If z=e^(xy^2),x=tcost,y=tsint,compute dz/dt at t=π/2?

1 Answer
Feb 20, 2018

- (pi/2)^3(π2)3

Explanation:

z=e^(xy^2)z=exy2

dz/dt= (del z)/(delx) dx/dt + (delz)/(dely) dy/dtdzdt=zxdxdt+zydydt

=e^(xy^2) (y^2) (cost -t sint) +e^(xy^2) (x) (2y dy/dt) ( sint +t cost)=exy2(y2)(costtsint)+exy2(x)(2ydydt)(sint+tcost)

=e^(xy^2) (y^2) (cost - tsin t)+e^(xy^2) 2xy (sint +tcost)^2=exy2(y2)(costtsint)+exy22xy(sint+tcost)2

At t=pi/2, cost =0, sint =1, x=0 and y= pi/2t=π2,cost=0,sint=1,x=0andy=π2

Therefore dz/dt =(pi/2)^2 (-pi/2)= -(pi/2)^3dzdt=(π2)2(π2)=(π2)3