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Unformatted text preview: 16C Final  Fall 2002 1. Find the general solution of the diﬀerential equation x2 y − 2xy = 9. 2. The rate of increase in sales S (in thousands of units) of palm pilots is proportional to the current level of sales and inversely proportional to the square of the time t. That is, dS kS =2 dt t where k is a constant. (a) Is the constant, k , positive or negative? (b) The saturation point for the market is 400,000 palm pilots. That is, the limit of S as t → ∞ is 400. After 1 year, 100,000 palm pilots have been sold. Find S as a function of time in years. 3. Sketch the region of integration and evaluate the following integral.
1 0 1 √ y x3 1 dx dy +2 4. Find the critical points of the function f (x, y ) = x + y + 1/(xy ) and determine whether they are relative minima, relative maxima, or saddle points. 5. Use Lagrange multipliers to ﬁnd the point on the plane x + 2y + 3z = 14 which is closest to the origin. (Hint: Minimize the square of the distance in order to avoid square roots.) 6. Find the ﬁrst 5 terms of the sequence of partial sums associated to the series ∞ (−1)n 2
n=0 7. Determine whether the following series converge or diverge. Justify your answers.
∞ (a)
n=1 3n3 + 1 2n3 + n2 − 5 ∞ (b)
n=1 πn− 2 5 1 16C Final  Fall 2002 ∞ √ (c)
n=1 n 4n 8. Determine whether the following geometric series converge. If they do, ﬁnd their sum.
∞ (a)
n=0 − π 2 n ∞ (b)
n=1 (−1)n+1 2n+1 5n−1 9. Find the radius of convergence of the following power series.
∞ (a)
n=0 n2 (x + 7)n n! ∞ (b)
n=0 (3x)n 2n 10. (a) Find the 3rd Taylor polynomial centered at 0 of f (x) = √ 3 1 . 1−x (b) Find the 6th Taylor polynomial centered at 0 of f (x) = √ 3 1 1 − x2 Extra credit: Suppose that a rubber ball, when dropped on a concrete patio, rebounds 90 percent of the distance it falls. Find the total distance, both up and down, traveled by the ball if it is dropped from a height of 6 feet. 2 ...
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This note was uploaded on 12/04/2009 for the course MATH 29325 taught by Professor Zhu during the Spring '09 term at UC Davis.
 Spring '09
 Zhu
 Calculus

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