Homework #10

# Homework #10 - jiwani(amj566 – Homework10 – Fouli...

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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 ( − π/ 2 , π/ 2). 1. x = − π 6 , π 6 2. x = π 6 correct 3. x = − π 3 4. x = − π 6 5. x = − π 3 , π 3 6. x = π 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 − π/ 2 + − π/ 6 π/ 2 for d 2 y/dx 2 thus shows that on ( − π/ 2 , π/ 2) the graph changes sign at x = π/ 6 only. Con- sequently, the graph changes concavity only at x = π 6 . 003 10.0 points Let f be the function defined by f ( x ) = x − cos 2 x, − π ≤ x ≤ π . Determine all interval(s) on which f is de- creasing. 1. [ − 5 π 12 , − π 6 ] , [ π 6 , 11 π 12 ] 2. [ − π 6 , − π 12 ] , [ π 6 , 11 π 12 ] 3. [ − 5 π 12 , − π 12 ] , [ 7 π 12 , 11 π 12 ] correct 4. [ − π, − 5 π 12 ] , [ 7 π 12 , π ] 5. [ − 5 π 12 , − π 8 ] , [ 3 π 8 , 11 π 12 ] Explanation: After differentiation, f ′ ( x ) = 1 + 2 sin2 x....
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## This note was uploaded on 07/10/2011 for the course KIN 321M taught by Professor Jensen during the Spring '11 term at University of Texas.

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Homework #10 - jiwani(amj566 – Homework10 – Fouli...

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