Fnd the derivative of the function. f(x) = (2x − 9)^4(x^2 + x + 1)^5

Given the system of differential equations

$$\begin{aligned} & \frac{d P}{d t}=2 P-10 \\ & \frac{d Q}{d t}=Q-0.2 P Q, \end{aligned}$$

determine whether P and Q are increasing or decreasing at the point

P=2, Q=3

If f(x) = e^x g(x), where g(0) = 3 and g'(0) = 1, find f '(0).

Find the remainder when f(x) = 2x^3 − 12x^2 + 11x + 2 is divided by x − 5.

7, −3, 3, −7

For each of the differential equations in Problem given below, find the values of c that make the general solution:

underdamped

$$s^{\prime \prime}+2 \sqrt{2} s^{\prime}+c s=0$$

Given the system of differential equations

$$\begin{aligned} & \frac{d P}{d t}=2 P-10 \\ & \frac{d Q}{d t}=Q-0.2 P Q, \end{aligned}$$

determine whether P and Q are increasing or decreasing at the point

P=6, Q=5

In Exercise given below, is the function a solution to $y^{\prime \prime}+b y^{\prime}+$ cy=0 whose characteristic equation is (r-2)(r+3)=0?

$$y=2 e^{-3 x}+3 e^{2 x}$$

Given the system of differential equations

$$\begin{aligned} & \frac{d x}{d t}=5 x-3 x y \\ & \frac{d y}{d t}=-8 y+x y, \end{aligned}$$

determine whether x and y are increasing or decreasing at the point

x=3, y=2

For each of the differential equations in Problem given below, find the values of b that make the general solution:

underdamped

$$s^{\prime \prime}+b s^{\prime}-16 s=0$$

For Problem given below, consider a conflict between two armies of x and y soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and t represents time since the start of the battle, then x and y obey the differential equations

$$\begin{aligned} & \frac{d x}{d t}=-a y \\ & \frac{d y}{d t}=-b x \quad a, b>0 . \end{aligned}$$

Solve the differential equation and hence show that the equation of the phase trajectory is

$$a y^2-b x^2=C$$

for some constant C. This equation is called Lanchester's square law. The value of C depends on the initial sizes of the two armies.

Suppose that $f$ is a one-to-one function such that $f(2)=5$. Find $f^{-1}(5)$.

Find the gradient vector field of f(x, y, z) = 9 sqrt(x^2 + y^2 + z^2).

Humans vs Zombies is a game in which one player starts as a zombie and turns human players into zombies by tagging them. Zombies have to "eat" on a regular basis by tagging human players, or they die of starvation and are out of the game. The game is usually played over a period of about five days. If we let H represent the size of the human population and Z represent the size of the zombie population in the game, then, for constant parameters a, b, and c, we have:

$$\begin{aligned} \frac{d H}{d t} & =a H Z \\ \frac{d Z}{d t} & =b Z+c H Z \end{aligned}$$

What is the relationship, if any, between a and c?

The motion of a mass on the end of a spring satisfies the differential equation

$$\frac{d^2 s}{d t^2}+2 \frac{d s}{d t}+3 s=0 .$$

Give the general solution to the differential equation.

Each graph in Figure mentioned represents a solution to one of the differential equations:

$x^{\prime \prime}+16 x=0$.