Mini-Lecture 7-08 - The tangent plane of a function.ipynb

Kernel: SageMath 9.1

def axes3d(x_range, y_range, z_range, **options):
    """Draw x, y, and z axes in 3D, and optionally a grid on the xy-plane

    'x_range' should have the form (x, xmin, xmax), and likewise for 'y_range'
    and 'z_range'. Here the 'x' can be either a symbolic variable, or a string.
    Either way, it is only used as a label for the corresponding axis.

    Options:
        'origin' - Locate the axes so that they all intersect at this point
                   (default (0,0,0))
        'grid' - Whether or not to draw a grid on the xy-plane (default True)
    All other options are as for the 'line3d' and 'arrow3d' commands, and are
    passed directly on to them.
    """

    x0, y0, z0 = options.pop("origin", (0, 0, 0))
    grid = options.pop("grid", True)
    gridoptions = options.copy()
    options.setdefault("color", "black")
    gridoptions.setdefault("color", "gray")
    x, xmin, xmax = x_range
    y, ymin, ymax = y_range
    z, zmin, zmax = z_range
    xmin, xmax = min(x0, xmin), max(x0, xmax)
    ymin, ymax = min(y0, ymin), max(y0, ymax)
    zmin, zmax = min(z0, zmin), max(z0, zmax)
    p = Graphics()
    p += arrow3d((xmin, y0, z0), (xmax, y0, z0), width=1, **options)
    p += arrow3d((x0, ymin, z0), (x0, ymax, z0), width=1, **options)
    p += arrow3d((x0, y0, zmin), (x0, y0, zmax), width=1, **options)
    p += text3d(str(x), (xmax + (xmax - xmin)*0.05, y0, z0), **options)
    p += text3d(str(y), (x0, ymax + (ymax - ymin)*0.05, z0), **options)
    p += text3d(str(z), (x0, y0, zmax + (zmax - zmin)*0.05), **options)
    xgridsize = 10**int(log(0.68*max(-xmin, xmax), 10))
    ygridsize = 10**int(log(0.68*max(-ymin, ymax), 10))
    zgridsize = 10**int(log(0.68*max(-zmin, zmax), 10))
    gridmin = ceil(xmin/xgridsize) * xgridsize
    for xval in srange(gridmin, xmax+xgridsize/100.0, xgridsize):
        #p += text3d("{:.0f}".format(float(xval)), (xval, ygridsize/5.0, 0), **options)
        if grid:
            p += line3d(((xval, ymin, 0), (xval, ymax, 0)), thickness=1, **gridoptions)
    gridmin = ceil(ymin/ygridsize) * ygridsize
    for yval in srange(gridmin, ymax+ygridsize/100.0, ygridsize):
        #p += text3d("{:.0f}".format(float(yval)), (xgridsize/5.0, yval, 0), **options)
        if grid:
            p += line3d(((xmin, yval, 0), (xmax, yval, 0)), thickness=1, **gridoptions)
    gridmin = ceil(zmin/zgridsize) * zgridsize
    #for zval in srange(gridmin, zmax+zgridsize/100.0, zgridsize):
        #p += text3d("{:.0f}".format(float(zval)), (xgridsize/5.0, 0, zval), **options)
    return p

xmin, xmax = -7.5, 7.5
ymin, ymax = -7.5, 7.5
zmin, zmax = 0, 6.5
f(x,y) = 3 + sin(2*(x-2)/3)*cos((y-3)/2) - 2*cos((x-2)/3)*sin((y-3)/5)

x0, y0 = 2, 3
z0 = f(x0,y0)
df_dx = diff(f, x)
df_dy = diff(f, y)
L(x, y) = f(x0, y0) + df_dx(x0, y0)*(x - x0) + df_dy(x0, y0)*(y - y0)

print((x0,y0,z0))
print(df_dx(x0, y0), df_dy(x0, y0))

p = axes3d((x, -7.5, 7.5), (y, -7.5, 7.5), ("z", 0, 6))
p += plot3d(f(x,y), (x, xmin, xmax), (y, ymin, ymax), opacity=0.7)
p += sphere((x0,y0,z0), 0.15, color="green")
p += plot3d(L(x,y), (x, x0-2, x0+2), (y, y0-2, y0+2), color="lightgreen", opacity=0.8)
p.show(frame=False)

(2, 3, 3)
2/3 -2/5

f(x,y) = 3 + sin(2*(x-2)/3)*cos((y-3)/2) - 2*cos((x-2)/3)*sin((y-3)/5)

p = axes3d((x, -7.5, 7.5), (y, -7.5, 7.5), ("z", 0, 6))
p += plot3d(f(x,y), (x, -7, 7), (y, -7, 7), opacity=0.7)
p.show(frame=False)

p += point((2, 3, f(2,3)), size=60, color="green")
p.show(frame=False)

df_dx = diff(f, x)
df_dx(2,3)

2/3

df_dy = diff(f, y)
df_dy(2,3)

-2/5

print(f(2,3))

3

L(x, y) = 3 + 2/3*(x - 2) + -2/5*(y - 3)

p += plot3d(L(x,y), (x, -1, 5), (y, 0, 6), color="lightgreen", opacity=0.9)
p.show(frame=False)