3 x x if x 3 if x 3 sin π 2 x π if x 3 d 3pts lim x

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3 - x x if x < 3 0 if x = 3 sin ( π 2 x - π ) if x > 3 . (d) [3pts] lim x →-∞ 1 + 3 - x x 6
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5. (12 points) The curve defined implicitly by ( x 2 - 2 x + y 2 ) 2 = x 2 + y 2 is called a lima¸ con trisectrix . This curve is pictured below, along with the y -intercepts, labeled A and B . Find the coordinates of the point where the tangent lines at A and B intersect. 7
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6. (14 points) Let f ( x ) = xe - x 2 . Answer the following questions: (a) [2pts] Determine all x - and y -intercepts for the curve. (b) [2pts] Determine any vertical asymptotes and horizontal asymptotes for the curve. (c) [2pts] Find all critical numbers for f ( x ). continued on next page 8
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6. continued. (d) [3pts] Find the intervals on which f ( x ) is increasing, and the intervals on which f ( x ) is decreasing. Determine x - and y -coordinates of all local minimum(s) and local maximum(s). (e) [3pts] Find the intervals on which f ( x ) is concave up and concave down. Find the x - and y -coordinates of all of the inflection points. continued on next page 9
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6. continued. (f) [2pts] Using (a)-(e), sketch the curve.
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  • Spring '08
  • WAlker
  • Math, TA, Mathematical analysis, Convex function, exact form

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