Question:

a) Let \(f\left( x,y \right)={{x}^{3}}+x{{y}^{2}}-2y\). Use the chain rule to find the rate of change of \(f\left( x,y \right)\), \(\frac{df}{dt}\) along the parametric curve \(x\left( t \right)={{t}^{3}}\), \(y\left( t \right)=2{{t}^{2}}\) at the time t = 1.

b) Let \(f\left( x,y \right)=xy-{{x}^{3}}\)

Obtain the

Taylor

series of the function \(f\left( x,y \right)\) about the point (x, y) = (1,1). Neglect terms of degree three and higher. Write your answer as a function of and \(\Delta x\) and \(\Delta y\)

c) Find the point on the plane \(2x+y-z=6\) which is closest to the origin by using Lagrange multipliers.