Two orthogonal vectors $u$ and $v$ are given. Compute the quantities $||u||^2$, $||v||^2$, and $||u+v||^2$. Use your results to illustrate the Pythagorean theorem.

$$\begin{equation*} u=\begin{bmatrix}1\\3\\2\end{bmatrix}\text{ and } v=\begin{bmatrix}-1\\1\\-1\end{bmatrix} \end{equation*}$$

Raul is explaining how he and his family are preparing for his siter´s birthday party. Read his description and answer the questions that follow in complete sentences.
¿Con que decora Raul?

Sketch the graph of the height of a particle against time if velocity is positive and acceleration is negative.

In living things, energy is stored in the molecule adenosine triphosphate, called ATP for short. ATP is said to act like a battery, storing energy when it is not needed and instantly releasing energy when it is required. Investigate how energy is stored and the role of ATP in biological processes. Write a report including at least three references.

$\angle A B D$ and $\angle B D C$ are complementary. Find the measures of both angles. $\mathrm { m } \angle A B D = ( 5 y + 1 ) ^ { \circ } , \mathrm { m } \angle B D C = ( 3 y - 7 ) ^ { \circ }$

Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)

$$\geq g ( x ) \geq 0$$

. How is this formula changed if the line

$$y=y_0$$

lies above R?

Graph the relation {(-2, 2), (-1, 1), (0, 1), (1, 1), (2, 0)}. Is the relation a function?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I have two points to work with, I use the formula for slope

$$\frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } }$$

to find the slope of the tangent line to the graph of a function

$$y = f ( x ) \text { at } ( a , f ( a ) ).$$

Consider the following reactions: $\mathrm { H } _ { 2 } ( g ) + \mathrm { I } _ { 2 } ( g ) \rightleftharpoons 2 \mathrm { HI } ( g ) \quad \text { and } \quad \mathrm { H } _ { 2 } ( g ) + \mathrm { I } _ { 2 } ( s ) \rightleftharpoons 2 \mathrm { HI } ( g )$ List two property differences between these two reactions that relate to equilibrium.

State the first step in solving each equation.

$$2 b + 9 = 3$$

Solve each equation on the interval

$$[0, 2\pi).$$

Use exact values where possible or give approximate solutions correct to four decimal places.

$$\tan x = 2 \cos x \tan x$$

Use a calculator or CAS to evaluate the following integrals. [T] $\int_{0}^{1} x \cdot e^{-x^{2}} d x$

Multiply. Simplify your answer.

$$\frac { a ^ { 2 } } { a } \left( a ^ { 2 } + 10 a + 25 \right)$$

A classical example of the Poisson distribution involves the number of deaths caused by horse kicks to men in the Prussian Army between 1875 and 1894. Data for 14 corps were combined for the 20-year period, and the 280 corps-years included a total of 196 deaths. After finding the mean number of deaths per corps-year, find the probability that a randomly selected corps-year has the fo llowing numbers of deaths: 0. The actual results consisted of these frequencies: 0 deaths (in 144 corps-years); l death (in 91 corps-years); 2 deaths (in 32 corps-years); 3 deaths (in 11 corps-years); 4 deaths (in 2 corps-years). Compare the actual results to those expected by using the Poisson probabilities. Does the Poisson distribution serve as a good tool for predicting the actual results?