This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 xaxis. √ 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) ...
View
Full
Document
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.
 Winter '08
 chuchel
 Calculus, Antiderivatives, Derivative, Integrals

Click to edit the document details