Decide whether each function defined as follows is a probability density function for the given interval. $f(x)=\sqrt{x}$; [4, 9]

Determine whether the Markov chain with matrix of transition probabilities P is absorbing. Explain.

$$P=\left[\begin{array}{ccc} {0.8} & {0.3} & {0} \\ {0.2} & {0.1} & {0} \\ {0} & {0.6} & {1} \end{array}\right]$$

Define a pure abstract base class called BasicShape. The BasicShape class should have the following members: Private Member Variable: area, a double used to hold the shape's area. Public Member Functions: getArea. This function should return the value in the member variable area. calcArea. This function should be a pure virtual function. Next, define a class named Circle. It should be derived from the BasicShape class. It should have the following members: Private Member Variables: centerX, a long integer used to hold the x coordinate of the circle’s center. centerY, a long integer used to hold the y coordinate of the circle’s center. radius, a double used to hold the circle's radius. Public Member Functions: constructor—accepts values for centerX, centerY, and radius. Should call the overridden calcArea function described below. getCenterX—returns the value in centerX. getCenterY—returns the value in centerY.calcArea—calculates the area of the circle (area = 3.14159 * radius * radius) and stores the result in the inherited member area. Next, define a class named Rectangle. It should be derived from the BasicShape class. It should have the following members: Private Member Variables: width, a long integer used to hold the width of the rectangle. length, a long integer used to hold the length of the rectangle. Public Member Functions: constructor—accepts values for width and length. Should call the overridden calcArea function described below. getWidth—returns the value in width. getLength—returns the value in length. calcArea—calculates the area of the rectangle (area = length * width) and stores the result in the inherited member area. After you have created these classes, create a driver program that defines a Circle object and a Rectangle object. Demonstrate that each object properly calculates and reports its area.

Write each expression in interval notation. Graph each interval. x<4

The two common programming methods in practice today are ______ and ______.

Solve each inequality and graph the solution. $8 \leq 3 r+1 \leq 13$

Find the domain and range of the function. Write your answer in interval notation. $g(x)=\sqrt{16-x^{4}}$

Find the derivative of each function. $y=-4 \sin 7 x$

Find F(s) if: (a) $f(t)=6 e^{-t} \cosh 2 t$ (b) $f(t)=3 t e^{-2 t} \sinh 4 t$ (c) $f(t)=8 e^{-3 t} \cosh t u(t-2)$

Determine whether the stochastic matrix P is regular. Then find the steady state matrix $\overline{X}$ of the Markov chain with matrix of transition probabilities P.

$$P=\left[\begin{array}{ccc} {\frac{1}{2}} & {\frac{1}{5}} & {1} \\ {\frac{1}{3}} & {\frac{1}{5}} & {0} \\ {\frac{1}{6}} & {\frac{3}{5}} & {0} \end{array}\right]$$

Write a program that calculates a car’s gas mileage. The program should ask the user to enter the number of gallons of gas the car can hold and the number of miles it can be driven on a full tank. It should then display the number of miles that may be driven per gallon of gas.

Find each integral. $\int \frac{\cos x}{\sqrt{\sin x}} d x$

The amount of a radioactive substance decreases exponentially, with a decay constant of 3% per month. a. Write a differential equation to express the rate of change. b.Find a general solution to the differential equation from part a. c. If there are 75 g at the start of the decay process, find a particular solution for the differential equation from part a. d.Find the amount left after 10 months.

The following definitions and program segments has errors. Locate as many as you can. vector numbers = { 1, 2, 3, 4 };