var('x,y')
f(x,y) = e^(-x^2 * y)
show(f)
df_dx = f.diff()[0]
show(df_dx)
df_dy = f.diff()[1]
show(df_dy)

show(f.diff(2))
show(f.diff(2)[1,0])

z0 = f(0,0)
print 'Z0:', z0
x_slope = df_dx(0,0)
y_slope = df_dy(0,0)
print x_slope, y_slope
x0 = 0
y0=0
Z(x,y) = z0 + x_slope*(x - x0) + y_slope*(y-y0)
show(Z)

f_plot = plot3d(f, (x, -1, 1), (y, -1, 1), color='red')
t_plot = plot3d(Z, (x, -1, 1), (y, -1, 1), color='blue')
f_plot + t_plot

#A -e
z0 = f(1,1)
print 'Z0:', z0
x_slope = df_dx(1,1)
y_slope = df_dy(1,1)
print x_slope, y_slope
x0 = 1
y0=1
Z(x,y) = z0 + x_slope*(x - x0) + y_slope*(y-y0)
show(Z)

f_plot = plot3d(f, (x, 0, 2), (y, 0, 2), color='red')
t_plot = plot3d(Z, (x, 0, 2), (y, 0, 2))
f_plot + t_plot

#a
var('x,y')
f(x, y) = 4
plot3d(f, (x, -2, 2), (y, -2, 2))

f(x,y) = x^2 + y^2

z0 = f(0,0)
print 'Z0:', z0
x_slope = f.diff()[0](0,0)
y_slope = f.diff()[1](0,0)
print x_slope, y_slope
x0 = 1
y0=1
Z(x,y) = z0 + x_slope*(x - x0) + y_slope*(y-y0)
show(Z)

f_plot = plot3d(f, (x, -1, 1), (y, -1, 1), color='red')
t_plot = plot3d(Z, (x, -1, 1), (y, -1, 1))
f_plot + t_plot

var('y')
f(x,y) = x^3 * y - x^2
plot_f = plot3d(f, (x, -1, 1), (y, -1 , 1))
plot_f

var('L, K')
Y(L, K) = 2*L^0.75 * K^0.25
show(Y)

dY_dL = Y.diff()[0]
dY_dK = Y.diff()[1]
show(dY_dL)
show(dY_dK)

print '(1,1) for dY/dL:', dY_dL(1 ,1)
print '(1,1) for dY/dK:', dY_dK(1 ,1)

print '(1000,1000000) for dY/dL:', dY_dL(1000,1000000)
print '(1000,1000000) for dY/dK:', dY_dK(1000,1000000)


print '(1000,1) for dY/dL:', dY_dL(1000,1)
print '(1000,1) for dY/dK:', dY_dK(1000,1)

print '(1,100000) for dY/dL:', dY_dL(1,100000)
print '(1,100000) for dY/dK:', dY_dK(1,100000)