Question

Let f: Z6 --> Z2 X Z3 be the function given by f([a]6) = ([a]2,[a]3). (a) Show that f is well-defined; that is, show that if [a]6=[b]6, then f([a]6) = f([b]6). (b) Prove that f is an isomorphism.

Answer #1

Consider the three groups G1 = Z12,
G2 = Z6 x Z2, and G3 =
Z4 x Z3. (The operation is addition in all
cases)
(a) Find an isomorphism between two of them
(b) explain why the third group is different

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

Let T ∈ L(C3) be the operator given by
T(z1,z2,z3)=(z1 +z2 −2z3,z1 +z2 −2z3,z1 +z2 −2z3).
Find a basis of C3 such that M(T ) is block diagonal with
upper-triangular blocks (as guaranteed by 8.29) and write the
matrix M(T ) in this basis.

Calculate ∫ ∫S f(x,y,z)dS for the given surface and function.
x2+y2+z2=144, 6≤z≤12; f(x,y,z)=z2(x2+y2+z2)−1.

Let f: Z -> Z be a function given by f(x) = ⌈x/2⌉ + 5. Prove
that f is surjective (onto).

5. Prove that the mapping given by f(x) =x^3+1 is a function
over the integers.
6. Prove that f(x) =x^3+is 1-1 over the integers
7. Prove that f(x) =x^3+1 is not onto over the
integers
8 Prove that 1·2+2·3+3·4+···+n(n+1)
=(n(n+1)(n+2))/3.

Given the function f(x, y, z) = (x2 + y2 +
z2 )−1/2
a) what is the gradient at the point (12,0,16)?
b) what is the directional derivative of f in the direction of
the vector u = (1,1,1) at the point (12,0,16)?

Let
F=(x2+y+2+z2)i+(ex2+y2)j+(3+x)k.
Let
a>0
and let
S
be part of the spherical surface
x2+y2+z2=2az+15a2
that is above the x-y plane. Find the flux of
F
outward across
S.

Let F be the defined by the function F(x, y) = 3 + xy - x - 2y,
with (x, y) in the segment L of vertices A (5,0) and B (1,4). Find
the absolute maximums and minimums.

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