Tangent Planes and Linear Approximations

Learning Objectives

After studying this material, you should be able to:

• Determine the equation of a plane tangent to a given surface $z = f(x, y)$ at a specific point $(x_0, y_0)$.
• Construct and use the Linearization $L(x, y)$ to approximate function values near a known point.
• Explain the formal definition of differentiability for functions of two variables, distinguishing it from the mere existence of partial derivatives.
• Calculate the total differential $dz$ to estimate small changes in a dependent variable.

Key Terms & Glossary

• Tangent Plane: A plane that "just touches" a surface at a point $P$. For a smooth surface, it contains all possible tangent lines to curves on the surface passing through $P$. Example: A flat piece of glass resting on top of a bowling ball.
• Linearization ($L(x, y)$): The linear function whose graph is the tangent plane. It provides the "best linear approximation" to the function near the point of tangency. Example: Using a flat map to calculate distances for a small park on the curved Earth.
• Differentiability: A property where the linear approximation becomes increasingly accurate as you move closer to the point. It requires the surface to be locally "flat" and smooth. Example: A smooth hillside is differentiable; a sharp mountain peak is not.
• Total Differential ($dz$): Represents the change in the height of the tangent plane ($z$) given small changes $dx$ and $dy$. Example: Estimating how much the volume of a box changes if each side is slightly elongated.

The "Big Idea"

In Single-Variable Calculus, we use the tangent line to approximate a curve. In Multivariable Calculus, we extend this to 3D: we use a tangent plane to approximate a surface. This allows us to simplify complex nonlinear functions into linear ones for local calculations, which is the foundation for optimization and numerical physics simulations.

Formula / Concept Box

Hierarchical Outline

• I. The Geometry of Tangent Planes
  • Smoothness: A tangent plane only exists if the surface is "smooth" (no sharp corners/creases).
  • Normal Vector: The coefficients of the tangent plane equation $(f_x, f_y, -1)$ relate to the vector normal to the surface.
• II. Linear Approximations
  • Local Linearity: $f(x, y) \approx L(x, y)$ when $(x, y)$ is close to $(a, b)$.
  • Error Analysis: The approximation error grows as you move further from the point of tangency.
• III. Differentiability and the Total Differential
  • Sufficient Condition: If $f_x$ and $f_y$ exist and are continuous near $(a, b)$, then $f$ is differentiable at $(a, b)$.
  • Differentials vs. Increments: $dz$ (differential) approximates $\Delta z$ (actual change).

Visual Anchors

Logic Flow: From Partials to Tangent Planes

Geometric Representation

Definition-Example Pairs

• Partial Derivative Interpretation: $f_x$ is the slope of the surface in the $x$-direction. Example: If you are standing on a hill and move strictly North, the steepness you feel is the partial derivative in the North direction.
• Total Differential Approximation: Using $dz$ to find errors. Example: A machine cuts a metal disk. If the radius $r$ has a tolerance of $\pm 0.1$mm, $dz$ tells us the estimated range of error for the disk's area.

Worked Examples

Example 1: Finding a Tangent Plane

Problem: Find the equation of the tangent plane to $z = x^2 + 3xy + y^2$ at the point $(1, 2)$.

Step 1: Find the z-value. $z_0 = f(1, 2) = 1^2 + 3(1)(2) + 2^2 = 1 + 6 + 4 = 11$.

Step 2: Compute partial derivatives. $f_x = 2x + 3y \implies f_x(1, 2) = 2(1) + 3(2) = 8$ $f_y = 3x + 2y \implies f_y(1, 2) = 3(1) + 2(2) = 7$

Step 3: Plug into the tangent plane formula. z - 11 = 8(x - 1) + 7(y - 2) z = 8x - 8 + 7y - 14 + 11 Final Equation: $z = 8x + 7y - 11$.

Example 2: Linear Approximation

Problem: Approximate $f(1.02, 1.97)$ for the function in Example 1.

Solution: Using L(x, y) = 11 + 8(x - 1) + 7(y - 2): L(1.02, 1.97) = 11 + 8(0.02) + 7(-0.03) L(1.02, 1.97) = 11 + 0.16 - 0.21 = 10.95.

Checkpoint Questions

1. Concept Check: If the partial derivatives $f_x$ and $f_y$ exist at a point, is the function guaranteed to be differentiable there? (Answer: No, the derivatives must also be continuous, or the limit of the error term must be zero).
2. Calculation: What is the total differential $dz$ for $z = e^{xy}$? (Answer: $dz = y e^{xy} dx + x e^{xy} dy$).
3. Visualization: In the equation of a plane $z = Ax + By + C$, what do $A$ and $B$ represent in terms of calculus? (Answer: $A$ is $f_x$ and $B$ is $f_y$).

[!TIP] When calculating $L(x, y)$, always pick a "base point" $(a, b)$ where the function and its derivatives are easy to calculate (integers or simple fractions).

[!WARNING] Differentiability in 2D is much stricter than in 1D. A surface can have partial derivatives in every direction but still not be differentiable if those tangent lines don't all lie in the same flat plane.