Mini-Lecture 7-04 - Linear approximation of a function f(X,Y).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

x, y, z = var("x, y, z")
f(x, y) = 0.5 + sin(x/1.5)*cos(y/2) + 1 - 2*cos(x/3)*sin(y/5) + 2
(x0, y0) = (1, -1)
z0 = f(x0, y0)
df_dx(x, y) = diff(f(x, y), x)
df_dy(x, y) = diff(f(x, y), y)
Deltaf_x(Deltax) = df_dx(x0, y0) * Deltax
Deltaf_y(Deltay) = df_dy(x0, y0) * Deltay
Deltaf(Deltax, Deltay) = df_dx(x0, y0) * Deltax + df_dy(x0, y0) * Deltay

axes = axes3d((x, -6.5, 6.5), (y, -6.5, 6.5), (z, -0.1, 6))
surface = plot3d(f(x, y), (x, -6, 6), (y, -6, 6),
                 plot_points=(40,40), color="lightskyblue", opacity=0.85)
pointXY = sphere((x0, y0, 0), 0.17, color="red")
point0 = sphere((x0, y0, z0), 0.17, color="darkgreen")
tangent_plane = parametric_plot3d((x0 + Deltax, y0 + Deltay, z0 + Deltaf_x(Deltax) + Deltaf_y(Deltay)),
                                  (Deltax, -3, 3), (Deltay, -3, 3), color="plum", opacity=0.5)
baseplot = axes + surface + point0

Learning goals:

• Know what the partial derivatives of a function $f(X, Y)$ are, and roughly how to compute them.

• Be able to compute a linear approximation to a function $f(X, Y)$, assuming that you know those partial derivatives.

Recall the set-up from a previous video:

We have a system of differential equations $$\begin{cases} X' = f(X,Y) \\ Y' = g(X,Y) \end{cases}$$

It has an equilibrium point at $(X^*, Y^*)$.

We want to find the formula for the linear approximation to the function $f$ at $(X^*, Y^*)$.

The graph of the function $f(X,Y)$ is some surface:

(axes + surface).show(aspect_ratio=(1,1,1), frame=False)

Here's the point $(X^*, Y^*)$:

(axes + pointXY).show(aspect_ratio=(1,1,1), frame=False)

And here's the corresponding point on the graph of $f$:
That point is $(X^*, Y^*, f(X^*, Y^*))$.

(baseplot + pointXY).show(aspect_ratio=(1,1,1), frame=False)

Suppose we choose another point near $(X^*, Y^*)$, but only by changing $X$.

Delta_X = 0.7
p = Graphics() + axes + pointXY
p += sphere((x0 + Delta_X, y0, 0), 0.17, color="fuchsia")
p.show(aspect_ratio=(1,1,1), frame=False)

Here's the point corresponding to that on the graph of $f$:

p = Graphics() + baseplot
p += sphere((x0 + Delta_X, y0, f(x0 + Delta_X, y0)), 0.17, color="fuchsia")
p.show(aspect_ratio=(1,1,1), frame=False)

Here are all the points we could get if we only change $X$, and hold $Y$ constant.

p = Graphics() + baseplot
p += parametric_plot3d((x, y0, z), (x, -6, 6), (z, 0, 5), color="red", opacity=0.7)
p.show(aspect_ratio=(1,1,1), frame=False)

That cross-section, viewed with just the $X$ and $Z$ axes (since $Y$ is being held constant):

@interact(show_tangent=checkbox(False, label="Show tangent line"))
def tangent_x(show_tangent):
    slope = df_dx(x0, y0)
    p = plot(f(x, y0), (x, -6, 6), color="lightskyblue", thickness=2)
    p += point((x0, z0), size=40, color="green")
    if show_tangent:
        p += line(((x0 - 3, z0 - slope*3), (x0 + 3, z0 + slope*3)), color="red")
    p.show(axes_labels=("$X$", "$Z$"), ymin=0, ymax=6.5, aspect_ratio=1, figsize=6)

Here's that same tangent line, on the surface:

p = Graphics() + baseplot
p += parametric_plot3d((x0 + Deltax, y0, z0 + Deltaf_x(Deltax)),
    (Deltax, -4, 4), color="red", opacity=0.85)
p.show(aspect_ratio=(1,1,1), frame=False)

Conclusion so far:

The slope, $\displaystyle \frac{\Delta Z}{\Delta X}$, of that tangent line can be found by setting $Y = Y^*$ (that is, plugging the constant $Y^* = -1$ into the equation for $f$) and then computing the derivative with respect to $X$ at $X^* = 1$.

This slope is called the partial derivative of $f$ with respect to $X$, at the point $(X^*, Y^*)$.

Definition: The partial derivative of the function $f(X, Y)$ at the point $(X^*, Y^*)$ is found by keeping $Y$ constant, and taking the derivative of $f$ with respect to $X$.

Notation: $\displaystyle \left. \frac{\partial f}{\partial X} \right|_{(X^*, Y^*)} =$ the partial derivative of $f$ with respect to $X$ at $(X^*, Y^*)$.

Tying this back in to linear approximation:

@interact(show_second=checkbox(False, label="Second point"), 
          show_deltas=checkbox(False, label=r"$\Delta X$, $\Delta f$"), 
          show_result=checkbox(False, label="Linear approx."))
def approx_x(show_second, show_deltas, show_result):
    slope = df_dx(x0, y0)
    p = plot(f(x, y0), (x, -2, 5), color="lightskyblue", thickness=2)
    if show_second:
        p += point((x0, 0), size=40, color="red")
        p += point((x0 + Delta_X, 0), size=40, color="fuchsia")
        p += point((x0 + Delta_X, f(x0 + Delta_X, y0)), size=40, color="fuchsia")
    p += point((x0, z0), size=40, color="green")
    p += line(((x0 - 3, z0 - slope*3), (x0 + 3, z0 + slope*3)), color="red")
    if show_deltas:
        p += line(((x0, z0), (x0 + Delta_X, z0), (x0 + Delta_X, f(x0 + Delta_X, y0))), 
                  color="darkgreen", linestyle="dashed")
        p += text(r"$\Delta X$", (x0 + Delta_X/2, z0 - 0.2), fontsize=14, color="darkgreen")
        p += text(r"$\Delta f$", (x0 + Delta_X + 0.3, (z0 + f(x0 + Delta_X, y0))/2), fontsize=14, color="darkgreen")
    if show_result:
        p += text(r"$\Delta f \approx \left( \left. \frac{\partial f}{\partial X} \right|_{_{(X^*, Y^*)}} \right) \cdot \Delta X$", 
                  (x0 + 1, (z0 + 2)/2), fontsize=20, color="black")
    p.show(axes_labels=("$X$", "$Z$"), ymin=2, ymax=6, aspect_ratio=1, figsize=6)

Suppose we choose another point near $(X^*, Y^*)$, but this time only changing $Y$.

Delta_Y = 0.8
p = Graphics() + axes + pointXY
p += sphere((x0, y0 + Delta_Y, 0), 0.17, color="fuchsia")
p.show(aspect_ratio=(1,1,1), frame=False)

Here's the point corresponding to that on the graph of $f$:

p = Graphics() + baseplot
p += sphere((x0, y0 + Delta_Y, f(x0, y0 + Delta_Y)), 0.17, color="fuchsia")
p.show(aspect_ratio=(1,1,1), frame=False)

Here are all the points we could get if we only change $Y$, and hold $X$ constant.

p = Graphics() + baseplot
p += parametric_plot3d((x0, y, z), (y, -6, 6), (z, 0, 5), color="red", opacity=0.7)
p.show(aspect_ratio=(1,1,1), frame=False)

That cross-section, viewed with just the $Y$ and $Z$ axes (since $X$ is being held constant):

@interact(show_tangent=checkbox(False, label="Show tangent line"))
def tangent_y(show_tangent):
    slope = df_dy(x0, y0)
    p = plot(f(x0, y), (y, -6, 6), color="lightskyblue", thickness=2)
    p += point((y0, z0), size=40, color="green")
    if show_tangent:
        p += line(((y0 - 3, z0 - slope*3), (y0 + 3, z0 + slope*3)), color="red")
    p.show(axes_labels=("$Y$", "$Z$"), ymin=0, ymax=6.5, aspect_ratio=1, figsize=6)

Here's that same tangent line, on the surface:

p = Graphics() + baseplot
p += parametric_plot3d((x0, y0 + Deltay, z0 + Deltaf_y(Deltay)),
    (Deltay, -4, 4), color="red", opacity=0.85)
p.show(aspect_ratio=(1,1,1), frame=False)

Another conclusion:

The slope, $\displaystyle \frac{\Delta Z}{\Delta Y}$, of that tangent line can be found by setting $X = X^*$ (that is, plugging the constant $X^* = 1$ into the equation for $f$) and then computing the derivative with respect to $Y$ at $Y^* = -1$.

This slope is called the partial derivative of $f$ with respect to $Y$, at the point $(X^*, Y^*)$.

Tying this back in to linear approximation:

@interact(show_second=checkbox(False, label="Second point"), 
          show_deltas=checkbox(False, label=r"$\Delta Y$, $\Delta f$"), 
          show_result=checkbox(False, label="Linear approx."))
def approx_y(show_second, show_deltas, show_result):
    slope = df_dy(x0, y0)
    p = plot(f(x0, y), (y, -5, 2), color="lightskyblue", thickness=2)
    if show_second:
        p += point((y0, 0), size=40, color="red")
        p += point((y0 + Delta_Y, 0), size=40, color="fuchsia")
        p += point((y0 + Delta_Y, f(x0, y0 + Delta_Y)), size=40, color="fuchsia")
    p += point((y0, z0), size=40, color="green")
    p += line(((y0 - 3, z0 - slope*3), (y0 + 3, z0 + slope*3)), color="red")
    if show_deltas:
        p += line(((y0, z0), (y0, f(x0, y0 + Delta_Y)), (y0 + Delta_Y, f(x0, y0 + Delta_Y))), 
                  color="darkgreen", linestyle="dashed")
        p += text(r"$\Delta Y$", (y0 + Delta_Y/2, f(x0, y0 + Delta_Y) - 0.2), fontsize=14, color="darkgreen")
        p += text(r"$\Delta f$", (y0 - 0.3, (z0 + f(x0, y0 + Delta_Y))/2), fontsize=14, color="darkgreen")
    if show_result:
        p += text(r"$\Delta f \approx \left( \left. \frac{\partial f}{\partial Y} \right|_{_{(X^*, Y^*)}} \right) \cdot \Delta Y$", 
                  (y0 - 1.4, (z0 + 2)/2), fontsize=20, color="black")
    p.show(axes_labels=("$Y$", "$Z$"), ymin=2, ymax=6, aspect_ratio=1, figsize=6)

Putting it all together:

We saw that when we kept $Y$ constant and just made a small change $\Delta X$ to the $X$ variable, the linear approximation was given by: $$\Delta f \approx \left( \left. \frac{\partial f}{\partial X} \right|_{(X^*, Y^*)} \right) \cdot \Delta X$$

Similarly, when we kept $X$ constant and just made a small change $\Delta Y$ to the $Y$ variable, the linear approximation was: $$\Delta f \approx \left( \left. \frac{\partial f}{\partial Y} \right|_{(X^*, Y^*)} \right) \cdot \Delta Y$$

So if we make both small changes, $\Delta X$ and $\Delta Y$, we should get $$\Delta f \approx \left( \left. \frac{\partial f}{\partial X} \right|_{(X^*, Y^*)} \right) \cdot \Delta X + \left( \left. \frac{\partial f}{\partial Y} \right|_{(X^*, Y^*)} \right) \cdot \Delta Y$$

Conclusions:

1. A function $f(X, Y)$ has not just a derivative, but two partial derivatives, one with respect to $X$ and another with respect to $Y$: $$\frac{\partial f}{\partial X} \qquad \text{and} \qquad \frac{\partial f}{\partial Y}$$

2. To compute the partial derivative of $f$ with respect to $X$ at $(X^*, Y^*)$, just hold $Y$ constant (or even substitute the constant $Y^*$ in place of $Y$), and take the derivative of $f$ with respect to $X$. Likewise for the partial derivative of $f$ with respect to $Y$.

3. The complete formula for the linear approximation of $f$ at $(X^*, Y^*)$ is $$\Delta f \approx \left( \left. \frac{\partial f}{\partial X} \right|_{(X^*, Y^*)} \right) \cdot \Delta X + \left( \left. \frac{\partial f}{\partial Y} \right|_{(X^*, Y^*)} \right) \cdot \Delta Y$$

Bonus content: Note that we just found two tangent lines to the surface at $(X^*, Y^*)$:

p = Graphics() + baseplot
p += parametric_plot3d((x0 + Deltax, y0, z0 + Deltaf_x(Deltax)),
    (Deltax, -4, 4), color="red", opacity=0.85)
p += parametric_plot3d((x0, y0 + Deltay, z0 + Deltaf_y(Deltay)),
    (Deltay, -4, 4), color="red", opacity=0.85)
p.show(aspect_ratio=(1,1,1), frame=False)

In fact, there are many tangent lines to the surface at $(X^*, Y^*)$:

p = Graphics() + baseplot
t = var("t")
for theta in srange(0, pi, pi/10):
    a, b = cos(theta), sin(theta)
    color="red" if a == 1 or a == 0 else "blue"
    p += parametric_plot3d((x0 + a*t, y0 + b*t, z0 + a*Deltaf_x(t) + b*Deltaf_y(t)),
        (t, -4, 4), color=color, opacity=0.85)
p.show(aspect_ratio=(1,1,1), frame=False)

All of them collectively form a plane, the tangent plane:

p = Graphics() + baseplot + tangent_plane
p.show(aspect_ratio=(1,1,1), frame=False)

We will be able to use those two slopes of the two original tangent lines (the two partial derivatives) to find an equation for the tangent plane.

p = Graphics() + baseplot + tangent_plane
p += parametric_plot3d((x0 + Deltax, y0, z0 + Deltaf_x(Deltax)),
    (Deltax, -4, 4), color="red", opacity=0.85)
p += parametric_plot3d((x0, y0 + Deltay, z0 + Deltaf_y(Deltay)),
    (Deltay, -4, 4), color="red", opacity=0.85)
p.show(aspect_ratio=(1,1,1), frame=False)