Classify the motion of the mass on a spring as either underdamped, critically damped, or overdamped, or nd the range of the parameter value that will give the desired result. Assume x = x(t), that is, x is a function of t. $x^{\prime \prime}+b x^{\prime}+4 x=0$, underdamped

Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=t+1,\ y=t^2+3t$$

$$\underline{Point}$$

$$t=-1$$

A pair of parametric equations and a value of the parameter $t$ are given. Find the rectangular coordinates of the point on the plane curve defined by the equations for the given value of $t$. $x=t^2-4, y=1-t^2 ; t=3$

A 5.30 MeV $\alpha$ particle is scattered through $60^{\circ}$ in passing through a thin gold foil. Calculate a) the distance of closest approach, D, for a head-on collision and b) the impact parameter, b, corresponding to the $60^{\circ}$ scattering.

In given problem, find the equation of the tangent line to the given curve at the given value of t without eliminating the parameter. Make a sketch.

$$x=2 \sec t, y=2 \tan t ; t=-\frac{\pi}{6}$$

Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=cos \theta,\ y=3\sin\theta$$

$$\underline{Point}$$

$$\theta=0$$

Use the Laplace transform to solve each system.

$$\begin{aligned} x^{\prime \prime}+y^{\prime \prime}=& e^{2 t} \\ 2 x^{\prime}+y^{\prime \prime}=&-e^{2 t} \\ x(0)=0, & y(0)=0 \\ x^{\prime}(0)=0, & y^{\prime}(0)=0 \end{aligned}$$

Given that $\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\quad\frac{dx}{dt}\neq0$ Eliminate the parameter and find $dy/dx$. Then compare your result with that part of (a).

$$x=2t$$

$$y=3t-1$$

A pair of parametric equations and a value of the parameter $t$ are given. Find the rectangular coordinates of the point on the plane curve defined by the equations for the given value of $t$. $x=\tan t, y=\csc t ; t=\frac{\pi}{3}$

Suppose the slope of the Phillips curve-the parameter $\bar{v}$- increases. How would the results differ from the Volcker disinflation example considered in the chapter? What kind of changes in the economy might influence the slope of the Phillips curve?

Parametric equations and a value for the parameter t are given. Determine the coordinates of the point on the plane curve defined by the parametric equations corresponding to the given value of $t$.

$$x=t^2+3, y=6-t^3 ; t=2$$

Identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate. A politician is interested in the proportion of voters in his district who think he is doing a good job.

Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=t^2+3t+2,\ y=2t$$

$$\underline{Point}$$

$$t=0$$

Classify the motion of the mass on a spring as either underdamped, critically damped, or overdamped, or nd the range of the parameter value that will give the desired result. Assume x = x(t), that is, x is a function of t. $x^{\prime \prime}+10 x^{\prime}+25 x=0$