Question

Consider the following.

Fourth roots of −4

(a) Use the formula
*z*_{k} =

n |
r |

cos

θ +
2πk |

n |

+ * i* sin

θ +
2πk |

n |

to find the indicated roots of the complex number. (Enter your
answers in trigonometric form. Let 0 ≤ *θ* <
2*π*.)

z_{0} |
= | |

z_{1} |
= | |

z_{2} |
= | |

z_{3} |
= |

(b) Write each of the roots in standard form.

z_{0} |
= | |

z_{1} |
= | |

z_{2} |
= | |

z_{3} |
= |

Answer #1

Let Θ ∼ Unif.([0, 2π]) and consider X = cos(Θ) and Y =
sin(Θ).
Can you find E[X], E[Y], and E[XY]?
clearly, x and y are not independent
I think E[X] = E[Y] = 0 but how do you find E[XY]?

Consider the following vector function.
r(t) =
6t2, sin(t) − t cos(t), cos(t) + t sin(t)
, t > 0
(a) Find the unit tangent and unit normal vectors
T(t) and
N(t).
T(t)
=
N(t)
=
(b) Use this formula to find the curvature.
κ(t) =

Consider the function on the interval (0, 2π). f(x) = sin(x)
cos(x) + 4. (A) Find the open interval(s) on which the function is
increasing or decreasing. (Enter your answers using interval
notation.) (B) Apply the First Derivative Test to identify all
relative extrema.

Consider the ellipse r(t) =〈3 cos(t),4 sin(t)〉, for 0 ≤ t ≤
2π.
(a) At what positions does ‖r′(t)‖ have maximum and minimum
values, that is, where is a particle moving along the ellipse
moving the fastest and slowest? Your answer will be vectors.
(b) At what positions does the curvature have maximum and
minimum values? Your answer will be vectors.

When plotted in the complex plane for , the function f () =
cos() + j0.1 sin(2) results in a so-called Lissajous figure
that resembles a two-bladed propeller.
a. In MATLAB, create two row vectors fr and fi corresponding to
the real and imaginary portions of f (), respectively, over a
suitable number N samples of . Plot the real portion against the
imaginary portion and verify the figure resembles a propeller.
b. Let complex constant w = x +...

3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into
irreducible polynomials in each of the following rings, justifying
your answers briefly:
(i) Z3 [x];
(ii) Q[x] (this can be done easily using an appropriate
theorem);
(iii) R[x] (hints: you may find it helpful to write γ = 2^(1/4),
the positive real fourth root of 2, and to consider factors of the
form x^2 + a*x + 2^(1/2);
(iv) C[x] (you may leave your answer in...

Consider the following collision problem from the game of golf.
A golf club driver is swung and hits a golf ball. The driver has a
shaft of length L and a head. To simplify the problem, we will
neglect the mass of the driver shaft entirely. The driver head we
will assume is a uniform density sphere of radius R and mass M. The
golf ball is a uniform density sphere of radius r and
mass m. Initially the ball is...

Applications I
Consider the following data representing the total time (in hours)
a student spent on reviewing for the Stat final exam and the actual
score on the final. The sample of 10 students was taken from a
class and the following answers were reported.
time score
0 23
4 30
5 32
7 50
8 45
10 55
12 60
15 70
18 80
20 100
Part 1: Use the formulas provided on the 3rd formula sheet to
compute...

Statistical Analysis for Business Applications I
Consider the following data representing the total time (in hours)
a student spent on reviewing for the Stat final exam and the actual
score on the final. The sample of 10 students was taken from a
class and the following answers were reported.
time score
0 23
4 30
5 32
7 50
8 45
10 55
12 60
15 70
18 80
20 100
Part 1: Use the formulas provided on the 3rd...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 11 minutes ago

asked 14 minutes ago

asked 14 minutes ago

asked 19 minutes ago

asked 37 minutes ago

asked 44 minutes ago

asked 47 minutes ago

asked 49 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago