: MATH 147 Assignment #3 Solutions 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. Solution It is clear that f ( x ) is continuous except possibly at x = ± 3. Observe that lim x →- 3- f ( x ) = lim x →- 3- x + 5 a =- 3 + 5 a = f (- 3) It follows that f ( x ) will be continuous at x =- 3 if and only if lim x →- 3 + f ( x ) =- 3 + 5 a . However, lim x →- 3 + f ( x ) = lim x →- 3-- ax + b = 3 a + b Therefore f ( x ) is continuous at x =- 3 if and only if- 3 + 5 a = 3 a + b or b = 2 a- 3 . We also know that lim x → 3- f ( x ) = lim x → 3-- ax + b =- 3 a + b 1 and that lim x → 3 + f ( x ) = lim x → 3 + 2 x + 10 b = 6 + 10 b = f (3) Therefore f ( x ) is continuous at x = 3 if and only if- 3 a + b = 6 + 10 b or- 3 a- 6 = 9 b. Substituting b = 2 a- 3 into the above equation leads to- 3 a- 6 = 18 a- 27 or a = 1 . This means that b = 2(1)- 3 =- 1. We have shown that f ( x ) is continuous if and only if a = 1 and b =- 1. 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 ) Solutions: i) Observe that sin | x | x = ( sin( x ) x if x ≥ sin(- x ) x if x < . 2 From this it follows that lim x →- sin | x | x = lim x →- sin(- x ) x = lim x →-- sin( x ) x =- 1 while lim x → + sin | x | x = lim x → + sin( x ) x = lim x → + sin( x ) x = 1 Since the two one sided limits are different, the limit does not exist. ii) First observe that on the interval [0 , 2] the expression | x | - | x- 2 | becomes x- (- ( x- 2)) = 2 x- 2. Therefore, for x ∈ [0 , 2], x 6 = 1 f ( x ) = 2 x- 2 x- 1 = 2. This observation shows that lim x → 1 | x |-| x- 2 | x- 1 = 2....
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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.

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m147A3sol - MATH 147 Assignment #3 Solutions 1) A function...

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