Evaluate $\int_{0}^{n} [[x ]] d x$, where n is a positive integer.

For a uniform random variable on the interval [0, B], find:

a. the cumulative distribution function F(x)

b. the median

Let $(a, b)=1$. Prove that $\left(a, b^n\right)=1$ for all positive integers $n$.

As seen , collar $D$ is released from rest and slides without friction downward a distance of $h=300 \mathrm{~mm}$ where it strikes a head fixed to the end of compound rod $A B C$. Rod segment (1) is made of aluminum $\left[E_1=70 \mathrm{GPa}\right.$, and it has a length of $L_1=800 \mathrm{~mm}$ and a diameter of $d_1=12 \mathrm{~mm}$. Rod segment (2) is made of bronze [ $\left.E_2=105 \mathrm{GPa}\right]$, and it has a length of $L_2=1,300 \mathrm{~mm}$ and a diameter of $d_2=16 \mathrm{~mm}$. What is the allowable mass for collar $D$ if the maximum normal stress in the aluminum rod segment must be limited to $200 \mathrm{MPa}$ ?

Anton van Leeuwenhoek, a $17^{\text {th }}$-century pioneer in the use of the microscope, described the microorganisms he saw as "animalcules" whose length was " 25 thousandths of an inch." How long were the animalcules in meters?

Write the numbers in given of the following sentences as decimals.
a. A flea has a body length about one sixteenth of an inch.
b. Venus orbits the sun about once every 224 and $\frac {7006} {10,000}$ days.

Two cannonballs, A and B, are fired from the ground with identical initial speeds, but with $\theta_{\mathrm{A}}$ larger than $\theta_{\mathrm{B}}$. Which cannonball reaches a higher elevation?

Complete each conversion. $\frac{3}{4}$ year=$\square$ weeks

To fill an order for 100 units of its product, a firm wishes to distribute production between its two plants, plant 1 and plant 2. The total-cost function is given by $c=f\left(q_{1}, q_{2}\right)=q_{1}^{2}+3 q_{1}+25 q_{2}+1000,$ where $q_{1}$ and $q_{2}$ are the numbers of units produced at plants 1 and 2, respectively. How should the output be distributed in order to minimize costs? (Assume that the critical point obtained corresponds to the minimum cost.)

Define the random variable of interest: X = number of _____. Is X binomial? Geometric? Justify your answer.

If $n$ is a positive integer, prove that

$$\int_0^1(\ln x)^n~dx = (-1)^nn!$$

A segment that is 1/2 of an inch long is how many sixteenths of an inch long?

In the "Fosbury flop" method of high jumping, named for the track and field star Dick Fosbury, an athlete's center of mass may pass under the bar while the athlete's body passes over the bar. Explain how this is possible.

Write with positive exponents.

$$9y^{-2}=\underline{\qquad\qquad\qquad}$$