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| m624 | Let f: R2 -> R be defined as in Problem 1-26. Show that Dxf (0, 0) exists for all x, although f is not even continuous at (0,0). |
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| m625 | Let f : R2 ->R be defined by
f(x, y) = { x|y| / √x2 + y2 (x, y) ≠ 0, 0 (x, y) =0. |
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| m626 | Let f : R2 ->R be defined by f (x, y) = |xy|. Show that f is not differentiable at 0 |
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| m627 | Let f: R->R 2. Prove that f is differentiable at a € R if and only if f 1 and f 2 are, and in this case
f 1(a) = ((f 1)1 (a) (f 2)1 (a)). |
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| m628 | Let f: Rn ->R be a function such that | f (x) | ≤ |x|2 . Show that f is differentiable at 0 |
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| m629 | Let f: Rn ->R. For x€ Rn,
a. Show that Deif (a) = Dif (a)..
b. Show that Dtxf (a) = Dxf(a)..
c. If f is differentiable at , show that Dxf(a) = Df(a)(x) (a) and therefore Dx + yf(a) = Dxf (a) + Dyf (a).. |
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| m630 | Let g: A ->Rp be as in Theorem 5-1. If f: Rn -> R is differentiable and the maximum (or minimum) of f on g-1 (0) occurs at , show that there are , such that |
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| m631 | Let g1, g2: R2 -> R be continuously differentiable and suppose D1 g2= D2 R1.. As in Problem 2-21, let |
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| m632 | Let g1, g2: R2-> R be continuous. Define f: R2->Rby f(x,y) =
(a) Show that D2f (x,y) = g2(x,y)
(b) How should f be defined so that D1f(x,y) =g1(x,y)?
(c) Find a function f: R2->R such that D1f (x,y)=x and D1f (x,y)=y |
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| m633 | Let Kn = {xЄRn: x1 = 0 and x2. . . x n−1 > 0}. If MCKn is a k-dimensional manifold and N is obtained by revolving M around the axis x1 = . = xn-1=0, show that N is a (k + 1) -dimensional manifold. Example: the tours (Figure 5-4). |
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| m634 | Let M be an (n – 1) -dimensional manifold in Rn. Let M (Є) be the set of end-points of normal vectors (in both directions) of length Є and suppose Є is small enough so that M(Є) is also an (n- 1)-dimensional manifold. Show that M(Є) is orientable (even if M is not). What is M(Є ) if M is the M"{o}bius strip? |
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| m635 | Let the random variable X be equally likely to assume any of the values 1/8, 1/4, or 3/8. Determine the mean and variance of X. |
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| m636 | Let the random variable X have a discrete uniform distribution on the integers. Determine the mean and variance of X. |
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| m637 | Let the random variable X have a discrete uniform distribution on the integers. Determine the mean and variance of X. |
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| m638 | Let U be the open set of Problem 3-11. Show that if f = X except on a set of measure 0, then f is not integrable on [0, 1] |
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| m639 | Let X denote the number of bits received in error in a digital communication channel, and assume that X is a binomial random variable with p _ 0.001. If 1000 bits are transmitted, determine the following:
(a) P(X =5)
(b) P(X < 2)
(c) P(X > 9)
(d) mean and variance of X |
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| m640 | List in order from least to the greatest. -8 7/8, 7 (to the 1st power), -5, |-6|, 4, |3|, - 8 5/8, - 100, 0, 1 (to the 7th power), 14/4, -67/8 |
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| m641 | Listed below are the 25 players on the opening-day roster of the 2013 New York Yankees Major League Baseball team, their salaries, and fielding positions.
Sort the players into two groups, pitchers (P) and non-pitchers (position players). Assume equal population standard deviations for the position players and the pitchers. Test the hypothesis that mean salaries of pitchers and non-pitchers are equal using the .01 significancelevel. |
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| m642 | Listed below are the 35 members of the Metro Toledo Automobile Dealers Association. We would like to estimate the mean revenue from dealer service departments. The members are identified by numbering them 00 through 34.
a. We want to select a random sample of five dealers. The random numbers are 05, 20, 59, 21, 31, 28, 49, 38, 66, 08, 29, and 02. Which dealers would be included in the sample?
b. Use the table of random numbers to select your own sample of five dealers.
c. A sample is to consist of every seventh dealer. The number 04 is selected as the starting point. Which dealers are included in thesample? |
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| m643 | Listed below are the 44 U.S. presidents and their age as they began their terms in office.
Use a statistical software package such as Excel or Minitab to help answer the following questions.
a. Determine the mean, median, and standard deviation.
b. Determine the first and third quartiles.
c. Develop a box plot. Are there any outliers? Do the amounts follow a symmetric distribution or are they skewed? Justify your answer.
d. Organize the distribution of ages into a frequencydistribution. |
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