sol to term test 1

sol to term test 1 - MAT 137Y 2008-2009 Winter Session...

This preview shows pages 1–2. Sign up to view the full content.

MAT 137Y, 2008-2009 Winter Session, Solutions to Term Test 1 1. Evaluate the following limits. (Do not prove them using the formal deﬁnition of limit.) (10%) (i) lim x 0 x sin x 1 - cos x . Multiplying top and bottom by 1 + cos x , we have lim x 0 x sin x ( 1 + cos x ) ( 1 - cos x )( 1 + cos x ) = lim x 0 x sin x ( 1 + cos x ) 1 - cos 2 x = lim x 0 x sin x ( 1 + cos x ) sin 2 x = lim x 0 x sin x · ( 1 + cos x ) = 1 · 2 = 2 . (10%) (ii) lim x 3 5 - x - x 2 - 7 x + 6 - 3 . Here we multiply top and bottom by the conjugates of both expressions to get lim x 3 5 - x - x 2 - 7 x + 6 - 3 5 - x + x 2 - 7 5 - x + x 2 - 7 ! ± x + 6 + 3 x + 6 + 3 ² = lim x 3 [( 5 - x ) - ( x 2 - 7 )]( x + 6 + 3 ) [( x + 6 ) - 9 ][ 5 - x + x 2 - 7 ] = lim x 3 ( 12 - x - x 2 )( x + 6 + 3 ) ( x - 3 )( 5 - x + x 2 - 7 ) = lim x 3 ( 3 - x )( 4 + x )( x + 6 + 3 ) ( x - 3 )( 5 - x + x 2 - 7 ) = - lim x 3 ( 4 + x )( x + 6 + 3 ) ( 5 - x + x 2 - 7 ) = - 21 2 . 2. (10%) (i) Find all solutions in the interval [ 0 , 2 π ) that satisfy the equation 2sin3 x - 1 = 0. We have 2sin3 x - 1 = 0 ⇐⇒ sin3 x = 1 2 . Solving for 3 x , we have either 3 x = π 6 + 2 k π or 3 x = 5 π 6 + 2 k π . Solving for x gives us the solutions x = π 18 + 2 3 k π , x = 5 π 18 + 2 3 k π . Therefore, the solutions in the interval [ 0 , 2 π ) that satisfy the original equation are x = π 18 , 13 π 18 , 25 π 18 , 5 π 18 , 17 π 18 , 29 π 18 . (10%) (ii) Solve the inequality | 2 x | + | x - 3 | < 5 and express your answer as a union of intervals.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 3

sol to term test 1 - MAT 137Y 2008-2009 Winter Session...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online