Investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b. $f ( x ) = ( x - a ) ^ { 2 } + b$

Investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b. $f ( x ) = \sqrt { b - ( x - a ) ^ { 2 } }$

$y=1 /\left(x^{2}+c\right)$ is a one-parameter family of solutions of the first-order $\text { DE } y^{\prime}+2 x y^{2}=0$. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. $y(-2)=\frac{1}{2}$

A hypocycloid is the path traced out by a point on a smaller circle of radius b that is rolling inside a larger circle of radius a>b. Find parametric equations for the hypocycloid and graph it for a=5 and b=3. Find an equation in terms of the parameter t for the slope of the tangent line to the hypocycloid and determine one point at which the tangent line is vertical. What interesting simplification occurs if a=2b?

Write a method in the HashIntSet class called removeAll that accepts another hash set as a parameter and ensures that this set does not contain any of the elements from the other set. For example, if the set stores [-2, 3, 5, 6, 8] and the method is passed [2, 3, 6, 8, 11] , your set would store [-2, 5] .

Seventy-two percent of Americans squeeze their toothpaste tube from the top. This and other not-so-serious findings are included in The First Really Important Survey of American Habits. Those outcomes are based on 7000 responses from the 25,000 questionnaires that were mailed.

b. As stated, the value of $72 \%$ refers to all Americans, so is that $72 \%$ a statistic or a parameter? Describe.

$y=1 /\left(x^{2}+c\right)$ is a one-parameter family of solutions of the first-order $\text { DE } y^{\prime}+2 x y^{2}=0$. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. $y(2)=\frac{1}{3}$

Identify the population, the parameter, the sample, and the statistic in the following setting. How much do gasoline prices vary in a large city? To find out, a reporter records the price per gallon of regular unleaded gasoline at a random sample of 10 gas stations in the city on the same day. The range (maximum - minimum) of the prices in the sample is 25 cents.

Simulate the world’s population from 1950 to 2050 using (a) the analytical solution, and (b) the fourth-order RK method with a step size of 5 years. Employ the following initial conditions and parameter values for your simulation: $p_{0}$ (in 1950) = 2,560 million people, $k_{g m}$ = 0.026/yr, and $p_{\max }$ =12,000 million people.

Find the general solution of the separable differential equation given. Note that this problem, maybe the general solution will be a relationship between x and y, rather than y = f(x).

$$x y \frac{d y}{d x}=\frac{\ln x}{\sqrt{1-y^{2}}}$$

(a) sketch the curve represented by the parametric equations, (b) use a graphing utility to verify your graph, and (c) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary.

$$\begin{array}{l}{x=4 \ln t} \\ {y=\frac{1}{2} t^{2}}\end{array}$$

$y=c_{1} \cos 2 x+c_{2} \sin 2 x$ is a two-parameter family of solutions of the second-order $\text { DE } y^{\prime \prime}+4 y=0$. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $y(0)=0, y(\pi)=0$

Let $X_{1}, \ldots, X_{n}$ be an i.i.d. sample from a Rayleigh distribution with parameter $\theta>0$: $f(x | \theta)=\frac{x}{\theta^{2}} e^{-x^{2} /\left(2 \theta^{2}\right)}, \quad x \geq 0$ a. Find the method of moments estimate of $\theta$. b. Find the mle of $\theta$. c. Find the asymptotic variance of the mle.

Eliminate the parameter t. Then use the rectangular equation to draw the plane curve represented by the given parametric equations. Use arrows to indicate the orientation of the curve corresponding to increasing values of $t$. (If an interval for $t$ is not specified, suppose that $-\infty<t<\infty$.)

$$x=3 \sin t, y=3 \cos t ; 0 \leq t<2 \pi$$