This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: garcia (jjg2564) HW 11 gualdani (56410) 1 This printout should have 17 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points If f is a function on ( 4 , 4) having exactly three critical points and the sign of f , f are given in 2 2 f > f < f < f < f < f > f < decide which of the following could be the graph of f . 1. 2 4 2 4 2 4 2 4 2. 2 4 2 4 2 4 2 4 3. 2 4 2 4 2 4 2 4 4. 2 4 2 4 2 4 2 4 5. 2 4 2 4 2 4 2 4 correct 6. 2 4 2 4 2 4 2 4 Explanation: For the given sign chart garcia (jjg2564) HW 11 gualdani (56410) 2 2 2 f > f < f < f < f < f > f < an inspection of the graphs shows that two of them fail to have exactly three critical points, leaving just four possible graphs for f . To distinguish among these we use the fact that (i) if f ( x ) > 0 on ( a, b ), then f ( x ) is in creasing on ( a, b ), while (ii) if f ( x ) < 0 on ( a, b ), then f ( x ) is decreasing on ( a, b ), and that (iii) if f ( x ) > 0 on ( a, b ), then the graph is concave UP on ( a, b ), while (iv) if f ( x ) < 0 on ( a, b ), then the graph is concave DOWN on ( a, b ). Consequently, again by inspection we see that the only possible graph for f is 2 4 2 4 2 4 2 4 002 10.0 points Which of the following is the graph of f ( x ) = x 2 x 2 16 ? Dashed lines indicate asymptotes. 1. 2 4 2 4 2 4 2 4 2. 2 4 2 4 2 4 2 4 3. 2 4 2 4 2 4 2 4 4. 2 4 2 4 2 4 2 4 cor garcia (jjg2564) HW 11 gualdani (56410) 3 rect 5. 2 4 2 4 2 4 2 4 6. 2 4 2 4 2 4 2 4 Explanation: Since x 2 16 = 0 when x = 4, the graph of f will have vertical asymptotes at x = 4; on the other hand, since lim x x 2 x 2 16 = 1 , the graph will have a horizontal asymptote at y = 1. This already eliminates some of the possible graphs. On the other hand, f (0) = 0, so the graph of f must also pass through the origin. This eliminates another graph. To decide which of the remaining graphs is that of f we look at the sign of f to determine where f is increasing or decreasing. Now, by the Quotient Rule, f ( x ) = 2 x ( x 2 16) 2 x 3 ( x 2 16) 2 = 32 x ( x 2 16) 2 . Thus f ( x ) > , x < , while f ( x ) < , x > , so the graph of f is increasing to the left of the origin and decreasing to the right of the origin. The only graph having all these properties is 2 4 2 4 2 4 2 4 Consequently, this must be the graph of f . 003 10.0 points If f is a continuous function on ( 4 , 4) such that (i) f has 3 critical points, (ii) f has 1 local maximum, (iii) f ( x ) > 0 on ( 4 , 2), (iv) f ( x ) < 0 on (0 , 2), (v) (0 , 1) is an inflection point, (vi) f ( x ) < 0 on (2 , 4), which one of the following could be the graph of f ? garcia (jjg2564) HW 11 gualdani (56410) 4 1....
View
Full
Document
 Fall '09
 Gualdani

Click to edit the document details