For the following exercise, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph.

$$g(x)=\left\{\begin{array}{l} \frac{3}{x-2}, x \neq 2 \\ 4, x=2 \end{array} ; g(0) ; g(-4) ; g(2)\right.$$

The positions of two cars in parallel lanes of a straight stretch of highway are plotted as functions of time in the Figure 2-33. Take positive values of x as being to the right of the origin. Qualitatively answer the following:

$(e)$ Sketch (no numbers) the velocity versus time curve for each car.

A 40-ft truck doing $50 \mathrm{mph}$ and a $60 -ft$ truck doing $40 \mathrm{mph}$ are traveling in the same direction in adjacent lanes on an interstate highway. How long (in seconds) does it take the faster truck to pass the slower truck?

Let R and $\Theta$ be independent random variables such that R has the Rayleigh density

$$f_{R}(r)=\left\{\begin{array}{ll} \sigma^{-2} r e^{-r^{2} / 2 \sigma^{2}}, & r \geq 0 \\ 0, & r<0 \end{array}\right.$$

and $\Theta$ is uniformly distributed on $(-\pi, \pi)$. Show that $X = R \cos \Theta$ and $Y=R \sin \Theta$ are independent random variables and that each has the normal density $n\left(0, \sigma^{2}\right)$.

An apparel shop sells skirts for $45 and blouses for$35. Its entire stock is worth $51,750, but sales are slow and only half the skirts and two-thirds of the blouses are sold, for a total of$30,600. How many skirts and blouses are left in the store?

Calculate the determinant of the given matrix

$$\left( \begin{array}{rrr}{-1} & {0} & {-1} \\ {1} & {1} & {2} \\ {2} & {1} & {3}\end{array}\right)$$

. Determine if the matrix has a nontrivial nullspace, and if it does find a basis for nullspace. Determine if the column vector is the matrix are linealy independent.

In slow-pitch softball, the distance of the paths between each pair of consecutive bases is 65 feet and the paths form right angles. Find the distance the catcher must throw a baseball from 3 feet behind home plate to second base.

Calcium oxide $(\mathrm{CaO})$ is used to remove sulfur dioxide generated by coal-burning power stations:

$$2 \mathrm{CaO}(s)+2 \mathrm{SO}_2(g)+\mathrm{O}_2(g) \longrightarrow 2 \mathrm{CaSO}_4(s)$$

Calculate the enthalpy change for this process if $6.6 \times 10^5 \mathrm{~g}$ of $\mathrm{SO}_2$ are removed by this process every day.

The joint density function for random variables $X$ and $Y$ is

$$f(x, y)= \begin{cases}C(x+y) & \text { if } 0 \leqslant x \leqslant 3,0 \leqslant y \leqslant 2 \\ 0 & \text { otherwise }\end{cases}$$

Find the value of the constant $C$.

Evaluate each piecewise function at the given values of the independent variable.

$$h ( x ) = \left\{ \begin{array} { c c c } { \frac { x ^ { 2 } - 9 } { x - 3 } } & { \text { if } } & { x \neq 3 } \\ { 6 } & { \text { if } } & { x = 3 } \end{array} \right.$$

$$h(0)$$

If you plot the point -8.85 on a number line, would you place it to the left or right of -8.8? Explain.

Determine whether the given matrices are linearly independent.

$$\left[ \begin{array} { r r } { 0 } & { 1 } \\ { 5 } & { 2 } \\ { - 1 } & { 0 } \end{array} \right] , \left[ \begin{array} { l l } { 1 } & { 0 } \\ { 2 } & { 3 } \\ { 1 } & { 1 } \end{array} \right],\left[ \begin{array} { r r } { - 2 } & { - 1 } \\ { 0 } & { 1 } \\ { 0 } & { 2 } \end{array} \right] , \left[ \begin{array} { r r } { - 1 } & { - 3 } \\ { 1 } & { 9 } \\ { 4 } & { 5 } \end{array} \right]$$

Find a function of the form $f(x)=a+b \cos c x$ that is tangent to the line y=1 at the point (0,1), and tangent to the line

$$y=x+\dfrac{3}{2}-\dfrac{\pi}{4}$$

at the point $\left(\dfrac{\pi}{4}, \dfrac{3}{2}\right)$

a. What is viscosity? b. Which has a higher viscosity, a fast-flowing liquid or a slow-flowing liquid? c. What can you infer about the amount of friction among the particles of a liquid that has low viscosity?