Tangent planes and linear approximations are powerful tools for understanding surfaces in 3D space. They allow us to simplify complex shapes by finding the best-fitting flat surface at a specific point, making calculations and predictions easier.

These concepts extend our knowledge of derivatives to multiple dimensions. By using partial derivatives, we can approximate function values, find normal lines, and make linear estimates for multivariable functions, all crucial skills in various fields.

Tangent Planes and Linear Approximations

Tangent plane equation for surfaces

• Tangent plane touches surface at single point and provides best linear approximation near that point (sphere tangent to plane)
• Components for tangent plane equation include point of tangency $(x_0, y_0, z_0)$ and partial derivatives $f_x(x_0, y_0)$ and $f_y(x_0, y_0)$
• General form of tangent plane equation: $z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$
• Calculate partial derivatives at point of tangency using techniques like power rule or chain rule
• Substitute calculated values into tangent plane equation for specific surface and point

Function approximation using tangent planes

• Tangent plane serves as local approximation of surface, most accurate near point of tangency (estimating height on topographic map)
• Apply tangent plane equation for approximation using point of interest $(x, y)$
• Estimate function value: $f(x, y) \approx f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$
• Assess approximation accuracy by considering distance from point of tangency (closer points yield better estimates)

Normal lines to surfaces

• Normal line perpendicular to tangent plane at point of tangency (plumb line on building)
• Components for normal line equation include point of tangency $(x_0, y_0, z_0)$ and direction vector $\langle f_x(x_0, y_0), f_y(x_0, y_0), -1 \rangle$
• Parametric equations of normal line: $x = x_0 + tf_x(x_0, y_0)$ $y = y_0 + tf_y(x_0, y_0)$ $z = z_0 - t$
• Calculate direction vector using partial derivatives evaluated at point of tangency
• Substitute computed values into normal line equations for specific surface and point

Linear approximations for multivariable functions

• Multivariable linear approximation extends single-variable concept to functions of several variables (economic modeling)
• Components include base point $(x_0, y_0, z_0)$, function value $f(x_0, y_0, z_0)$, and partial derivatives at base point
• Linear approximation formula: $f(x, y, z) \approx f(x_0, y_0, z_0) + f_x(x_0, y_0, z_0)(x - x_0) + f_y(x_0, y_0, z_0)(y - y_0) + f_z(x_0, y_0, z_0)(z - z_0)$
• Calculate partial derivatives at base point using appropriate differentiation techniques
• Substitute values into formula and evaluate for points near base point
• Accuracy of approximation decreases as distance from base point increases (climate predictions)