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The winter 2011 exam 2 for math 106, covering topics such as integration and probability density functions. The exam includes questions on evaluating integrals using substitution and without using the table of integrals, finding the expected value of a random variable, and finding the 4th order taylor polynomial and estimating e−1 using it. Additionally, the document includes some useful trigonometric identities.
Typology: Exams
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Note: Read directions carefully and show all your work. Partial credit will be assigned based upon the correctness, completeness, and clarity of your answers.
Name:
ID #:
Question 1) Evaluate the following integrals
a) (^) ∫
sin^4 x cos^3 x dx.
b) (^) ∫
sin^2 x cos^2 x dx.
Question 2) Evaluate the integral (^) ∫ 3 x + 5 x^2 − 4 dx.
Question 4) Let T be the number of minutes that it takes a randomly selected person to solve a certain puzzle. The probability density function of T is
f (x) =
3 /x^4 if x ≥ 1 0 if x < 1
a) What is the probability that T > 2?
b) The expected number of minutes (the mean) that it takes for a random person to solve the puzzle is given by∫ ∞ −∞ xf^ (x)^ dx. Find the expected number of minutes it takes a random person takes to solve the puzzle.
Question 5) Consider the function f (x) = e−^3 x.
a) Find P 4 (x), the 4th^ order Taylor polynomial, of f (x) centered at x = 0.
b) Use P 4 (x) to find an estimate for e−^1.
c) Use Taylor’s Theorem to approximate the error of your estimate from part (b) on the interval [0, 1 /3]. Recall that error bounds for estimates using a Taylor Polynomial Pn(x) may be determined using:
|f (x) − Pn(x)| ≤
Kn+ (n + 1)! |x − x 0 |n+1,
where Kn+1 is a constant such that |f (n+1)(x)| ≤ Kn+1 for all x in [0, 1 /3].
Question 7) Determine whether the improper integral
∫ (^) ∞
1
dx x^2 +
x
converges using a comparison test.
Useful Trigonometric Identities:
