Bard Math 212 - Multivariable Maxima and Minima

42 days ago by ethan

The Graphs of Some Function of Two Variables,
Showing Various Types of Critical Points

f(x, y) = 2(x - 1)^2 + (y - 2)^2 + 4
x, y = var('x,y') plot3d(2*(x - 1)^2 + (y - 2)^2 + 4, (x,-1,3), (y,0,4), aspect_ratio=1) 
       

f(x, y) = 2(x - 1)^2 - (y - 2)^2 + 4
x, y = var('x,y') plot3d(2*(x - 1)^2 - (y - 2)^2 + 4, (x,-1,3), (y,0,4), aspect_ratio=1) 
       

f(x, y) = \sin x\sin y
x, y = var('x,y') plot3d(sin(x)*sin(y), (x,-10,10), (y,-10,10), aspect_ratio=1) 
       

f(x, y) = \sin (x + y)
x, y = var('x,y') plot3d(sin(x + y), (x,-10,10), (y,-10,10), aspect_ratio=1) 
       

f(x, y) = x^4 + y^4
x, y = var('x,y') plot3d(x^4 + y^4, (x,-1.5,1.5), (y,-1.5,1.5), aspect_ratio=1) 
       

f(x, y) = x^3 + y^3
x, y = var('x,y') plot3d(x^3 + y^3, (x,-1.5,1.5), (y,-1.5,1.5), aspect_ratio=[2,2,1]) 
       

f(x, y) = x^2 + 4y^2 - 4xy + 2
x, y = var('x,y') plot3d(x^2 + 4*y^2 - 4*x*y + 2, (x,-5,5), (y,-5,5)) 
       

The graph of f(x, y) = \tan y is in purple.

The point (0, 0, 0) is in red.
x, y = var('x,y') W = plot3d(tan(y), (x,-pi/4,pi/4), (y,-pi/4,pi/4), color='purple', opacity=0.8) S = sphere((0,0,0),size=0.02, color='red', aspect_ratio=[1,1,1]) show(W + S, figsize=5)