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Question 2
(a) Given that A ~ = (x + 2y + 4z)ˆ
i + (2x - 3y- z)ˆj +(4x-y + 2z)kˆ
(i). Show that the vector field A ~ is irrotational
(ii). Find the scalar potential φ such that A~ = ∇φ, if φ(0, 0, 0) = 1
A restaurant operator in Accra has found out that during the lockdown,if she sells a plate of her food for Ghc 20 each,she can sell 300 plates but for each Ghc5 she raises the price,10 less plates are sold. A.Draw a table relating 5 different price levels with their corresponding number of plates sold.
B.Use the table to find the slope of the demand
C.Find the equation of the demand fraction.
D.Use your equation to determine the price in Ghc if she sells one plate of food to maximize her revenue
Evaluate and classify the critical point of the function f(x, y) = xy
If A~ = 2yˆi - zˆj + 3xkˆ
(i). Find the unit vectors eˆr, eˆθ and eˆz of a cylindrical coordinates in termsof ˆi, ˆj and kˆ.
(ii). Solve for ˆi, ˆj and kˆ in terms of eˆr, eˆθ and eˆz
(iii). Represent the vector A~ in cylindrical coordinates and determine Ar, Aθ
and Az
(a) Find the linearization of f(x, y) = e^(x) cos y at the point (0, π/2)
2. f(x,y,z)= 1/(√ (4-x^2-y^2-z^2)
all 77 rooms in a motel will be rented each night if the maanager charges $39 or less per room. if he charges $(39 +y) per room then 2y rooms will remain vacant. If each rented room costs the manager $5 per day and each unrented room $3 per day in overhead , how much should the manager charge per room to maximize his profit. find;
i) no of rented rooms
ii)the revenue
iii)total cost in overhead
x→∞
Determine the first order partial derivative of the following function:
"F(x,y)=\\int_{y}^x cos(e^t)dt"
2−y
2
.
