4641.sagews

Caderno 2 - Ex. 15 (e)

f(x,y) = 2*(x-y)^2 - 2*(x^4+y^4)

G = plot3d( f(x,y), (x,-2,2), (y,-2,2) )
P1 = point( [0,0,f(0,0)], color="red", size=10)
P2 = point( [-1,1,f(-1,1)], color="black", size=10)
P3 = point( [1,-1,f(1,-1)], color="black", size=10)
show(G+P1+P2+P3)

Caderno 2 - Ex. 17 (c)

f(x,y) = 4*x^2*exp(y) - 2*x^4 - exp(4*y)

H = f.diff(2); pretty_print(H)

\newcommand{\Bold}[1]{\mathbf{#1}}\left($$\begin{array}{rr} \left( x, y \right) \ {\mapsto} \ -24 \, x^{2} + 8 \, e^{y} & \left( x, y \right) \ {\mapsto} \ 8 \, x e^{y} \\ \left( x, y \right) \ {\mapsto} \ 8 \, x e^{y} & \left( x, y \right) \ {\mapsto} \ 4 \, x^{2} e^{y} - 16 \, e^{\left(4 \, y\right)} \end{array}$$\right)

H(-1,0)

[-16 -8] [ -8 -12]

H(1,0)

[-16 8] [ 8 -12]

G = plot3d(f(x,y),(x,-2,2),(y,-1,1))
P1 = point( [-1,0,f(-1,0)], color="red", size=10)
P2 = point( [1,0,f(1,0)], color="red", size=10)
show(G+P1+P2)

Caderno 2 - Ex. 17 (d)

f(x,y) = -y * log(x) + y^2 + log(2*x)

solve( [f.diff(x)==0,f.diff(y)==0], (x,y))

[[x == e^2, y == 1]]

H = f.diff(2); H(exp(2),1)

[ 0 -e^(-2)] [-e^(-2) 2]

G = plot3d( f(x,y), (x,4,10), (y,0,2) )
P = point( [exp(2),1,f(exp(2),1)], color="red", size=10)
L = line( [ [4,1,1+log(2)],[10,1,1+log(2)] ], color="black",size=20)
show(G+P+L)

Cadermo 2 , Ex 19

y=var('y')

plot3d( (x^2-y^2)^2 + 2*(x^2 - y^2), (x,-2,2), (y,-2,2) )

Caderno 2 - Ex 20

C1(x1) = 0.02*x1^2 + 4*x1 + 500
C2(x2) = 0.05*x2^2 + 4*x2 + 275
f(x1,x2) = 15*(x1+x2) - C1(x1) - C2(x2)

G = plot3d( f(x1,x2), (x1,100,300), (x2,50,200) )
P = point( [275,110,f(275,110)], color="red", size=10) 
show(G+P)
print f(275,110)

1342.50000000000

500000*2

1000000