4. Let r(?) = �?, 4 3 ? 3/2, ?2 �. (a) Find T, N, and B at the
point corresponding to ? = 1. (b) Find the equation of the
osculating plane at the point corresponding to ? = 1. (c) Find the
equation of the normal plane at the point corresponding to ? =
1
If u(t) = < sin(5t),
cos(5t), t > and
v(t) = < t, cos(5t),
sin(5t) >, use the formula below to find the given
derivative.
d/dt[ u(t) * v(t)] = u'(t) * v(t) + u(t)* v'(t)
d/dt [ u(t) x v(t)] = ?
s(t)= 5t + 3/t^2 be the position in feet of a
particle after t seconds for t ≥ 1.
(a) Compute the average velocity from t = 1 to t = 3. Include
units in your answer
(b) Where is the velocity = 0?
(c) Show the accelaration for t ≥ 1 is positive.
(d) What are the units for the acceleration?
Let
f (t) =
{
7
0 ≤ t ≤ 2π
cos(5t)
2π < t ≤ 4π
e3(t−4π)
t > 4π
(a)
f (t) can be written in the form
g1(t) +
g2(t)U(t −
2π) +
g3(t)U(t −
4π)
where U(t) is the Heaviside function. Enter the
functions g1(t),
g2(t), and
g3(t), into the answer box below (in
that order), separated with commas.
(b)
Compute the Laplace transform of
f (t).
Let G = Z4 × Z4, H = ⟨([2]4, [3]4)⟩.
(a) Find a,b,c,d∈G so that G is the disjoint union of the 4
cosets a+H,b+
H, c + H, d + H. List the elements of each coset.
(b) Is G/H cyclic?
Let f(t) =t^2−1 and g(t) =e^t.
(a) Graph f(g(t)) and g(f(t)).
(b) Which is larger,f(g(5)) or g(f(5))? Justify your answer.
(c) Which is larger, (f(g(5)))′or g(f(5))′? Justify your
answer.
Let G be a group of order 42 = 2 * 3 * 7
(a) Let P7 be a Sylow 7-subgroup of G and let P3 be a Sylow
3-subgroup of G . What are the orders of P3 and P7?
(b) Prove that P7 is the unique Sylow 7-subgroup of G and that
P7 is normal.
(c) Prove that P3P7 is a subgroup of G
(d) Prove that P3P7 is a normal subgroup of G .
(e) Let P2...
5.
(a) Let σ = (1 2 3 4 5 6) in S6. Show that G = {ε, σ, σ^2, σ^3,
σ^4, σ^5} is a group using the operation of S6. Is G abelian? How
many elements τ of G satisfy τ^2 = ε? τ^3 = ε? ε is the identity
permutation.
(b) Show that (1 2) is not a product of 3-cycles. Must be
written as a proof!
(c) If a^4 = 1 and ab = b(a^2) in a...