1.6 Practice Prelim

1.6 Practice Prelim - Parts of the first(I and second(II...

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Parts of the first (I) and second (II) exams from Durrett, Spring 1998 . I.2. Differentiate the following (a) 11 + 7 x + 5 x 3 + 3 x 2 + 2 x (b) x 2 +3 x 3 +2 x +4 (c) ( x 3 + 3 x )( x 2 + 1) 4 (d) p 1 + 2 + 3 x I.3. Peekaboo Streak ( the Winter Olympics were in 1998 ) is at the top of a ski jump. We know that her height as a function of t will be a polynomial f ( t ) = a + bt + ct 2 + dt 3 + et 4 Use the following information to determine the constants a , b , c , d , and e . (i) Her height at time t = 0 is 70. (ii) Her velocity at time t = 0 is 0. (iii) Her acceleration at any time t is 16 - t 2 . II.2. Sketch the graph of the following function f ( x ) = x 3 3 - 5 x 2 2 + 4 x Indicate the set of values where it is (a) increasing, (b) convex (concave up). II.3. Bruno knows from his college daze that the function g ( x ) = x ln x has either a maximum or minimum at some point x > 0. Use the first derivative to find the location of this point and the second derivative test to tell whether it is a minimum or maximum.
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