practice test final answers

practice test final answers - MAT 111 Practice Final...

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Unformatted text preview: MAT 111 Practice Final Solutions Spring 2010 1 1. (20 points) Find the following limits: (a) lim x →∞ sin 2 x x 2 Since the numerator is bounded and denominator goes to infinity, the limit is zero. (b) lim t → sin 2 3 t t 2 multiplying and dividing by 9, we get = 9 lim t → sin 3 t 3 t 2 = 9 2 (c) lim x → + √ x [1 + sin 2 (2 /x )] We see that ≤ sin 2 (2 /x ) ≤ 1 so, 1 ≤ 1 + sin 2 (2 /x ) ≤ 2 And so, √ x ≤ √ x [1 + sin 2 (2 /x )] ≤ 2 √ x Now, lim x → + √ x = 0 = lim x → + 2 √ x So by squeeze theorem, the limit of the quantity in the middle is also zero. (d) lim h → √ 1+ h- 1 h Multiplying and dividing by √ 1 + h + 1, we get = lim h → ( √ 1 + h ) 2- 1 2 h ( √ 1 + h + 1) = 1 + h- 1 h ( √ 1 + h + 1) = lim h → 1 √ 1 + h + 1 = 1 / 2 3 2. (10 points) Differentiate the following. (a) y = ( x 2 + 2) √ 1 + tan x = 2 x √ 1 + tan x + ( x 2 + 2) 1 2 √ 1 + tan x sec 2 x (b) y = tan √ x +2 sin 2 x = sec 2 √ x + 2 sin 2 x · 1 2 √ x +2 · sin 2 x- 2 sin x cos x √ x + 2 sin 4 x 4 3. (20 points) A boat is pulled into a dock by a rope attached to the bow of the boat3....
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This note was uploaded on 12/01/2010 for the course PSY 110 taught by Professor Winters during the Spring '08 term at University of Miami.

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practice test final answers - MAT 111 Practice Final...

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