Math16B - Spring 09 Final Review Questions

Math16B - Spring 09 Final Review Questions - y (0) = 2....

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MATH 16B - SPRING 2009 - FINAL EXAM REVIEW PROBLEMS (1) Let f ( x,y ) = 3 x 2 + 6 xy + 4 y 2 + 4 y . Find all maxima, minima, and saddle points of f ( x,y ). (2) Let R = { ( x,y ) | 0 y ln( x ) , 1 x e } . Compute Z Z R y dx dy (3) Use two iterations of Newton’s method with an initial guess of x 0 = 3 to find an approximation for 10. Express your answer as a fraction p/q with p and q whole numbers. (4) Let f ( t ) be the solution for the initial value problem y 0 = t 2 + t + 1 y 3 / 2 y (0) = 9 . (a) Use Euler’s method with n = 1 steps to approximate f (1). (b) Solve the initial value problem. (5) Solve the initial value problem y 0 + 2 t y = sin t y ( π ) = 7 . (6) Find (in notation) an infinite series that converges to Z 1 0 e 2 x 2 - 1 x dx. (7) A continuous random variable X has cumulative distribution function F ( x ) = 0 if x < 0 e x - 1 if 0 x ln 2 1 if x > ln(2) . Compute E ( X ). (8) Compute the following integral if it converges, or show that it diverges if not: Z 0 e 2 x e 4 x + 6 e 2 x + 9 dx. (9) Let f ( x ) = x 3 e x 4 . Compute f (10) (0). (10) Does the sum X k =2 1 k (ln k ) 4 converge? Justify your answer. (11) Sketch the solution to y 0 = sin(3 y ) satisfying
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Unformatted text preview: y (0) = 2. Include any extreme values (maxima and minima), asymptotes, concavity, and inection points. (12) Find the maximum value of the function f ( x,y ) = 2 x + 4 y subject to the constraint x 2 + y 2 = 1. (13) Is the function f ( x ) = if x < sin( x ) if 0 x 5 / 2 if x > 5 / 2 a probability density function? Justify your answer. (14) Use Simpsons rule with n = 2 subdivisions to approximate Z 3 1 1 x dx. Express your answer as a fraction p/q with p and q whole numbers. (15) In a certain class, 25% of students attend lecture on any given day. If all of the students complete an online survey about their attendance for the most recent lecture, what is the probability that the rst student to complete the survey who attended lecture was the second student to complete the survey overall? 1...
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This note was uploaded on 05/12/2010 for the course CHEM 3A taught by Professor Fretchet during the Fall '08 term at University of California, Berkeley.

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