Expand sin x as a finite series in powers of x , with remainder in Lagrange's form. Hence find the series for sin x .

Using Maclaurin's series expand the following function in powers of x 

e^{sin x}

If xy = 4 , find the maximum and minimum value of 4x + 9y

Find the partial differential equation of eliminating f from z = xy + (x²+y²+z²)

Eliminate the arbitrary function f and g from z = f(x+iy) + g(x-iy) and form a partial differential equation.

Form partial differential equation by eliminating arbitrary function 

z =  x + y + f(xy)

Eliminating arbitrary constants form partial differential equation from the 

z = ax + by

Solve the following partial differential equation 

(1).         pq + p + q = 0

(2).         pq = 1

Prove Green's theorem in a plane.

Using double integral find the area enclosed by the curve y= 2x² and y²=4x

Find the area bounded by the parabola y²= 4 - x and y² = 4 - 4x as a double integral and evaluate it.