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

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MAT 111 Practice Final Solutions Spring 2010 1

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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 0 sin 2 3 t t 2 multiplying and dividing by 9, we get = 9 lim t 0 sin 3 t 3 t 2 = 9 2
(c) lim x 0 + x [1 + sin 2 (2 /x )] We see that 0 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 0 + x = 0 = lim x 0 + 2 x So by squeeze theorem, the limit of the quantity in the middle is also zero. (d) lim h 0 1+ h - 1 h Multiplying and dividing by 1 + h + 1, we get = lim h 0 ( 1 + h ) 2 - 1 2 h ( 1 + h + 1) = 1 + h - 1 h ( 1 + h + 1) = lim h 0 1 1 + h + 1 = 1 / 2 3

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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 boat

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