Calculus Answers

1)Use the method of Language multipliers,find the maximum value of the function f(x,y,z)=xyz subject to the constraint g(x,y,z)=2xy+2yz+zx=12,x>0,y>0,z>0
2)Use polar coordinates to find the volume of the soild inside the sphere x^(2)+y^(2)+z^(2)=16 and outside the cylinder x^(2)+y^(2)=4
3)Given the vector flied
F(x,y,z)=(y^(2)-4z,2xy +6yz ,-4x+5y^(2))
a)Find a potential function of F
b)Evaluate the line intergral

x/[(x+1)(x^3 +1)] decompose this partial fraction.

decomposition of this partial fraction x-1/[(x-2)(x^2 +3)] ≡ [A/(x-2)]+[Bx+C/(x^2 +3)] ? why we use Bx+C here ??? and what will happen when it is (x^3 +3) or (x^4 +3) instead of (x^2 +3)

A consumer utility function given u(x,y)=in(x+2y-y2/2 )find indirect utility function of consumer.

For the following equations determine the domain, co-domain and range:
f(x) = (x-1)^2 + 2x
g(x) = cos(2x) + 1

Using f(x) & g(x) calculate:
f + g
f - g
f * g
g/f

Find the area under the curve given by f(x) = 3x2 + 5x between x = 3 and x = 7

limit Δx tends to 0; Δy / Δx = dy/dx. why we use Δx tends to 0 here ?? what does it mean?

(a) Differentiate the following function
y = x^2 e^x
(b) A cosmetic company is planning the introduction of a promotion of a new lipstick line. The marketing research department after test marketing the new line in a carefully selected city found that the demand in the city is approximately given by p = 12e^(-x), where x which should be within this range, 0 ≤ x ≤ 2 were thousand lipsticks sold per week at a price of Kenya shillings. At what price will the weekly revenue be at maximum? What is the maximum weekly revenue

Find the geometric series whose sum is 1 and first term is 0.4

(a) Given the Cobb Douglas production function
Y= AK^(2/3) L^(2/3)
Where, Y= Output, A=level of technology, K = capital, L= Labour
Find
The marginal product of labour (MPL)
The marginal product of capital (MPK)
Given y = (〖3x〗^2+5x)/(x^3+10x^2 ), find dy/dx
Compute ∫_3^7▒(2x+5)dx