Find all values of c such that the domain of f(x) = x+3 / x²+3cx+6 is the set of all real numbers.

Plot the points and find the slope of the line that passes through the points. State whether the line rises, falls, is horizontal, or is vertical. (-1, 0), (6, -2)

plot the points and find the slope of the line passing through the pair of points. (-3, -2), (1, 6)

Use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. f(x) = x^4 - 2x²

Find the open intervals on which the function is increasing or decreasing.

$$y = x \sqrt{16-x^2}$$

The radius r of a circle is increasing at a rate of 3 inches per minute. Find the rates of change of the area when (a) r=6 inches and (b) b=24 inches.

Volume The radius r of a sphere is increasing at a rate of 2 inches per minute.

(a) Find the rate of change of the volume when r = 6 inches and r= 24 inches.

(b) Explain why the rate of change of the volume of the sphere is not constant even though dr/dt is constant.

The radius r of a sphere is increasing at a rate of 3 inches per minute. (a) Find the rates of change of the volume when r = 9 inches and r = 36 inche s. (b) Explain why the rate of change of the volume of the sphere is not constant even though dr/ dt is constant.

Find the points at which the graph of the equation has a vertical or horizontal tangent line. 4x^2 + y2 - 8x + 4y + 4 = 0

To escape Earth’s gravitational field, a rocket must be launched with an initial velocity called the escape velocity.A rocket launched from the surface of Earth has velocity (in miles per second) given by v = √2GM/r + v_0² - 2GM/R ≈ √192000/r + v_0² - 48 where v_0 is the initial velocity, r is the distance from the rocket to the center of Earth, G is the gravitational constant, M is the mass of Earth, and R is the radius of Earth (approximately 4000 miles). (a) Find the value of v_0 for which you obtain an infinite limit for r as v approaches zero. This value of v_0 is the escape velocity for Earth.

Find the inverse of $f(x)=x+b$ ( $b$ constant $)$. How is the graph of $f^{-1}$ related to the graph of $f$ ?

To escape Earth’s gravitational field, a rocket must be launched with an initial velocity called the escape velocity. A rocket launched from the surface of Earthhas velocity v (in miles per second) given by v = √2GM/r + v_0^2 - 2GM/R ≈ √192000/r + v_0^2 - 48 where v_0 is the initial velocity, r is the distance from the rocket to the center of Earth, G is the gravitational constant, M is the mass of Earth, and R is the radius of Earth (approximately 4000 miles). (c) A rocket launched from the surface of a planet has velocity v (in miles per second) given by v = √10600/r + v_0^2 - 6.99. Find the escape velocity for this planet. Is the mass of this planet larger or smaller than that of Earth? (Assume that the mean density of this planet is the same as that of Earth.)

Express the interval in terms of an inequality involving absolute value. [1, 5]

Show that $f$ and $g$ are inverse functions in graphically.

$$f(x)=\frac{1}{1+x}, \quad x \geq 0, \quad g(x)=\frac{1-x}{x}, \quad 0<x \leq 1$$