# Test3mac2311 page 4 of 5 8 8 pts assume f is

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TEST3/MAC2311 Page 4 of 5 ______________________________________________________________________ 8. (8 pts.) Assume f is continuous everywhere. If f ( x ) x ( x 2) 4 find all the critical points of f and at each stationary point apply the second derivative test to determine relative extrema, if possible. If the second derivative test fails at a critical point, apply the first derivative test to determine the true state of affairs there. ______________________________________________________________________ 9. (12 pts.) Evaluate each of the following limits. If a limit fails to exist, say how as specifically as possible. (a) lim x → π sin( x ) x π (b) lim x 1 3 x x (c) lim x 0 x tan( x ) x 3

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TEST3/MAC2311 Page 5 of 5 ______________________________________________________________________ 10. (20 pts.) When f is defined by and . (a) (3 pts.) What are the critical point(s) of f and what is the value of f at each critical point? (b) (3 pts.) Determine the open intervals where f is increasing or decreasing. (c) (3 pts.) Determine the open intervals where f is concave up or concave down. (d) (3 pts.) List any inflection points or state that there is none. (e) (3 pts.) Locate any x-intercepts of f , or state that there isn’t any. (f) (5 pts.) Carefully sketch the graph of f below by plotting a few essential points and then connecting the dots appropriately. y x ______________________________________________________________________ Silly 10 point Bonus Problem: Show ln( x 1) x if x 0. [Say where your work is, for it won’t fit here!]
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