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M408L_HW01

M408L_HW01 - hyun(hh7953 HW01 gogolev(57440 This print-out...

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hyun (hh7953) – HW01 – gogolev – (57440) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine if lim x → ∞ braceleftBig ln(5 + 8 x ) - ln(4 + x ) bracerightBig exists, and if it does find its value. 1. limit = ln 5 4 2. limit = ln 8 correct 3. limit does not exist 4. limit = ln 4 5 5. limit = ln 1 8 Explanation: By properties of logs, ln(5 + 8 x ) - ln(4 + x ) = ln parenleftbigg 5 + 8 x 4 + x parenrightbigg = ln parenleftbigg 5 /x + 8 4 /x + 1 parenrightbigg . But lim x → ∞ 5 /x + 8 4 /x + 1 = 8 . Consequently, the limit exists and limit = ln 8 . 002 10.0 points Determine if lim x → ∞ sin 1 parenleftBigg 3 + 3 x 5 + 2 x parenrightBigg exists, and if it does, find its value. 1. limit = π 3 correct 2. limit = π 6 3. limit = π 2 4. limit does not exist 5. limit = π 4 6. limit = 0 Explanation: Since lim x → ∞ 3 + 3 x 5 + 2 x = 3 2 , we see that lim x → ∞ sin 1 parenleftBigg 3 + 3 x 5 + 2 x parenrightBigg exists, and that the limit = sin 1 3 2 = π 3 . 003 10.0 points Determine if lim x → ∞ tan 1 parenleftbigg 3 + x 4 + 3 x parenrightbigg exists, and if it does, find its value. 1. limit = π 4 2. limit = π 3 3. limit = π 6 correct 4. limit = 0

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hyun (hh7953) – HW01 – gogolev – (57440) 2 5. limit does not exist 6. limit = π 2 Explanation: Since lim x → ∞ 3 + x 4 + 3 x = 1 3 , we see that lim x → ∞ tan 1 parenleftbigg 3 + x 4 + 3 x parenrightbigg exists, and that the limit = tan 1 1 3 = π 6 . 004 10.0 points Express the function f ( x ) = sin 2 x + 5 cos 2 x in terms of cos 2 x . 1. f ( x ) = 3 + 2 cos 2 x correct 2. f ( x ) = 3 - 2 cos 2 x 3. f ( x ) = - 3 + 2 cos 2 x 4. f ( x ) = - 2 - 3 cos 2 x 5. f ( x ) = - 2 + 3 cos 2 x 6. f ( x ) = 2 - 3 cos 2 x Explanation: Since sin 2 x = 1 2 (1 - cos 2 x ) , cos 2 x = 1 2 (1 + cos 2 x ) , we can rewrite f ( x ) as f ( x ) = 1 2 (1 - cos 2 x ) + 5 2 (1 + cos 2 x ) . Consequently, f ( x ) = 3 + 2 cos 2 x .
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