h(x,y)=
cos(x) + sin(y) sin(x) - cos(2y)Part (a). hap(x,y)=
-1 0 1 x-pi - * -1 1 0 y-piPart (b). (u,v)=h(1.1*pi,0.8*pi) is approximately (-0.372,-1.314).
Part (c). -ê2 is the direction of fastest increase for u at (pi,pi).
The ``best affine approximation at x0'' is just Eap(x)=E(x0)+ E'(x0)(x-x0), which is
-3 2 -4 0 x-1 5 + 2 4 0 * y-2 3 0 0 1 z-3
Part (a). The affine approximation is
Bap(x)=
-1 0 0 -2 x-1 0 + -2 0 2 * y 1 2 0 0 z-1Part(b) The directional derivative along (1,-3, 0)/10^(1/2) is
0 -2/10^(1/2) 2/10^(1/2)Part(c). B(2,-1,1) is approximately
-1 -2 2
The chain rule for this problem is dP/dt=dP/dz*z'(2,0)*v, where z'(2,0)=[-4 0] and v=[500 0]T. Plugging in:
dp/dt=700*e-2[-4 0]*[500 0]T =7.1×105