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Condition |
free/or 0.5$ |
| m504 | If f : Rn -> R is differentiable and f (0) = 0, prove that there exist gi: Rn -> R such that f (x) = |
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| m505 | If f: Rn -> Rn, the graph of f is {(x, y): y = f (x)}. Show that the graph of is an -dimensional manifold if and only if is differentiable. |
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| m506 | If g: Rn -> Rn and detg1 (x) ≠ 0, prove that in some open set containing we can write g = to gn 0 ∙ ∙ ∙ o g1, 0.., where is of the form gi(x) = (x1, ∙ ∙ ∙ Fi (x) , ∙ ∙ ∙ Xn), and T is a linear transformation. Show that we can write g = gn o ∙ ∙ ∙ 0g1 if and only if g1 (x) is a diagonal matrix. |
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| m507 | If is continuous, show that |
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| m508 | If A is a Jordan measurable set and ε > 0, show that there is a compact Jordan measurable set C C A such that ∫ A − C1 < ε. |
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| m509 | If M C R n is an orientable (n - 1)-dimensional manifold, show that there is an open set A C Rn and a differentiable g: A-> R1 so that M = g-1 (0) and g1 (x) has rank 1 for x ЄM. |
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| m510 | If M is an -dimensional manifold-with-boundary in Rn, define μ as the usual orientation of M x = Rnx (the orientation μ so defined is the usual orientation of M. If xЄ∂M, show that the two definitions of n (x) given above agree. |
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| m511 | If M is an -dimensional manifold (or manifold-with-boundary) in R n, with the usual orientation, show that ∫ fdx1 ^ . ^ dx n, as defined in this section, is the same as ∫ M f, as defined in Chapter 3. |
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| m512 | If M is an -dimensional manifold in Rn, with the usual orientation, show that dV = dx1^ . . . ^ dxn, so that the volume of M, as defined in this section, is the volume as defined in Chapter 3. (Note that this depends on the numerical factor in the definition of w ^ n.) |
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| m513 | If M is a k-dimensional manifold with boundary, prove that ∂M is a (k - 1) -dimensional manifold and M - ∂M is a k=dimensional manifold. |
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| m514 | If M is an oriented one-dimensional manifold in RN and c: [0, 1] ->M is orientation-preserving, show that |
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| m515 | If M1CRN is an -dimensional manifold-with-boundary and M 2 C M1 - ∂M1 is an -dimensional manifold with boundary, and M1, M2 are compact, prove that |
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| m516 | If P (A) = 0.2, P (B) = 0.2, and A and B are mutually exclusive, are they independent? |
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| m517 | If P (A) = 0.3, P (B) = 0.2, and P (A U B) = 0.1, determine the following probabilities:
(a) P (A’)
(b) P (A U B)
(c) P (A` ∩ B)
(d) P (A ∩ B’)
(e) P [(A U B’)]
(f) P (A` U B) |
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| m518 | If P (A|B) = 0.3, P (B) = 0.8, P (A) = 0.3, and are the events B and the complement of A independent? |
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| m519 | If P (A|B) = 0.4, P (B) = 0.8, P (A) = 0.5, are the events A and B independent? |
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| m520 | If P (A|B) = 1, must A = B? Draw a Venn diagram to explain your answer. |
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| m521 | If plotted, the following sales series would appear curvilinear. This indicates that sales are increasing at a somewhat constant annual rate (percent). To fit the sales, therefore, a logarithmic equation should be used.
a. Determine the logarithmic equation.
b. Determine the coordinates of the points on the logarithmic straight line for 2005 and 2011.
c. By what percent did sales increase per year, on the average, during the period from 2004 to 2014?
d. Based on the equation, what are the estimated sales for2015? |
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| m522 | A dry cleaning establishment claims that a new spot remover will remove more than 70% of the spots to which it is applied. To check this claim, the spot remover will be used on 12 spots chosen at random. If fewer than 11 of the spots are removed, we shall not reject the null hypothesis that p = 0.7: otherwise, we conclude that p > 0.7.
(a) Evaluate a, assuming that p = 0.7.
(b) Evaluate 8 for the alternative p = 0.9. |
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| m523 | If the height and the base of a triangle are doubled, what happens to the area? Explain |
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