Calculus Answers

Find the slope of the tangent at (3,1,1)to the curve of intersection of x=3 and z= x^2y-3xy^2+1.

Evaluate: lim x→∞ [(√ 2x^2 +3x- 2)-(√ 2x^2-3x+2)]

Apply Inverse function theorem to check the local invertibility of the following
function f: R^2 → R^2 given by at the point (0, π).
f(x,y)=(y cosx, x-y+2)

Find the directional derivate of
f(x,y)={2xy/(x^2+y^2), (x,y)≠(0,0) ; 0, (x,y)=(0,0)
at (0,0) in the direction of (i) θ=π/2, (ii) θ=π/4

If , z = e^x sin y where x = st^2 and y = s^2 t, find ∂z/∂s and ∂z/∂t
.

Find the moment of inertia I2 for the solid above the xy-plane bounded by the
paraboloid z = x^2 + y^2 and the cylinder , x^2 + y^2 = 9 assuming the mean density to
be constant C.

Prove that the functions g(x, y)= (2x-3y)/(4x+5y) and h(x,y)=x/y, y≠0, y≠-(4/5)x are functionally dependent.

Find the values of a and b for which the following limit exists.
lim x→0 (ae^x +e^-x +bx)/(1-cosx)

true/false. justify your answer.
lim (x,y)→(0,0) (x^2-y^2)/(x^2+y^2) does not exist.

true/false.justify your answer.
The work done by the Force F(x, y) = (−y,x) in moving a particle along the
boundary of the ellipse 9x^2 + 4 − y^2 = 36 is 6.