Units

# Unit 2: Tangent Plane

## The Concept

For a 2D curve [latex]y=f(x)[/latex], there is at most one **tangent line** to a point [latex](x_0, y_0)[/latex] on the curve. The **equation of tangent line** to 2D curve [latex]y=f(x)[/latex] at point [latex](x_0, y_0)[/latex] is

[latex]y=y_0+f'(x_0)(x-x_0)[/latex].

The **tangent plane** in 3D is an extension of the above tangent line in 2D. For a 3D surface [latex]z=f(x,y)[/latex], there are infinitely many tangent lines to a point [latex](x_0, y_0, z_0)[/latex] on the surface; these tangent lines lie in the same plane and they form the tangent plane at that point.

Recall that two lines determine a plane in 3D space. Thus, one usually uses two special tangent lines to determine a tangent plane and these two tangent lines are related to the partial derivatives (i.e., [latex]f_x[/latex] and [latex]f_y[/latex]) of the surface function [latex]z = f(x,y)[/latex]. The** equation of the tangent plane** to surface [latex]z = f(x,y)[/latex] at point [latex](x_0, y_0, z_0)[/latex] is

[latex]z = z_0 + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y-y_0)[/latex].

## The Plot

Now, you should engage with the 3D plot below to understand the tangent plane^{[1]}. Follow the steps below to apply changes to the plot and observe the effects:

- Input a 3D surface function in the function box in the plot. The function can be a single variable function or a double variable function.
- Adjust point [latex]P[/latex] using the sliders or by dragging the point on the graph below.
- The tangent plane equation will be depicted on the plot.

## Self-Checking Questions

Check your understanding by solving the following questions^{[2]}:

- Find the equation of the tangent plane to the surface defined by the function [latex]x^2+10xyz+y^2+8z^2=0,P(−1,−1,−1)[/latex]
- Find the equation of the tangent plane to the surface defined by the function [latex]h(x,y) = ln(x^2) + y^2[/latex] at Point [latex](x_0,y_0) = (3,4)[/latex].

- Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0. ↵
- Gilbert Strang, Edwin “Jed” Herman, OpenStax, Calculus Volume 3, Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA). ↵