Calculus Answers

true/false.justify your answer.
The set {(x, y, z) | x > 0, y > 0, z < 0} is a domain in R^3 .

true/false.justify your answer.
The level curves of z = y / x are hyperbolas.

true/false.justify your answer.
The function f (x, y, z) = e^(xyz) is integrable over [0, 1]×[0, 1]×[0,1]

Find the second Taylor polynomial of f (x, y) = 1 + 3 x^2 y − 4 y^3 at (1, 1)

Near what points may the surface
x^3 + 3y^2 + 8xz^2 − 3z^3y =1
be represented as a graph of a differentiable function z = k (x, y)? Justify your
answer

Find the centre of gravity of a mass in the shape of a semicircular disc of radius 4,
if the density at (x, y) is (2y)/(x^2+y^2)

Compute 5 (98) using Binomial theorem

Find the volume of the solid bounded by the planes z=0, x=1, x=2, y=-1, y=1 and the surface z=x^2 + y^2

If u = x cosy, v= ye^x , w = sinxz , find the Jacobian ∂(u,v,w)/∂(x,y,z), and evaluate it at (2,0,π/3).

Find the second Taylor polynomial of f(x,y)=1+3x^2y-4y^3 at (1,1).