Para cada uno de los verbos a continuación, trata de dar por lo menos un modismo que conozcas o que hayas oido. Luego escribe una oración con ese modismo. Puedes consultar un diccionario si es necesario.

MODELO

• meter

Escribes meter la pata

Héctor metió la pata cuando dijo que Daniel le caia mal.

echar

Alcohol wiped across a tabletop rapidly disappears. What happens to the temperature of the tabletop? Why?

Creating uses. List five garments in your wardrobe. Describe two new ways to wear each one.

Solve $\left(x^{2}+4\right) \frac{d y}{d x}=2 x-8 x y$ subject to y(0)=-1.

What is an example of alliteration in The Hill We Climb?

For given sentence, write the verb form that agrees with the subject.

A few, however, (has, have) never won.

A Taylor series usually gives better and better approximations for values of a function the more terms you use. For some series this is true only for certain values of $x$. For instance, the series for the natural logarithm,

$$\ln x=\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n}(x-1)^n$$

converges to $\ln x$ only for $0<x \leq 2$. If $x$ is outside this interval of convergence, the series does not converge to a real number. It diverges and thus cannot represent $\ln x$. In this problem, you will investigate the ratio of a term in this series to the term before it. You will also try to discover a way to find, from this ratio, whether the series converges.

a. The formula for $t_n$, the $n$th term in the series for $\ln x$, is

$$t_n=(-1)^{n+1} \frac{1}{n}(x-1)^n$$

Let $r_n$ be the ratio $\left|t_n+1 / t_n\right|$. Find a formula for $r_n$ in terms of $x$ and $n$.

b. Calculate $r_{10}$ for $x=1.2, x=1.95$, and $x=3$.

c. Let $r$ be the limit of $r_n$ as $n$ approaches infinity. Find an equation for $r$ in terms of $x$.

d. Evaluate $r$ for $x=-0.1, x=0, x=0.9$, $x=1.9, x=2$, and $x=2.1$.

e. Make a conjecture: "The series converges to $\ln x$ whenever the value of $x$ makes $r-$ ?and diverges whenever the value of $x$ makes $r-$ ?-."

f. If your conjecture is correct, you can use it to show that the series converges if $x$ is in the interval $0<x<2$. Check your conjecture by showing that it gives this interval.

Multiplying a vector by a matrix is a much more complicated process than multiplying a vector by a scalar. However, in certain special cases, matrix and scalar multiplication can lead to the same result. If $\mathbf{A} \vec{v}=\lambda \vec{v}$ where $\mathbf{A}$ is a matrix, $\vec{v}$ a nonzero vector, and $\lambda$ a scalar, then $\vec{v}$ is said to be an eigenvector of $\mathbf{A}$, and $\lambda$ is said to be an eigenvalue of $\mathbf{A}$. For instance, let

$$\mathbf{A}=\left(\begin{array}{cc}-2 & -1 \\ 8 & 7\end{array}\right), \overrightarrow{v_1}=(1,-8)$$

, and $\lambda_1=6$. We have:

$$\begin{array}{ll} \mathbf{A} \vec{v}_1=\left(\begin{array}{cc} -2 & -1 \\ 8 & 7 \end{array}\right) \cdot\left(\begin{array}{c} 1 \\ -8 \end{array}\right)=\left(\begin{array}{c} 6 \\ 1 \\ -48 \end{array}\right)=\left(\begin{array}{c} 6 \\ -48 \end{array}\right) . \\ \lambda_1 \vec{v}_1= & 6 \cdot\left(\begin{array}{c} 6 \\ -8 \end{array}\right) \end{array}$$

We see that multiplying $\vec{v}_1$ by $\lambda_1=6$ works out the same as multiplying $\vec{v}_1$ by $\mathbf{A}$; we say that $\vec{v}_1$ is an eigenvector of $\mathbf{A}$ with an eigenvalue of $\lambda_1=6$.

(a) Show that $\vec{v}_2=(1,-1)$ is an eigenvector of $\mathbf{A}$. What is the eigenvalue?

(b) Show that $\vec{v}_3=(-3,3)$ is an eigenvector of $\mathbf{A}$. What is the eigenvalue?

(c) If $\vec{v}$ is an eigenvector of $\mathbf{A}$, explain why the vectors $\vec{v}$ and $\mathbf{A} \vec{v}$ are parallel.

Is it fair to say that the synthesis of NADPH in chloroplasts is merely the reverse of NADH oxidation in mitochondria? Explain your answer.

Hypotheses are best described as:

A. assumptions.

B. replications.

C. explanations.

D. predictions.

For the following exercise,

Use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

$a=14, b=13, c=20$; find angle $C$.

Involves practices including asking questions and constructing explanations

Place a check next to each sentence in which the subject and verb agree. _____ One new product are beefalo.

If you double the temperature of a blackbody, by what factor will the total energy radiated per second per square meter increase ?