16C-SQ02 - integer to the next like the interval[3 4...

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Total points for this page: Math 16C Final, Spring 2002 1) For each series, i) Determine if it converges. ii) Justify your reasoning. iii) Find the sum, if possible. (3 points each) a) 2.1 ( 29 n n ! n = 0 b) 0.4 ( 29 n n = 1 c) 1 n n 4 ( 29 n = 1 d) n - 1 n n = 1 2) Find the Taylor series for f ( x ) = (3 x + 7) –2 centered at c = –1 by following steps a–d. a) Find the first three derivatives of f . (3 points) b) Evaluate f (–1), f (–1), f (–1), and f ′′′ (–1). (2 points) c) Find a formula for f ( n ) (–1). (2 points) d) Use the Taylor's Theorem to write the answer. (2 points) 3) The function g ( x , y ) = 3 x 2 y – 4 xy + y 2 has three critical points. Find them, and classify each one. (Classify means determine what type of critical point each one is.) (11 points) 4) Newton's method. Consider h ( x ) = x 4 x – 1. Using derivatives and related analysis, it can be proven that this function has exactly two roots. (A root is a value x where h ( x ) = 0.) a) Determine the two unit intervals that contain roots. A unit interval is an interval from one
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Unformatted text preview: integer to the next, like the interval [3, 4]. Explain your reasoning. (5 points) b) For the larger root, make an initial guess x 1 . Use Newton's Method to find x 2 . Then write a formula for x 3 , but do not simplify. (5 points) 5) Explain how you would estimate cos(0.5) to within 0.001 using a calculator with no trig functions. (4 points) 6) Solve the differential equation dy dx = x + 1 e y , subject to the condition y (2) = ln 8. Solve your answer for y , a function of x . (4 points) Total points for this page: Extra Credit. Only 4 points, but give it a try if you want an A+: You are told that g ( x , y ) is a function of two variables, and you are given the two partial derivatives: g x = sin y + y 2 ( e x ) – 1 g y = x (cos y ) + 2 y ( e x ) + 2 y . Find all possible functions g . Hint: do something like antidifferentiation, but keep in mind that these are functions of two variables....
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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.

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16C-SQ02 - integer to the next like the interval[3 4...

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