Problems

 

NOTE: The multiple choice answers to problem (8) in the version of the exam distributed in class did not contain the correct answer. This has been corrected in the current version.

(1)

10 POINTS   Find the sixth Taylor polynomial $p_{6}(x)$ of $f(x) = \sin (x^{3}) - x$ at $x = 0$.

(a) $1 + x - x^{3}/6$     (b) $-x + x^{3}$     (c) $-x + x^{3}/6$     (d) $1 + x^{3}$     (e) $x^{3}/6$

SOLUTION KEY: 3.1         SOLUTION: 4.1

(2)

10 POINTS   Suppose that the scores of all people who took the general GRE test in 1997 are normally distributed with a mean score of 1900 points and a standard deviation of 300 points. Which of the following is closest to the lowest score of a person who falls in the top $20 \%$ of scores?

(a) $2302$     (b) $1958$     (c) $2000$     (d) $2152$

(e) none of the above

SOLUTION KEY: 3.2         SOLUTION: 4.2

(3)

10 POINTS   Let $X$ be a continuous random variable taking values in the interval $[0,1]$ and having a probability density function $f(x) = c\sqrt{x^{3}}$. Find the expected value of $X$.

(a) $2/5$     (b) $5/7$     (c) $7/2$     (d) $2/7$      (e) $5/2$

SOLUTION KEY: 3.3         SOLUTION: 4.3

(4)

10 POINTS   If

$$a_{n} = \frac{4 - 3n^{2} + 6\sqrt{n}}{1 - \sqrt[3]{1 - 27n^{6} + 8n}},$$

then $\lim_{n \to \infty} a_{n}$ is

(a) $-3/2$     (b)     (c) $-1$     (d) $1$     (e) $-3/4$

SOLUTION KEY: 3.4         SOLUTION: 4.4

(5)

10 POINTS   Let $f(x) = e^{2x}$. what is the minimal degree of the Taylor polynomial at $x = 0$ guaranteed by the Taylor remainder theorem to approximate $f(-1/5)$ with an error not exceeding $0.01$.

(a) $2$     (b) $3$     (c) $5$     (d) $4$     (e) $1$

SOLUTION KEY: 3.4         SOLUTION: 4.4

(6)

10 POINTS   The number of miles per hour by which drivers violate the speed limit on a certain country road is an exponential random variable with a mean of $5$ miles per hour. What is the probability that a randomly picked driver will violate the speed limit by no more than $15$ miles per hour?

(a) $1 - e^{-15}$     (b) $1 - e^{-3}$     (c) $e^{-1/5}$      (d) $e^{-3}$     $1 - e^{-15}$

SOLUTION KEY: 3.6         SOLUTION: 4.6

(7)

10 POINTS   Let $Z$ be the standard normal random variable. Suppose that a pair of fair dice is cast $180$ times and that the number $x$ of tossings for which the sum of the two uppermost faces is equal to $7$ is recorded. What is the standard normal probablility that you would have to calculate to approximate the probability that $x \leq 62$.

(a) $P(Z \leq 6.5)$     (b) $P(Z \gt 6.1)$     (c) $P(Z \leq 2.5)$    

(d) $P(Z \leq 4.5)$     (e) $P(Z \leq 6.1)$

SOLUTION KEY: 3.7         SOLUTION: 4.7

(8)

10 POINTS   If the degree one Taylor polynomial of $f(x) = \sqrt{1 + x}$ at $x = 3$ is used to estimate $f(3.08)$ then the approximation is

(a) $2.22$     (b) $1.9$     (c) $1.09$     (d) $2.04$     (e) $2.02$

SOLUTION KEY: 3.8         SOLUTION: 4.8

(9)

10 POINTS   The number of users of a certain computer network who chose their birthdate as a login password is a random variable with a mean $\mu = 300$ and a standard deviation $\sigma = 16$. Use Chebishev's inequality to find the probabiblity that the number of users who chose their birthdate as a password is between $276$ and $324$.

(a) $5/9$     (b) $3/4$     (c) $7/16$     (d) $1/3$      (e) $1/16$

SOLUTION KEY: 3.9         SOLUTION: 4.9

(10)

10 POINTS   Among the $9$ new releases in a certain video library there are $4$action movies. If a customer rents three of the new releases picked at random, what number of action movies does she expect to get.

(a) $2$     (b) $3/4$     (c) $4/3$     (d) $8/5$     (e) $5/4$

SOLUTION KEY: 3.10         SOLUTION: 4.10