Traza una línea entre las casillas (squares) y escribe una historia coherente. Observa el significado de las palabras, los verbos y las expresiones de tiempo para guiarte. ¡Hay dos historias mezcladas! tuve un accidente.

Find $\lim _{x \rightarrow+\infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x).$ If the limiting value is infinite, indicate whether it is $+ \infty$ or $-\infty. \quad f(x)=\frac{2 x+1}{3 x^{2}+2 x-7}$

In this exercise, determine the convergence or divergence of the series.

$$\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^n n}{n^2-3}$$

True or False: A square is a rhombus. Explain your choice.

Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than $0.001$. Then find the zero(s) using a graphing utility and compare the results. \

$$f(x)=x^3+3$$

Use the Integral Test to determine the convergence or divergence of the series.

$$\sum_{n=1}^{\infty} \frac{n}{n^4+1}$$

Find the curl and divergence of the given vector field. $y^{2} \mathbf{i}+4 x^{2} y \mathbf{j}$

(Taylor approximations and root finding.) Newton’s method $x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$ approximates solutions of f (x) = 0 near the input $x_0.$ a. If f and g are inverse functions, explain why a solution of g(x) = a is the value f (a) of f. b. Let $p_{N}(x)$ be the N th degree Maclaurin polynomial of $e^{x} .$ Use Newton’s method to approximate solutions of $p_{N}(x)-2=0$ for N = 4, 5, 6. c. Explain why the approximate roots of $p_{N}(x)-2=0$ are approximate values of ln(2).

Find the indicated roots of the given quadratic equations by finding $x_3$ from Newton’s method. Compare this root with that obtained by using the quadratic formula.

$x^2 + 4x + 2 = 0$ (between $-4$ and $-3$)

Mention two uses of the Clapeyron equation.

In this exercise, assume that all variables represent positive real numbers.

In this exercises, rationalize each denominator.

$$\frac{6}{\sqrt[3]{4}}$$

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. $-7^{-2}$

In given problem, first find the general solution (involving a constant C ) for the given differential equation. Then find the particular solution that satisfies the indicated condition.

$\frac{d y}{d x}=x^{-3}+2 ; y=3$ at x=1

Find (a) the divergence and (b) the curl of the vector field F.

$$\mathbf{F}(x,y,z)=yz^2\mathbf{i}+x^2z\mathbf{j}$$