Given the following rational function:

            F(X) = (2X^2 - 20) / [( X - 5 )^2]

1. Find all horizontal and vertical asymptotes

2. Find the first derivative of F(X) and simplify

3. Find the critical values for X and determine if they are at a maximum or minimum, using the First Derivative Sign Test.

4. Find the Second Derivative and Use it to confirm your answers to part c. You may keep the 2nd derivative in “rough” form and simply substitute in the X value found in part c to see if you confirm a max or a min.

4x-2y-2z=2
-x-5y+2z=-8
4x-5y+z=11

Please make sure you show all steps numerically, becuse I want to be sure I know how to arrive at the same numbers.

A couple is deciding to invest in the laundromat business. There are two laundromat stores available for sale. They can only afford to buy one of them.

    The annual flow of income from each of the available laundromats is given below:

Laundromat 1: I’(t) = dI(t)/dt = 9000e^(.04t)

Laundromat 2: I’(t) = dI(t)/dt = 12500

The couple has decided to use the Present Value of each of the Laundromats after 8 years, at an annual interest rate of 10%, to compare the value of both investments.

They will buy the Laundromat with the highest Present Value.

1. Find the Present Value of each Laundromat

2. Which Laundromat should the couple buy? Explain.

         Hint: PV(t) = Definite ∫ I’(t)e^(-rt)dt, taken between ( a<t<b)

          Here a=0; b=8

                   r = 10%/100 = .10

I don't understand how to use beats using sum-to-product identity, or this assignment at all.

1. Dealing with vector fields is very common in engineering and science applications. Vector fields are often used to model a moving fluid throughout space, magnetic or gravitational force, and etc.

a. Provide two examples of vector field for engineering or science applications in two and three dimensions

b. Define a conservative vector field. Verify whether the given examples for vector fields in Part 1(a) are conservative? [15 marks]

c. Interpret the fundamental theorem of line integrals for conservative vector fields. Explain this with an example of a force field (i.e. vector field) and compute the work done by the force field in moving a particle from one point to another point along a curved path in three dimensional space. [20 marks

The function f(x, y) = 10−x 2−4y 2+2x has one critical point. Find that critical point and show that it is not a saddle point. Indicate whether this critical point is a maximum or a minimum, and find that maximum or minimum value.

Evaluate the line integral ∫CF⋅d r where F=〈4sinx,cosy,xz〉 and C is the path given by r(t)=(t^3,3t^2,3t) for 0≤t≤1 ∫CF⋅d r=

this as a whole question 1, answer all parts please

a)Show that the derivative of f(x) = 6+4x^2 is f(x)'=8x by using the definition of the derivative as the limit of a difference quotient.

b)If the area A = s^2 of an expanding square is increasing at the constant rate of 4 square inches per second, how fast is the length s of the sides increasing when the area is 16 square inches?

c)Find the intervals where the graph of y = x^3-5x^2+2x+4 is concave up and concave down, and find all the inflection points.

d)Find all the relative maximum and/or relative minimum values and points of F(x) = (x^4/3)-2x^2

e)Find all the relative maximum and/or relative minimum values and points of F(x)=x^4-4x on the closed interval [0,4]

f)A particle moves along the x-axis with an acceleration given by a(t)=4t + 7, where t is measured in seconds and s (position) is measured in meters. If the initial position is given by s(0) = 4 and the initial velocity is given by v(0) = 7 then find the position of the particle at t seconds.

g)Find the maximum value of xy if it is required that 7x + 1y = 62

Explain how to use derivatives to find the top of a hill or the bottom of a valley on a graph?

What does it even mean for a derivative to not exist and why is this involved? And what does that have to do with a derivative of zero?

The rate of growth of the population of rabbits in China is proportional to the current rabbit population. The population after t years is R(t). Write the differential equation for which R(t) is a solution. Your equation should involve an unknown constant

Initially, there are 100 rabbits but the population is increasing at a rate of 20 per year. Use this information to find the unknown constant in part That is, write the differential equation (without an unknown constant) for which R(t) is a solution

Find the population of rabbits in year t. That is, find R(t

Find the time t in which there are 1, 000 rabbits

What is the partial fraction decomposition of 5x^2/((x+1)(x^2+3x+2)(x^2+4))

proof a circle is divided into n congruent arcs (n ?? 3), the tangents drawn at the endpoints of these arcs form a regular polygon.

Test the series for convergence or divergence.

∞∑n=1(−1)nn4n

Identify bn.

A 600​-room hotel can rent every one of its rooms at $90 per room. For each​ $1 increase in​ rent,

3 fewer rooms are rented. Each rented room costs the hotel​ $10 to service per day. How much should the hotel charge for each room to maximize its daily​ profit? What is the maximum daily​ profit?