Solve the following exercise about the solution of Ax=b. Follow the steps in the text to $x_p$ and $x_n$. Start from the augmented matrix with last column b.

Suppose row 3 of U has no pivot. Then that row is The equation Ux=c is only solvable provided The equation Ax=b (is) (is not) (might not be) solvable.

Solve the nonlinear inequality. Express the solution using interval notation, and graph the solution set.

$$(x+2)(x-1)(x-3) \leq 0$$

A wheel, starting from rest, rotates with a constant angular acceleration of $2.00 \text{~rad/s}^2$. During a certain $3.00 \text{~s}$ interval, it turns through $90.0\text{~ rad}$. What is the angular velocity of the wheel at the start of the $3.00 \text{~s}$ interval?

Describe elevator trips in parallel shafts. Assume that the passenger and freight elevators move at steady speeds and can start and stop instantly. The trips are nonstop unless otherwise indicated. Both elevators start at the same time. A notation like (4, 600) means that 4 minutes into a trip, the elevator is 600 feet above the bottom of the elevator shaft. Rough sketches on a coordinate system may be helpful.

In the Ellis Building,

Passenger: Starts 400 ft up, travels to (2, 40)

Freight: (0.5, 100) to (3, 250)

a. When, if ever, are the two elevators at the same height during this trip? What height is that?

b. How fast does each elevator travel?

Derive an expression for $r_d$ in terms of K, $V_{to}, V_{GSQ},$ and $I_{DQ}$ for an NMOS transistor operating in the triode region.

Discuss the state of polarization when the $x$ and $y$ components of the electric field are given by the following equabions: (a)

$$\begin{aligned} & E_x=E_1 \cos (\omega t+k z) \\ & E_y=\frac{1}{\sqrt{2}} E_0 \cos (\omega t+k z+\pi) \end{aligned}$$

(b)

$$\begin{aligned} & E_x=E_0 \sin (\omega t+k z) \\ & E_y=E_0 \cos (\omega t+k z) \end{aligned}$$

(c)

$$\begin{aligned} & E_s-E_0 \sin \left(k z-\cos +\frac{\pi}{3}\right) \\ & E_y-E_0 \sin \left(k z-\cot -\frac{\pi}{6}\right) \end{aligned}$$

(d)

$$\begin{aligned} & E_x=E_0 \sin \left(k z-\theta t+\frac{\pi}{4}\right) \\ & E_y=\frac{1}{\sqrt{2}} E_0 \sin (k z-\omega t) \end{aligned}$$

In each case, plot the rotation of the tip of the electric vector on the plane $z=0$. [Ans: (a) Linearly polarized, (b) right circularly polarined, (c) left circularly polarized, and (d) left elliptically polarized.]

What angle is equal to its complementary angle?

How did Romans use technology to improve life in the empire?

For the following exercises, start with the graph of $f(x)=4^{x}.$ Then write a function that results from the given transformation. Shift f(x) 3 units downward

A neutral solid metal sphere of radius 0.1 m is at the origin, polarized by a point charge of $6 \times 10^{-8}\ \mathrm{C}$ at location $\langle- 0.3,0,0\rangle\ \mathrm{m} .$ At location $\langle 0,0.07,0\rangle\ \mathrm{m},$ what is the electric field contributed by the polarization charges on the surface of the metal sphere? How do you know?

Use graphing technology to graph the equation given below.

$$y=\frac{216}{x^3}$$

Solve each problem.

In triangle D E F, m $\angle$ E is three times $m \angle$ D, and m $\angle$ F is $9^{\circ}$ less than m $\angle$ E. What is the measure of each angle?

Find the exact solution, using common logarithms, and a two-decimal-place approximation of each solution, when appropriate.

$$2^{-x}=8$$

How did technology help industry expand?