Let F(x, y, z)  =  (6x⁵ ln(5y² + 3) + 10z⁴) i  +  ((( 10yx⁶)/(5y² + 3))+ 8z) j  +  (40xz³ + 8y − 6π sin πz) k
and let  r(t)  =  (t³ + 1) i  +  (t² + 2) j  +  t³ k ,  0  ≤  t  ≤  1. Evaluate  

F · dr (please explain steps, thank you :)

PopSolution Inc is a small IT Solution company based in Central Jakarta, Indonesia. They have just finished their sales pitch meeting with their latest potential client, an automobile sales company based in South Jakarta. The company was very satisfied with their portofolios, and asked them to provide a final quotation on the price that will be agreed upon. As for the project detail, the client asked for a simple web application with 3 main functions: (1) a homepage showing the company’s credibility and location, (2) a page to list all cars the company is selling, and (3) a Contact Us page where future customers can fill their data to further be contacted by the company’s representatives. Since the transaction is too substantial to be handled in the application, all purchase will be done outside of the web application. Every month, PopSolution spends 30.000.000 IDR for the overhead cost of all its employees. The organizational average productivity for the system of similar type is 25 FP/person-month. This means that with one person doing the job of 50 Function Point, it will take one month, with 2 persons doing the job, it will take half a month to finish. As the new member of the development team, you are asked to prove your expertise on software estimation by providing the team with Function Point Analysis (FPA) on your next internal meeting. Your tasks are:

Calculate the Unadjusted Function Point, Value Adjustment Factors, and the Function Point.

9) Suppose that t years from now, one investment will be generating profit at the rate of P 1 ′ = 40 + t 2 hundred dollars per year, while a second investment will be generating profit at the rate of P 2 ′ ( t ) = 100 + 4 t hundred dollars per year. For how many years does the rate of profitability of the second investment exceed that of the first? Compute the total net excess profit for the time period.

Suppose that ac + bd = 0. Show that vectors [a, b] and [c, d] are perpendicular. The number ac + bd is called the dot product of the vectors [a, b] and [c, d].

Show all your work. Let f(x) = x 5 e x 3 (i) Use the Taylor series for e x around 0 to find the Taylor series for f(x) (ii) Use (i) to find f (20)(0), f (21)(0), f (22)(0), f (23)(0)

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)?