Use the Laplace transform to find the solution y(t) to the IVP y′′+3y′+2y=2e−tt with initial condition y(0)=1,y′(0)=2

Calculate
L−1[6s−2(s+1)(5s2+3s+2)]

Let f(x) = x / (4−x^2) .

(a) Find the domain and intercepts of f(x).

(b) Find all asymptotes and limits describing the end behavior of f(x).

(c) Find all local extrema and the intervals on which f(x) is increasing or decreasing.

(d) Find the inflection points and the concavity of f(x).

(e) Use this information to sketch the graph of f(x)

A linear cost function is

C(x) = 7x + 800. (Assume C is measured in dollars.)

(a) What are the slope and the C-intercept?

(b) What is the marginal cost

MC ?

MC =

What does the marginal cost mean?

If production is increased by this many units, the cost decreases by $1.Each additional unit produced reduces the cost by this much (in dollars).    If production is increased by this many units, the cost increases by $1.Each additional unit produced costs this much (in dollars).

(c) What are the fixed costs?
$

(d) How are your answers to parts (a), (b), and (c) related?
slope = fixed costs, and C-intercept = marginal cost

= marginal cost    

= marginal costslope = marginal cost, and C-intercept = fixed costs
(e) What is the cost of producing one more item if 50 are currently being produced?
$

What is the cost of producing one more item if 100 are currently being produced?

Find the general solution for the given differential equation
9y′′′−27y′′+4y′−12y=x+4xe3x9y′′′−27y′′+4y′−12y=x+4xe3x

NOTE: Write your answer clearly in below type:
yg=yc+ypyg=yc+yp

Consider the vector field F(x,y,z) =〈2xycos(z) + 1,x²cos(z) +ze^yz,yeyz−x²ysin(z)〉. Use the component test to determine if F is conservative. If it is, find a potential function f.

When solving a system of equatios using the additio or substitution method, how can you tell if a system has infinitely many or no solutions? What's the relationship between the graph in these situations? Show all work neatly.

What's the difference between numerical method to calculate differential (backward, central, forward)? At what condition do I have to use forward over central and backward, etc? If there's boundary condition that permits the complete use of central method, what will happen if I still use forward, backward, and central regardless there's boundary condition that permits the use of central method?
note: I'm not used to describe pure mathematical problem in english, so I'm sorry if my question is hard to understand. feel free to add comment under this question and I'll explain it again, hopefully in clearer manner. thank you

Given r(t) = <2 cos(t), 2 sin(t), 2t>. • What is the arc length of r(t) from t = 0 to t = 5. SET UP integral but DO NOT evaluate • What is the curvature κ(t)?

Let z = 4 - 4i. find the sixth roots of z and represent them
in the plane.

f(x)=e^x/(3+e^x)

Find the first derivative of f.

Use interval notation to indicate where f(x) is increasing.

List the x coordinates of all local minima of f. If there are no local maxima, enter 'NONE'.

List the x coordinates of all local maxima of f. If there are no local maxima, enter 'NONE'.

Find the second derivative of f:

Use interval notation to indicate the interval(s) of upward concavity of f(x).

Use interval notation to indicate the interval(s) of downward concavity for f(x)

List the x values of the inflection points of f. If there are no inflection points, enter 'NONE'.

For the cost and price functions​ below, find

​(​a) the​ number, q, of units that produces maximum​ profit;

​(​b) the​ price, p, in dollars per unit that produces maximum​ profit; and

​(​c) the maximum​ profit, P, in dollars.

C(q)=100+20qe^-0.01q ; p=60e^-0.01q

​(a) The​ number, q, of units that produces maximum profit is $__

​(Do not round until the final answer. Then round to the nearest whole number as​ needed.)

​(b) The​ price, p, per unit that produces maximum profit is $__

​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

​(c) The maximum profit is $__

​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

The balance in an investment account years t after the account is opened is given by 7500(1.065t). Compute the average rate of change for 2.5 ≤ t ≤ 3.5. (Round your answer to two decimal places.) Δ balance/ Δt=?

Use the model for projectile motion, assuming there is no air resistance and g=32 feet per second per second.

Determine the maximum height and range of a projectile fired at a height of 8 feet above the ground with an initial speed of 300 feet per second and at an angle of 45degrees above the horizontal. (Round your answers to three decimal places.)

(a) maximum height (feet)

(b) maximum range (feet)

Complete the following table: (Use Table 15.1) (Do not round intermediate calculations. Round your answers to the nearest cent.)