sample-midterm2-with-solutions-v2

2x2 1 1 x 2 5 determine the following limits a cot2q

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Unformatted text preview: 2 ) 2 - 3 x 2 · (1 + x 2 ) 2 ÷ ç ÷ è ø -5 = -(1 + x 2 ) 2 · (1 + x 2 - 3 x 2 ) = 2x2 - 1 5 (1 + x 2 ) 2 Q5. = 2x2 - 1 ( 1 + x 2 )5 Determine the following limits: a. cot(2q ) . q ® 0 csc(q ) lim b. 3x 5 + x - 50 x ®¥ 6 x 4 - 2 x 2 + 10 lim Solution a. lim q ®0 cot(2q ) cos(2q ) sin(q ) cos(2q ) sin(q ) 1 = lim × = lim × × q ® 0 sin(2q ) q ®0 csc(q ) 1 1 1 sin(2q ) = = lim q ®0 cos(2q ) sin(q ) 2q 1 × × × 1 sin(2q ) 2 q 1 cos(2q ) sin(q ) 2q × lim × lim lim q ®0 q ® 0 sin(2q ) 2 q ®0 1 q = 1 1 ×1 × 1 × 1 = . 2 2 b. 2 b. 1 50 ö 1 50 ö æ æ x5 ç 3 + 4 - 5 ÷ ç lim 3+ lim 4 - lim 5 ÷ x®¥ x x®¥ x x®¥ 3 x + x - 50 x xø ø = lim è = lim x · è lim 4 2 x ® ¥ 6 x - 2 x + 10 x®¥ x®¥ 2 10 ö 2 10 ö æ æ x4 ç 6 - 2 + 4 ÷ ç lim 6 - lim 2 + lim 4 ÷ x®¥ x x®¥ x x xø è è x¬¥ ø () 5 ( 30 ) ((6 + 0 - 0)) - +0 = lim x × x ®¥ = ( lim x ) × 1 2 x ®¥ =¥ Q6. Find all numbers c that satisfy the conclusion of the Mean Value Theorem for f (...
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