Homework #10 - jiwani (amj566) Homework10 Fouli (58395) 1...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: jiwani (amj566) Homework10 Fouli (58395) 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine the increasing and decreasing properties of the function f ( x ) = ( x 1) 4 5 ( x + 2) 1 5 on its natural domain. 1. inc: [ 2 , 7 5 ] , dec: [ 7 5 , ) 2. inc: [ 2 , 7 5 ] [1 , ) , dec: [ 7 5 , 1] 3. inc: ( , 7 5 ] [1 , ) , dec: [ 7 5 , 1] cor- rect 4. inc: [ 7 5 , 1] , dec: [ 2 , 7 5 ] [1 , ) 5. inc: ( , 2] [1 , ) , dec: [ 2 , 1] Explanation: The natural domain of f is ( , ) and f is differentiable everywhere on its domain except at x = 2 , 1. Now by the Product Rule, f ( x ) = 1 5 braceleftBig 4 parenleftBig x + 2 x 1 parenrightBig 1 5 + parenleftBig x 1 x + 2 parenrightBig 4 5 bracerightBig . Now 4 parenleftBig x + 2 x 1 parenrightBig 1 5 + parenleftBig x 1 x + 2 parenrightBig 4 5 = 4( x + 2) + x 1 ( x 1) 1 5 ( x + 2) 4 5 = 5 x + 7 ( x 1) 1 5 ( x + 2) 4 5 . Thus f ( x ) = x + 7 5 ( x 1) 1 5 ( x + 2) 4 5 . Since ( x + 2) 4 5 is positive everywhere except at x = 2, where it is zero, we have only to look at the sign chart + + 7 5 1 for x + 7 5 ( x 1) 1 5 to determine where f is increasing or decreas- ing. Consequently, f is inc: ( , 7 5 ] [1 , ) , dec: [ 7 5 , 1] . 002 10.0 points Find all values of x at which the graph of y = x 2 + 4 sin x changes concavity on ( / 2 , / 2). 1. x = 6 , 6 2. x = 6 correct 3. x = 3 4. x = 6 5. x = 3 , 3 6. x = 3 7. there are no values of x Explanation: The graph changes concavity at x when d 2 y/dx 2 changes sign at x . Now after differentiating twice we see that d 2 y dx 2 = d dx parenleftBig 2 x + 4 cos x parenrightBig = 2 4 sin x . The sign chart jiwani (amj566) Homework10 Fouli (58395) 2 / 2 + / 6 / 2 for d 2 y/dx 2 thus shows that on ( / 2 , / 2) the graph changes sign at x = / 6 only. Con- sequently, the graph changes concavity only at x = 6 . 003 10.0 points Let f be the function defined by f ( x ) = x cos 2 x, x . Determine all interval(s) on which f is de- creasing. 1. [ 5 12 , 6 ] , [ 6 , 11 12 ] 2. [ 6 , 12 ] , [ 6 , 11 12 ] 3. [ 5 12 , 12 ] , [ 7 12 , 11 12 ] correct 4. [ , 5 12 ] , [ 7 12 , ] 5. [ 5 12 , 8 ] , [ 3 8 , 11 12 ] Explanation: After differentiation, f ( x ) = 1 + 2 sin2 x....
View Full Document

Page1 / 10

Homework #10 - jiwani (amj566) Homework10 Fouli (58395) 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online