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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 inﬂection 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 π/ 2 if x > 5 π/ 2 a probability density function? Justify your answer. (14) Use Simpson’s 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 ﬁrst student to complete the survey who attended lecture was the second student to complete the survey overall? 1...
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