14.4 - Tangent Planes and Linearization

Recall how we can approximate a curve with a tangent line.

python graph

Python

import micropip await micropip.install("numpy") await micropip.install("matplotlib") import numpy as np import matplotlib.pyplot as plt # Define the function for the curve (y = f(x)) def f(x): return np.sin(x) # Derivative of the function (f'(x)) for the slope of the tangent line def f_prime(x): return np.cos(x) # Define the x values and the point where the tangent will be drawn x = np.linspace(0, 2 * np.pi, 400) y = f(x) x_tangent = np.pi / 3 # Point of tangency y_tangent = f(x_tangent) # Define the tangent line equation: y_tangent + f'(x_tangent) * (x - x_tangent) tangent_line = y_tangent + f_prime(x_tangent) * (x - x_tangent) # Plot the curve plt.plot(x, y, label=r"$f(x) = \sin(x)$") # Plot the tangent line plt.plot(x, tangent_line, "--", label="Tangent Line", color="orange") # Highlight the point of tangency plt.scatter([x_tangent], [y_tangent], color="red", zorder=5) plt.text(x_tangent, y_tangent, f"({x_tangent:.2f}, {y_tangent:.2f})", color="red") # Add labels and title plt.title("Curve with Tangent Line") plt.xlabel("x") plt.ylabel("y") plt.legend() plt.grid(True) # Show the plot plt.show()

Output

This applies the same in the 3D dimension.

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Tangent Planes in 3d

Info

Tangent Plane should include every tangent line to the surface at $(x_0,y_0,f^{\prime }(x_0,y_0))$

Representing tangent planes in 3d is very similar, except the tangent plane will be just that, a plane that represents the curvature at one exact point on the given plane.

python graph

Python

import micropip await micropip.install("numpy") await micropip.install("matplotlib") import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D # Define the function for the surface (z = f(x, y)) def f(x, y): return np.sin(np.sqrt(x**2 + y**2)) # Partial derivatives of the surface for the tangent plane def df_dx(x, y): r = np.sqrt(x**2 + y**2) return x * np.cos(r) / r if r != 0 else 0 def df_dy(x, y): r = np.sqrt(x**2 + y**2) return y * np.cos(r) / r if r != 0 else 0 # Generate meshgrid for the surface plot x = np.linspace(-5, 5, 400) y = np.linspace(-5, 5, 400) X, Y = np.meshgrid(x, y) Z = f(X, Y) # Define the point of tangency x_tangent, y_tangent = 2, 2 z_tangent = f(x_tangent, y_tangent) # Compute the tangent line (or tangent plane in 3D) dfdx = df_dx(x_tangent, y_tangent) dfdy = df_dy(x_tangent, y_tangent) # Define the tangent plane: z = z_tangent + dfdx * (x - x_tangent) + dfdy * (y - y_tangent) tangent_plane = z_tangent + dfdx * (X - x_tangent) + dfdy * (Y - y_tangent) # Create the plot fig = plt.figure() ax = fig.add_subplot(111, projection="3d") # Plot the surface ax.plot_surface(X, Y, Z, cmap="viridis", alpha=0.8, rstride=100, cstride=100) # Plot the tangent plane at the point of tangency ax.plot_surface(X, Y, tangent_plane, color="orange", alpha=0.5) # Highlight the point of tangency ax.scatter([x_tangent], [y_tangent], [z_tangent], color="red", s=50) ax.text( x_tangent, y_tangent, z_tangent, f"({x_tangent}, {y_tangent}, {z_tangent:.2f})", color="red", ) # Set labels and title ax.set_title("3D Surface with Tangent Plane") ax.set_xlabel("X") ax.set_ylabel("Y") ax.set_zlabel("Z") plt.show()

Output

Finding the Tangent Plane for a given Equation

To find the tangent plane, we need a few things:

• The equation of the surface
  • ex $z=2x^2+y^2$
• The point of tangency
  • ex $(1,1,3)$

Once you have these, you can find the tangent plane by following these steps:

1. Find $Z_0$ by plugging in the point of tangency into the surface equation.

$z\_0=2(1{)}^2+(1{)}^2=3$

2. Find the partial derivatives of the surface equation.

For f(x):

$f(x)=4x|\_(1,1)=4$

For f(y):

$f_y=2y{|}_{(1,1)}=2$

3. Plug in the point of tangency and the partial derivatives into the equation of the tangent plane.