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sample_final - MAT16b Sample Final Problem 1. (15 pts) Find...

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Unformatted text preview: MAT16b Sample Final Problem 1. (15 pts) Find the following indefinite integrals (antiderivatives): (a) (b) (c) x ln(x) dx ln(x) dx x 1 dx x ln2 (x) Problem 2. (15 pts) Find the following definite integrals: π /2 (a) 0 x cos(x) dx ln(2) (b) 0 π /2 ex dx ex − 1 (c) 0 sin(x) cos(x) dx Problem 3. (20 pts) Which of the following improper integrals converge and which diverge? ∞ (a) 0 2 t2 e−t dt 1 dt (t − 1)2 1 √ dt t e−t dt −∞ 1 3 (b) 1 1 (c) 0 0 (d) (e) −1 √ 1 dt 1 − x2 Problem 4. (10 pts) Find all of the critical points of the function f (x) = 2sin(x) in the interval [0, 2π ). 1 Problem 5. (15 pts) Find the antiderivative of the function f ( x) = x2 − 1 . x2 + 2 x − 8 Problem 6. (15 pts) Let X be a random variable with values in the interval [0, ∞) whose probability density function is f (x) = e−x . What is the expected value of X? Problem 7. (10 pts) What is the area of the region bounded by the graph of y = x3 − 3x2 − 4x and the x-axis. √ Problem 8. (10 pts) Find a constant C such that f (x) = C x is a probability density function on the interval [0, 4]. √ If X is a random variable on [0, 4] with probability density function is C x, where C is the constant you just found, what is the probability that X is between 1 and 4? Problem 9. (10 pts) If f (x) > 0 for all x, show that the the antiderivative of cos(x)f (x) − f (x) sin(x) [f (x)]2 is sin(x) . f (x) ...
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This note was uploaded on 10/08/2010 for the course MATH 16b taught by Professor Chuchel during the Winter '08 term at UC Davis.

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sample_final - MAT16b Sample Final Problem 1. (15 pts) Find...

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