| № |
Condition |
free/or 0.5$ |
| m163 | a. If f : R -> R satisfies f1 (a) ≠ 0 for all a €R, show that f is 1-1 on all of R. |
buy |
| m164 | a. If M is a k-dimensional manifold in Rn and k < n, show that M has measure 0.
b. If M is a closed -dimensional manifold with boundary in Rn, show that the boundary of M is ∂M. Give a counter-example if M is not closed.
c. If M is a compact -dimensional manifold with boundary in Rn, show that M is Jordan-measurable. |
buy |
| m165 | a. Let Ί: Rn -> Rn be self-adjoint with matrix A = (aij), so that aij = aji. If f (x) = <Tx, x> =Σ aij xixj, show that Dkf (x) = 2 Σj = 1 akjxj. By considering the maximum of <Tx, x>on Sn-1 show that there is xЄSn-1 and ^ ЄR with Tx = ^x.
b. If V = {yЄRn: <x, y> = 0}, show that Ί(v) CV and Ί: V and Ί: V -> V is self-adjoint.
c. Show that Ί has a basis of eigenvectors. |
buy |
| m166 | a. Let f : R -> R be defined by
F (x) = {x 2 sin 1/x) x ≠ 0,
0 x = 0. |
buy |
| m167 | a. Let f: Rn -> R be a continuously differentiable function. Show that f is not 1-1.
b. Generalize this result to t the case of a continuously differentiable function f: Rn -> Rm with m < n. |
buy |
| m168 | a. Let g: Rn -> Rn be a linear transformation of one of the following types: |
buy |
| m169 | a. Show that an unbounded set cannot have content 0. |
buy |
| m170 | a. Show that the set of all rectangles [a1, b1] x . x [an, bn] where each ai and each bi are rational can be arranged into a sequence (i.e. form a countable set).
b. If A C Rn is any set and O is an open cover of A, show that there is a sequence U1, U2, U3,. of members of O which also cover A. |
buy |
| m171 | a. Show that Theorem 5-5 is false if M is not required to be compact.
b. Show that Theorem 5-5 holds for noncom-pact M provided that w vanishes outside of a compact subset of M. |
buy |
| m172 | a. Show that this length is the least upper bound of lengths of inscribed broken lines. |
buy |
| m173 | a. Suppose that f: (0, 1) -> R is a non-negative continuous function. Show that ∫ (0, 1) exists if and only if lim Є-> ∫ c 1-c f exists.
b. Let An = [1 - 1/2n, 1 - 1/2n +1] Suppose that f: (0, 1) ->R satisfies ∫Arf = (-1)n/n and f(x) = 0 for all x Є Un An. Show that ∫(0,1)f does not exist, but limЄ->∫(Є, 1 - Є)f = log 2. |
buy |
| m174 | ABC Auto Insurance classifies drivers as good, medium, or poor risks. Drivers who apply to them for insurance fall into these three groups in the proportions 30%, 50%, and 20%, respectively. The probability a “good” driver will have an accident is .01, the probability a “medium” risk driver will have an accident is .03, and the probability a “poor” driver will have an accident is .10. The company sells Mr. Brophy an insurance policy and he has an accident. What is the probability Mr. Brophy is:
a. A “good” driver?
b. A “medium” risk driver?
c. A “poor” driver? |
buy |
| m175 | An absolute k-tensor on v is a function Vk ->R of the form |w| for w Є Ak (V). An absolute k-form on M is a function such that n (x) is an absolute k-tensor on Mx. Show that ∫Mn can be defined, even if M is not orientable. |
buy |
| m176 | According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $1,994. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $450.
a. What percent of the adults spend more than $2,500 per year on reading and entertainment?
b. What percent spend between $2,500 and $3,000 per year on reading and entertainment?
c. What percent spend less than $1,000 per year on reading and entertainment? |
|
| m177 | According to a recent survey, Americans get a mean of 7 hours of sleep per night. A random sample of 50 students at West Virginia University revealed the mean length of time slept last night was 6 hours and 48 minutes (6.8 hours). The standard deviation of the sample was 0.9 hour. At the 5% level of significance, is it reasonable to conclude that students at West Virginia sleep less than the typical American? Compute the p-value. |
|
| m178 | According to a study by the American Pet Food Dealers Association, 63% of U.S. house-holds own pets. A report is being prepared for an editorial in the San Francisco Chronicle. As a part of the editorial, a random sample of 300 households showed 210 own pets. Do these data disagree with the Pet Food Dealers Association’s data? Use a .05 level of significance. |
|
| m179 | According to a study by the Federal Reserve Board, the rate charged on credit card debt is more than 14%. Listed below is the interest rate charged on a sample of 10 credit cards.
Is it reasonable to conclude the mean rate charged is greater than 14%? Use the .01 significancelevel. |
|
| m180 | According to an IRS study, it takes a mean of 330 minutes for taxpayers to prepare, copy, and electronically file a 1040 tax form. This distribution of times follows the normal distribution and the standard deviation is 80 minutes. A consumer watchdog agency selects a random sample of 40 taxpayers.
a. What is the standard error of the mean in this example?
b. What is the likelihood the sample mean is greater than 320 minutes?
c. What is the likelihood the sample mean is between 320 and 350 minutes?
d. What is the likelihood the sample mean is greater than 350 minutes? |
|
| m181 | According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 24.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 6.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.
a. Find the probability that the return for common stocks will be greater than 0%.
b. Find the probability that the return for common stocks will be less than 20%. |
|
| m182 | According to media research, the typical American listened to 195 hours of music in the last year. This is down from 290 hours 4 years earlier. Dick Trythall is a big country and western music fan. He listens to music while working around the house, reading, and riding in his truck. Assume the number of hours spent listening to music follows a normal probability distribution with a standard deviation of 8.5 hours.
a. If Dick is in the top 1% in terms of listening time, how many hours did he listen last year?
b. Assume that the distribution of times 4 years earlier also follows the normal probability distribution with a standard deviation of 8.5 hours. How many hours did the 1% who listen to the least music actually listen? |
buy |
