Use a CAS to solve the initial value problem. Plot the solution curves.

$$y^{\prime}=\frac{1}{\sqrt{4-x^{2}}}, \quad y(0)=2$$

Completa estas oraciones con las formas verbales que se usan en el texto.

La canción protesta se compone con el propósito de ____________ problemas sociales o políticos.

Consider the functions f(x)=a sin 2x and g(x)=(sin x)/a, where a>0 is a real number. Graph the two functions on the interval $[ 0 , \pi / 2 ]$, for $a = \frac { 1 } { 2 }$, 1, and 2.

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. Find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). $\frac{d y}{d x}=(y-1)^{2}, \quad y(0)=1$

When using rectangles to approximate the area of a region between the curves $y=f(x)$ and $y=g(x)$, how do you choose whether to represent the height of the rectangle as $f(x)-g(x)$ or $g(x)-f(x)$?

Find the distance between each pair of points. a. (17, 29) and (-1, -1) b. (x, y) and (-1, -1)

Show that $\int_{1}^{\infty} \frac{1}{x^{p}} d x$ converges if p>1 and diverges if $p \leq 1$.

Find the lengths of the curves. $x=\ln \left(1+t^{2}\right), y=2 \tan ^{-1} t,(0 \leq t \leq 1)$

In this problem, find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as $t$ increases.

$$4 y^{\prime \prime}-y=0, \quad y(-2)=1, \quad y^{\prime}(-2)=-1$$

For each of the following stirred tank reactions, carry out the following analysis. a. Write an initial value problem for the mass of the substance. b. Solve the initial value problem. A 2000-L tank is initially filled with a sugar solution with a concentration of 40 g/L. A sugar solution with a concentration of 10 g/L flows into the tank at a rate of 10 L/min. The thoroughly mixed solution is drained from the tank at a rate of 10 L/min.

Use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. dy/dx = -x/y and y = 3 when x = 4

Find the slope of the line tangent to the following polar curves at the given points. $r=6+3 \cos \theta ;(3, \pi)$ and (9, 0)

Find the arr length of the following curves on the given interval

$$y=\ln \left(x-\sqrt{x^{2}-1}\right), \text { for } 1 \leq x \leq \sqrt{2}$$

Simplify each complex fraction. $\frac{\frac{1}{x+1}-\frac{1}{x}}{\frac{1}{x}}$