if (X_{1},...,X_{n}) are independent random variables, each with the density function f, show that the joint density of (X_{(1)},...,X_{(n)} is n!f (x_{1})..f(x_{n}), x_{1}<x_{2}<...<x_{n}

A force acting on a particle moving in the xy plane is given by $F=(2yi+x^{2}j)N$, where x and y are in meters. The particle moves from the origin to a final position having coordinates x=4.65 m and y=4.65 m, as in Figure.

a) Calculate the work done by F along OAC
b) Calculate the work done by F along OBC
c) Calculate the work done by F along OC
d) Is F conservative or nonconservative?

A sled with rider having a combined mass of 120 kg travels over the perfectly smooth icy hill shown in the accompanying figure. How far does the sled land from the foot of the cliff (in m)?

A 5.00 kg package slides 1.50 m down a long ramp that is inclined at ${\displaystyle {12.0}^{\circ }}$ below the horizontal. The coefficient of kinetic friction between the package and the ramp is k =0.310. Calculate
(a) the work done on the package by friction (b) the workdone on the package by gravity (c) the work done on the package bythe normal force (d) the total work done on the package (e) Ifthe package has a speed of 2.20 m/s at the top of the ramp, what isits speed after sliding 1.50 m down the ramp?

A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors ${\displaystyle \hat{i}}$ and ${\displaystyle \hat{j}}$.
1. (Enter in box 1) $\frac{m}{s^2}\hat{i}+(\text{ Enter in box 2 })\frac{m}{s^2}\hat{j}$
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) ${\displaystyle \frac{m}{s^2}\hat{i}+}$
5. ( Enter in box 5) ${\displaystyle \frac{m}{s^2}\hat{j}}$

Evaluate the following using Limits of Theorem
$\underset{x\to 16}{lim}(\frac{x^2-256}{4-\sqrt{x}})$

For an arbitrary n in Z, the cyclic subgroup ${\displaystyle ⟨n⟩}$ of Z, generated by n under addition, is the set of all multiples of n. Describe the subgroup ${\displaystyle ⟨m⟩\cap ⟨n⟩}$ for arbitrary m and n in Z.

Let ${\displaystyle V}$ be the variety of algebras ${\displaystyle (A,\cdot )}$ satisfying the identitie
${\displaystyle x\cdot x\approx x}$ and ${\displaystyle (x\cdot y)\cdot z\approx (z\cdot y)\cdot x}$
Let ${\displaystyle W}$ be the subvariety of V defined by the additional identity ${\displaystyle y\cdot (x\cdot y)\approx }$
Determine ${\mathbf{F}}_{\mathbf{W}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$. Write out a Cayley table.